PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 84(98) (2008), 1–36 DOI: 10.2298/PIM0898001J

SYMMETRIES AND INTEGRABILITY

Božidar Jovanović

Abstract. This is a survey on finite-dimensional integrable dynamical systems related to Hamiltonian G-actions. Within a framework of noncommutative integrability we study integrability of G-invariant systems, collective motions and reduced integrability. We also consider reductions of the Hamiltonian flows restricted to their invariant submanifolds generalizing classical Hess–Appel’rot case of a heavy rigid body motion.

Contents

1. Introduction 1 2. Noncommutative Integrability 4 3. Symmetries and Reductions 9 4. Integrable Systems Related to Hamiltonian Actions 12 5. Reductions and Integrability 18 6. Partial Integrability 22 References 32

1. Introduction

1.1. Hamiltonian Systems. Let (M, ω) be 2n-dimensional connected symplectic manifold. By X_{h} we shall denote the Hamiltonian vector field of a function h and by {·, ·} the canonical Poisson bracket on M defined by

dh_{x}(ξ) = ω_{x}(ξ, X_{h}(x)), ξ ∈ T_{x}M and {f_{1}, f_{2}} = ω(X_{f_{2}}, X_{f_{1}}) = df_{1}(X_{f_{2}}),

respectively (we follow the sign convention of Arnold [3]). The equations:

(1.1)  ẋ = X_{h}(x) ⟺ ḟ = {f, h}, f ∈ C^{∞}(M)

are called Hamiltonian equations with the Hamiltonian function h.

A function f is an integral of the Hamiltonian system (constant along trajectories of (1.1)) if and only if it commutes with h: {h, f} = 0.

2000 Mathematics Subject Classification: 70H06, 37J35, 53D25.

1.2. Natural Mechanical Systems. The basic examples of Hamiltonian systems are natural mechanical systems (Q, κ, v), where Q is a configuration space, κ is a Riemannian metric on Q (also regarded as a mapping κ : T Q → T^{∗}Q) and v : Q → ℝ is a potential function.

Let q = (q^{1}, …, q^{n}) be local coordinates on Q. The motion of the system is described by the Euler–Lagrange equations

(1.2)  \frac{d}{dt} \frac{∂l}{∂q̇^{i}} = \frac{∂l}{∂q^{i}} , i = 1, …, n,

where the Lagrangian is l(q, q̇) = \frac{1}{2}(κ_{q}q̇, q̇) − v(q) = \frac{1}{2} ∑_{ij} κ_{ij}q̇^{i}q̇^{j} − v(q).

Equivalently, we can pass from velocities q̇^{i} to the momenta p_{j} by using the standard Legendre transformation p_{j} = κ_{ij}q̇^{i}. Then in the coordinates q^{i}, p_{i} of the cotangent bundle T^{∗}Q the equations of motion read:

(1.3)  \frac{dq^{i}}{dt} = \frac{∂h}{∂p_{i}} ,  \frac{dp_{i}}{dt} = −\frac{∂h}{∂q^{i}} ,  i = 1, …, n,

where h is the Legendre transformation of l

(1.4)  h(q, p) = \frac{1}{2} ∑_{i,j=1}^{n} κ^{ij}p_{i}p_{j} + v(q),

interpreted as the energy of the system. Here κ^{ij} are the coefficients of the tensor inverse to the metric.

This system of equations is Hamiltonian on T^{∗}Q endowed with the canonical symplectic form ω = ∑_{i=1}^{n} dp_{i} ∧ dq^{i}. The corresponding canonical Poisson bracket is given by

(1.5)  \{f, g\} = ∑_{i=1}^{n} (\frac{∂f}{∂q^{i}} \frac{∂g}{∂p_{i}} − \frac{∂g}{∂q^{i}} \frac{∂f}{∂p_{i}} ) .

For v ≡ 0, the equations (1.3) are the geodesic flow equations of the Riemannian manifold (Q, κ).

1.3. Quadratures. Recall that integration by quadratures of a system of differential equations ẋ = X(x) in some domain V ⊂ ℝ^{n}{x} is the search for its solutions by a finite number of “algebraic” operation (including inversion of the functions) and “quadratures”, i.e., calculations of the integrals of known functions.

The Lie theorem says that if we have n linearly independent vector fields X_{1}, …, X_{n} that generate a solvable Lie algebra under commutation and [X_{1}, X_{i}] = λ_{i}X_{i}, then the differential equation ẋ = X_{1}(x) can be integrated by quadratures in V (e.g., see [2]).

By modifying the Lie theorem, Kozlov obtained the following result which gives the sufficient conditions to the integration by quadratures of Hamiltonian systems with invariant relations (see [42, 2, 43]).

Consider ℝ^{2n} = {(q^{1}, …, q^{n}, p_{1}, …, p_{n})} endowed with the canonical Poisson bracket (1.5). Suppose that we have n-functions f_{1}, …, f_{n} that generate solvable Lie algebra with respect to Poisson bracket

{f_{1}, f_{j}} = c_{1,j}^{1}f_{1}, {f_{2}, f_{j}} = c_{2,j}^{1}f_{1} + c_{2,j}^{2}f_{2}, …, {f_{n}, f_{j}} = c_{n,j}^{1}f_{1} + ⋯ + c_{n,j}^{n}f_{n}

and which are independent on the level-set

M_{c} = {(q, p) ∈ ℝ^{2n} | f_{1} = c_{1}, …, f_{n} = c_{n}}.

Theorem 1.1 (Kozlov [43]). Suppose

c_{1,j}^{1}c_{1} = 0, c_{2,j}^{1}c_{1} + c_{2,j}^{2}c_{2} = 0, …, c_{n,j}^{1}c_{1} + ⋯ + c_{n,j}^{n}c_{n} = 0

for all j. Then M_{c} is an invariant manifolds for Hamiltonian systems with Hamiltonians f_{1}, …, f_{n}. The solutions of the Hamiltonian systems that lie on M_{c} can be found by quadratures.

In a special case one gets Liouville’s theorem: if a Hamiltonian system with n degrees of freedom has n independent first integrals in involution than (at least locally) it can be integrated by quadratures.

1.4. Liouville Integrability. One of the central problems in Hamiltonian dynamics is whether the equations (1.1) are completely integrable or not. The usual definition of complete integrability is as follows:

Definition 1.1. Hamiltonian equations (1.1) are called completely integrable if there are n Poisson-commuting smooth integrals f_{1}, …, f_{n} whose differentials are independent in an open dense subset of M. The set of integrals F = {f_{1}, …, f_{n}} is called a complete involutive (or commutative) set of functions on M.

Remark 1.1. The last condition needs to be commented. The functional independence of the integrals can be meant in three different senses. The differentials of f_{1}, …, f_{n} can be linearly independent:

(i) on an open everywhere dense subset,

(ii) on an open everywhere dense subset of full measure, (iii) everywhere except for a piece-wise smooth polyhedron. Instead of these conditions one can assume all the functions to be real analytic.

Then it suffices to require their functional independence at least at one point (we will call such a situation analytic integrability).

If the system is completely integrable, by Liouville’s theorem it can be integrated by quadratures. Moreover, the global regularity of dynamics in the case of complete integrability follows from the following classical Liouville–Arnold theorem:

Theorem 1.2 (Liouville–Arnold [48, 3]). Suppose that the equations (1.1) have n Poisson-commuting smooth integrals f_{1}, …, f_{n} and let M_{c} = {f_{1} = c_{1}, …, f_{n} = c_{n}} be a common invariant level set.

(i) If M_{c} is regular (the differentials of f_{1}, …, f_{n} are independent on it), compact and connected, then it is diffeomorphic to the n-dimensional Lagrangian torus.

(ii) In a neighborhood of M_{c} there are action-angle variables I, φ mod 2π such that the symplectic form becomes ω = ∑_{i=1}^{n} dI_{i} ∧ dφ_{i} and the Hamiltonian function h depends only on actions I_{1}, …, I_{n}. Thus the Hamiltonian equations are linearized

φ̇_{1} = ω_{1}(I) = \frac{∂h}{∂I_{1}} , …, φ̇_{r} = ω_{r}(I) = \frac{∂h}{∂I_{n}} ,

i.e., dynamics on invariant tori I_{i} = const is quasi-periodic.

The topology of toric foliation in many low-dimensional integrable systems is well known (e.g., see [79, 68, 9, 4] and references therein). The obstructions to the existence of global action-angle variables are studied in [24].

Even the system is integrable, the dynamics on the singular set (where the differentials of integrals f_{1}, …, f_{n} are dependent) can be quite complicated. For example, the flows given in [21, 10, 22] are integrable by smooth integrals, and by Taimanov’s theorem [77] can not be integrable by analytic ones.

1.5. Outline of the paper. This paper is a survey in integrable dynamical systems related to Hamiltonian G-actions including integrability of collective motions, reduced integrability and partial integrability. It turns out that most natural framework to use here is non-commutative integrability developed by Mischenko and Fomenko [61] and Nekhoroshev [63].

In the case of non-commutative integrability (sometimes known as superintegrability or degenerate integrability) invariant manifolds are isotropic but non-Lagrangian. In section 2 we show that non-commutative integrability always implies usual Liouville integrability by means of smooth commuting function (see [12]).

In section 3 we recall on the basic notion of Hamiltonian G-actions, Marsden– Weinstein reduction of symmetries and Lagrange–Routh reduction of natural mechanical systems.

Following [12], the construction of integrable systems related to Hamiltonian G-actions is presented in section 4. We also show that integrability of the reduced systems is essentially equivalent to the integrability of the original systems in section 5 (see [83, 40]).

In section 6 we study reductions of the Hamiltonian flows restricted to their invariant submanifolds (see [41]). If the reduced system is integrable, the original system may not be. The typical situation is that invariant submanifold is foliated on invariant tori, but not with quasi-periodic motions. We refer to such systems as partially integrable. The classical example is the Hess–Appel’rot case of a heavy rigid body motion about a fixed point [39, 1].

Many interesting problems related to the torus actions, algebraic complete integrability as well as separation of variables are not discuss here.

2. Noncommutative Integrability

2.1. Noncommutative Integration. If f_{1} and f_{2} are integrals of (1.1), then so are an arbitrary smooth function F (f_{1}, f_{2}) and the Poisson bracket {f_{1}, f_{2}}. Therefore without loss of generality we can assume that integrals of (1.1) form an algebra F with respect to the Poisson bracket. For simplicity we shall assume that F is functionally generated by functions f_{1}, …, f_{l} so that

{f_{i}, f_{j}} = a_{ij}(f_{1}, …, f_{l}).

Suppose that

dim F_{x} = dim span {df_{i}(x) | i = 1, …, l} = l, x ∈ U,
dim ker{·, ·}|_{F_{x}} = r, x ∈ U,

holds for an open dense set U ⊂ M. Let ϕ : M → ℝ^{l} be the moment mapping: ϕ(x) = (f_{1}(x), …, f_{l}(x)) and let Σ = ϕ(M ∖ U). Then we have the following noncommutative integration theorem (see Mishchenko and Fomenko [61], Nekhoroshev [63] and Brailov [17]):

Theorem 2.1. Suppose that: dim F_{x} + dim ker{·, ·}|_{F_{x}} = dim M, for x ∈ U. Let c ∈ ϕ(M) ∖ Σ be a regular value of the moment map. Then:

(i) M_{c} = ϕ^{−1}(c) is an isotropic submanifold of M and the equations (1.1) on M_{c} can be (locally) solved by quadratures;

(ii) Compact connected components T_{c}^{r} of M_{c} are diffeomorphic to r-dimen-sional tori. The dynamics on T_{c}^{r} is quasi-periodic, i.e., can be linearized in appropriate angle coordinates φ_{1}, …, φ_{r}:

φ_{1}(t) = ω_{1}t, …, φ_{r}(t) = ω_{r}t.

Let us just point out some steps in the proof of the theorem. Under the hypotheses of Theorem 2.1, there exist r linearly independent commuting vector fields X_{1} = X_{h}, X_{2}, …, X_{r} on M_{c}. They can be obtained as linear combinations of the Hamiltonian vector fields X_{f_{1}}, …, X_{f_{l}} from the conditions ω(X_{i}, X_{f_{j}}) = 0, i = 1, …, r, j = 1, …, l. Since commutative algebras are solvable we can apply the Lie theorem to locally integrate by quadratures the system ẋ = X_{h}(x). On a compact connected component T_{c}^{r} of M_{c}, the vector fields X_{1}, …, X_{r} are complete. Therefore T_{c}^{r} is diffeomorphic to an r-dimensional torus and the motion on the torus is quasi-periodic (the proof of this fact is just the same as in the usual Liouville theorem).

Recently, the obstructions for noncommutative integration of Hamiltonian systems within the framework of differential Galois theory have been studied in [49, 50].

It is important to note that the assumptions of Theorem 2.1 imply integrability of the considered system in the usual, commutative sense.

Theorem 2.2. [12] Under the assumptions of Theorem 2.1, the Hamiltonian system (1.1) is Liouville integrable, i.e., it admits n Poisson-commuting C^{∞}-smooth integrals g_{1}, …, g_{n}, independent on an open dense subset of M.

Proof. On the image of M under the moment mapping ϕ(M) ⊂ ℝ^{l}{y_{1}, …, y_{l}} we can introduce a “Poisson structure” {·, ·}_{F} by the formulas:

{y_{i}, y_{j}}_{F} = a_{ij}(y_{1}, …, y_{l}).

Note that rank{·, ·}_{F}(y) = l − r = 2q for all y ∈ ϕ(M) ∖ Σ and that (ϕ(M) ∖ Σ, {·, ·}_{F}) is a Poisson manifold.

From the definition of the Poisson bracket {·, ·}_{F} it follows that if smooth functions F, G : ℝ^{l} → ℝ are in involution with respect to {·, ·}_{F} on ϕ(M), then their liftings f = F ∘ ϕ and g = G ∘ ϕ commute on M:

(2.1)  {f, g}(x) = {F, G}_{F}(ϕ(x)) = 0.

Let z ∈ ϕ(M) ∖ Σ. By the theorem on the local structure of Poisson brackets (see [82, 46]), there is a neighborhood U(z) ⊂ ϕ(M) ∖ Σ of z and l independent smooth functions G_{1}, …, G_{l} : U(z) → ℝ, G_{i}(z) = 0 such that

{G_{i}, G_{i+q}}_{F} = 1 = −{G_{i+q}, G_{i}}_{F} i = 1, …, q

and the remaining Poisson brackets vanish. Let a ball B^{α}(ϵ_{α}) belong to U(z), where B^{α}(ϵ_{α}) = {y ∈ U(z) | G_{1}^{2} + ⋯ + G_{l}^{2} < ϵ_{α}}. Then, starting from the n involutive functions on B^{α}:

h_{1} = G_{1}^{2} + G_{1+q}^{2}, …, h_{q} = G_{q}^{2} + G_{2q}^{2}, h_{q+1} = G_{2q+1}^{2}, …, h_{n} = G_{l}^{2}

we can construct a smooth set of nonnegative functions F_{1}^{α}, …, F_{n}^{α} : ℝ^{l} → ℝ that are independent on an open dense subset of B^{α}(ϵ), equal to zero outside B^{α}(ϵ_{α}), in involution on ϕ(M), and satisfies inequalities F_{i}^{α} < e^{−ϵ_{α}} together with all derivatives. To this end we use the construction suggested by Brailov for Darboux symplectic balls (see [79]).

Let g : ℝ → ℝ be smooth nonnegative function, such that g(x) is equal to zero for |x| > ϵ_{α}, monotonically increases on [−ϵ, 0] and monotonically decreases on [0, ϵ]. Let h(y) = g(h_{1}(y) + ⋯ + h_{n}(y)). This function could be extended by zero to the whole manifold. Now, we can define F_{i}^{α} by: F_{i}^{α} = h · h_{i}. Obviously, {F_{i}^{α}, F_{j}^{α}}_{F} = 0. These functions are independent inside B^{α}. Also, we can choose g in such a way that F_{i}^{α} satisfies the inequalities F_{i}^{α} < e^{−ϵ_{α}} together with all derivatives.

In the same way we can construct a countable family of open balls {B^{α}(ϵ_{α})}, B^{α} ∩ B^{β} = ∅, and functions {F_{1}^{α}, …, F_{l}^{α}} with the above properties, such that B = ⋃_{α} B^{α}(ϵ_{α}) is an open everywhere dense set of ϕ(M) ∖ Σ. Let us define the functions F_{1}, …, F_{n} : ℝ^{l} → ℝ by:

F_{i}(y) = \begin{cases}F_{i}^{α}, & y ∈ B^{α} ⊂ B \\ 0, & y ∈ ℝ^{l} ∖ B, i = 1, …, n \end{cases}

By (2.1), the functions g_{1} = F_{1} ∘ ϕ, …, g_{n} = F_{n} ∘ ϕ will have the desired properties. □

Theorem 2.2 says that the r-dimensional invariant tori T^{r} can be organized into larger, n-dimensional Lagrangian tori T^{n} that are level sets of a commutative algebra of integrals. Since the tori T^{n} are fibered into invariant tori T^{r}, the trajectories of (1.1) are not dense on T^{n}. In this sense, the system (1.1) is degenerate. Thus, establishing the fact of non-commutative integrability of the system give us more information on the behavior of its integral trajectories than we could obtain from the usual Liouville integrability. Note that, contrary to the case of non-degenerate integrable systems, the fibration by Lagrangian tori is neither intrinsic nor unique.

Let P_{0} = ϕ(M)∖Σ, M_{0} = ϕ^{−1}(P_{0}). If all invariant submanifolds M_{c} = ϕ^{−1}(c), c ∈ P_{0} are compact and connected, then ϕ : (M_{0}, {·, ·}) → (P_{0}, {·, ·}_{F}) is a Poisson morphism and isotropic fibration. This fibration is symplectically complete, i.e., the symplectic orthogonal distribution to the tangent spaces of the fibres is a foliation (see [46]).

2.2. Generalized Action-Angle Variables. With notations of Theorem 2.1, let T_{c}^{r} be a compact connected component of the invariant level set M_{c}.

Theorem 2.3 (Nekhoroshev [63]). In a neighborhood of T_{c}^{r} there are generalized action-angle variables p, q, I, φ mod 2π, defined in a toroidal domain O = 𝕋^{r}{φ} × B_{σ}{I, p, q},

B_{σ} = {(I_{1}, …, I_{r}, p_{1}, …, p_{k}, q_{1}, …, q_{k}) ∈ ℝ^{l} | ∑_{i=1}^{r} I_{i}^{2} + ∑_{j=1}^{k} q_{j}^{2} + p_{j}^{2} ⩽ σ^{2}}

such that the symplectic form becomes

ω = ∑_{i=1}^{r} dI_{i} ∧ dφ_{i} + ∑_{i=1}^{k} dp_{i} ∧ dq_{i},

and the Hamiltonian function h depends only on I_{1}, …, I_{r}. The Hamiltonian equations take the following form in action-angle coordinates:

(2.2)  φ̇_{1} = ω_{1}(I) = \frac{∂h}{∂I_{1}} , …, φ̇_{r} = ω_{r}(I) = \frac{∂h}{∂I_{r}} , İ = ṗ = q̇ = 0.

The obstructions to the existence of global generalized action-angle variables are studied in [63, 23, 30]. The action-angle variables in the case of non-compact invariant manifolds are given in [32].

Definition 2.1. [7] The Hamiltonian system (2.2) defined in the toroidal domain O = 𝕋^{r} × B_{σ} is 𝕋^{r}-dense if the set of points (I_{0}, p_{0}, q_{0}) ∈ B_{σ} for which the trajectories of (2.2) are dense on the torus {I = I_{0}, p = p_{0}, q = q_{0}} is everywhere dense in the ball B_{σ}.

The frequencies ω_{1}(I), …, ω_{r}(I) corresponding to the dense trajectories (2.2) are rationally independent. For example, a non-degenerate system (det(\frac{∂ω}{∂I}) ≠ 0 on an open dense set of O) is 𝕋^{r}-dense.

Any smooth first integral of a 𝕋^{r}-dense system is a function of the variables I, q, p only.

Now, let h = h(I_{1}, …, I_{r}) be an arbitrary differentiable Hamiltonian function defined on a toroidal domain O = 𝕋^{r}{φ} × B_{σ}{I, p, q}. Then we have

Theorem 2.4 (Bogoyavlenski [7]). There exists a family of balls B_{τ} ⊂ B_{σ} such that the union ⋃_{τ} B_{τ} is dense in B_{σ} and the following properties hold.

(i) In the toroidal domain O_{τ} = 𝕋^{r}×B_{τ} there exists a canonical transformation: {I, p, q, φ} → {I^{τ} , p^{τ} , q,^{τ} , φ^{τ} }, that transforms the system (2.2) to the form

φ̇_{1}^{τ} = ω_{1}^{τ} = \frac{∂h}{∂I_{1}^{τ}} , …, φ̇_{r̃(τ)}^{τ} = ωτ ; r̃ = \frac{∂h}{∂Iτ ; r̃} ,

φ̇_{r̃+1}^{τ} = 0, …, φ̇_{r}^{τ} = 0, İ^{τ} = ṗ^{τ} = q̇^{τ} = 0, h = h(I_{1}^{τ} , …, Ir̃), r̃ = r̃(τ) ⩽ r. The system is 𝕋r̃-dense in O_{τ} (regarded as the product 𝕋r̃ × (𝕋^{r−r̃} × B_{τ})).

(ii) Moreover, if h(I_{1}, …, I_{r}) is analytic and the maximal dimension of the closures of trajectories is equal to r̃, then such canonical transformation exists globally and the system is 𝕋r̃-dense in O.

2.3. Complete Algebras of Functions. Let F be an algebra of functions, closed under the Poisson brackets, on the symplectic manifold (M, ω). Let F_{x} be the subspace of T_{x}^{∗}M generated by differentials of functions in F and let K_{x} ⊂ F_{x} be the kernel of Poisson structure restricted on F_{x}:

(2.3)  F_{x} = span {df_{(}x) | f ∈ F}, K_{x} = ker{·, ·}|_{F_{x}}.

Suppose that dim F_{x} = l, dim K_{x} = r holds on an open dense subset U ⊂ M. We shall denote U by reg F or by reg M (regular points of F). The numbers l and r are usually denoted by ddim F and dind F and are called differential dimension and differential index of F.

Definition 2.2. The algebra F is said to be complete if:

(2.4)  ddim F + dind F = dim M.

Note that instead of algebras one usually consider sets of functions closed under the Poisson bracket. The notions of completeness, ddim and dind are defined just in the same way as above.

For example, suppose that independent functions f_{1}, …, f_{l} generate a finite dimensional Lie algebra 𝔤 = F = ⨁_{i=1}^{l} ℝf_{i} under the Poisson brackets:

{f_{i}, f_{j}} = ∑_{k=1}^{l} c_{ij}^{k}f_{k},

c_{ij}^{k} are constants. Then the numbers ddim F and dind F coincide with the dimension and the index of the Lie algebra 𝔤.

By Theorem 2.1, we can give the following definition:

Definition 2.3. The Hamiltonian system (1.1) is completely integrable in the noncommutative sense if it possesses a complete algebra F of integrals.

In Definition 2.2 we do not require that F is generated by l = ddim F functions. We shall briefly explain this. Let x_{0} belong to reg F. Then there are integrals f_{1}, …, f_{l} ∈ F that are independent in x_{0}, where l = ddim F. Let V be the open set where the functions f_{1}, …, f_{l} are independent. Since f_{1}, …, f_{l} are integrals of (1.1) the trajectory of (1.1) that has initial position in V remains in V. Indeed, the phase flow of (1.1) preserves the form df_{1} ∧ ⋯ ∧ df_{l}. Therefore we can consider the restriction of (1.1) to V and apply Theorem 2.1 to integrate it.

Definition 2.4. Let F be an algebra of functions. We shall say that A ⊂ F is a complete subalgebra if ddim A + dind A = ddim F + dind F.

Mishchenko and Fomenko stated the conjecture that non-commutative integrable systems are integrable in the usual commutative sense by means of integrals that belong to the same functional class as the original non-commutative algebra F of integrals (in other words, one can find complete commutative subalgebra A of F). In the analytic case, when F = span_{ℝ}{f_{1}, …, f_{l}} is a finite-dimensional Lie algebra, the conjecture has been proved by Mishchenko and Fomenko in the semisimple case [60] and just recently by Sadetov [74] for arbitrary Lie algebras (see also [81]).

Thus, according to Theorem 2.2, the following general conjecture remains:

Conjecture 2.1. Suppose that on a real-analytic symplectic 2n-dimensional manifold M we have a Hamiltonian system ẋ = X_{h}(x) completely integrable by means of a infinitely dimensional non-commutative algebra F of integrals which are real analytic functions. Then the system possesses n commuting real analytic integrals.

2.4. Noncommutative Integrability on Poisson Manifolds. One can easily formulate the above setting for Hamiltonian systems ḟ = {f, h} on a Poisson manifold (M, {·, ·}). Instead of (2.4), the algebra of functions F is said to be complete if ddim F + dind F = dim M + corank {·, ·}. Also, in this case, regular compact connected components level-sets of functions within F are tori of dimension dim M − ddim F = dind F − corank {·, ·}.

The equivalent definition of the completeness is as follows. Let Λ be the bivector field on M associated to the Poisson structure {f_{1}, f_{2}}(x) = Λ_{x}(df_{1}(x), df_{2}(x)). We say that F is complete at x if the space F_{x} defined by (2.3) is coisotropic:

(2.5)  F_{x}^{Λ} ⊂ F_{x}.

Here F_{x}^{Λ} is skew-orthogonal complement of F_{x} with respect to Λ:

F_{x}^{Λ} = {X ∈ T_{x}^{∗}M | Λ_{x}(F_{x}, X) = 0}.

The algebra F is complete if it is complete at a generic point x ∈ M. It is clear that in this case K_{x} = F_{x}^{Λ}, for a generic x ∈ M.

Remark 2.1. Specially, for symplectic manifolds it is natural to state the completeness criterium in terms of the symplectic structure. Set

W_{x} = {X_{f}(x) | f ∈ F}, D_{x} = {X ∈ T_{x}M | (F_{x}, X) = 0}.

The condition (2.5) is equivalent to the coisotropy of W_{x} (W_{x}^{ω} ⊂ W_{x}) and isotropy of D_{x} (D_{x}^{ω} ⊃ D_{x}) in the symplectic linear space T_{x}M.

3. Symmetries and Reductions

3.1. Hamiltonian G-Actions. Let a connected Lie group G act on 2n-dimen-sional connected symplectic manifold (M, ω). The action is Hamiltonian if G acts on M by symplectomorphisms and there is a well-defined momentum mapping:

(3.1)  Φ : M → 𝔤^{∗}

(𝔤^{∗} is a dual space of the Lie algebra 𝔤) such that one-parameter subgroups of symplectomorphisms are generated by the Hamiltonian vector fields of functions

(3.2)  ϕ_{ξ}(x) = (Φ(x), ξ), ξ ∈ 𝔤

and

(3.3)  ϕ_{[ξ_{1},ξ_{2}]} = {ϕ_{ξ_{1}}, ϕ_{ξ_{2}}}.

Then Φ is equivariant with respect to the given action of G on M and the co-adjoint action of G on 𝔤^{∗}:

Φ(g · x) = Ad_{g}^{∗}(Φ(x)).

In particular, if μ belongs to Φ(M), then the co-adjoint orbit O(μ) belongs to Φ(M) as well.

As the important example, consider the G action on the configuration space Q. The action can be naturally extended to the Hamiltonian action on (T^{∗}Q, ω):

g · (q, p) = (g · q, (dg^{−1})^{∗}p)

with the momentum mapping Φ given by (e.g., see [46])

(3.4)  (Φ(q, p), ξ) = (p, ξ_{q}), ξ ∈ 𝔤.

Here ξ_{q} is the vector given by the action of one-parameter subgroup exp(tξ) at q.

3.2. Symplectic Reduction. Now, let G be a Lie group with a free and proper Hamiltonian action on a symplectic manifold (M, ω) with the momentum mapping (3.1). Assume that η is a regular value of Φ, so that M_{η} = Φ^{−1}(η) and M_{O_{η}} = Φ^{−1}(O_{η}) are smooth manifolds. Here O_{η} = G/G_{η} is the coadjoint orbit of η. The manifolds M_{η} and M_{O_{η}} are G_{η}-invariant and G-invariant, respectively.

There is a unique symplectic structure ω_{η} on N_{η} = M_{η}/G_{η} satisfying

ω|_{M_{η}} = dπ_{η}^{∗}ω_{η}

where π_{η} : M_{η} → N_{η} is the natural projection (Marsden and Weinstein [51]).

According to Noether’s theorem, if h is a G-invariant function, then the momentum mapping Φ is an integral of the Hamiltonian system (1.1). In addition, its restriction to the invariant submanifold M_{η} projects to the Hamiltonian system

(3.5)  ẏ = X_{H_{η}}

on the reduced space N_{η} with H_{η} defined by

(3.6)  h|_{M_{η}} = H_{η} ∘ π_{η}.

3.3. Poisson Reduction. An alternative description of the reduced space is as follows. Let {·, ·} be the canonical Poisson bracket on (M, ω). Then the manifold M/G carries the induced Poisson structure {·, ·}^{G} defines as follows. Let

π : M → M/G

be the natural projection. In what follows, by capital letters we shall denote the functions on M/G and with small letters corresponding G-invariant functions on M:

f = F ∘ π.

Then

{F_{1}, F_{2}}^{G} ∘ π = {f_{1}, f_{2}}.

The mapping N_{η} → M_{O_{η}}/G which assigns to the G_{η}-orbit of x ∈ M_{η} the Gorbit through x in M_{O_{η}} establish the symplectomorphism between N_{η} and M_{O_{η}}/G (e.g., see [67]). In particular, M_{O_{η}}/G is the symplectic leaf in (M/G, {·, ·}^{G}).

The Hamiltonian flow (1.1) projects to the Hamiltonian flow

(3.7)  Ḟ = {F, H}^{G},  F ∈ C^{∞}(M/G)

on the reduced space M/G. The system (3.5) is actually the restriction of (3.7) to the symplectic leaf M_{O_{η}}/G.

3.4. Cotangent Bundle Reduction. Let G be a connected Lie group acting freely and properly on Q and ρ : Q → B = Q/G be the canonical projection. Then 0 is the regular value of the cotangent bundle momentum mapping (3.4) and the reduced space (Φ^{−1}(0)/G, ω_{0}) is symplectomorphic to T^{∗}(Q/G), e.g., see [67].

Now, let (Q, κ, v) be a G-invariant natural mechanical system. That is G acts by isometries and the potential is the pull back v = V ∘ρ of the potential V defined on Q/G. Therefore, the corresponding Hamiltonian function (1.4) is G-invariant and the system can be reduced to T^{∗}(Q/G). The reduced system is also natural mechanical system on Q/G with the potential V and metric which has the clear geometrical interpretation.

Let

(3.8)  V_{q} = {ξ_{q} | ξ ∈ 𝔤}

be the tangent space to the fibber G · q (vertical space at q) and V = ⋃_{q} V_{q} be the vertical distribution.

Consider the horizontal distribution H = ⋃_{q} H_{q} ⊂ T Q orthogonal to V with respect to the metric κ. Equivalently, H is the zero level-set of the tangent bundle momentum mapping Φ_{l}:

(3.9)  H = Φ_{l}^{−1}(0), (Φ_{l}(q, q̇), ξ) = (\frac{∂l}{∂q̇} , ξ_{q}) = (κ_{q}q̇, ξ_{q}), ξ ∈ 𝔤.

The horizontal distribution H is G-invariant and the quotient space H/G can be naturally identified with the tangent bundle T (Q/G). The restriction of the metric κ to H define the submersion metric K on the reduced space Q/G via

K(X_{1}, X_{2})_{ρ(q)} = κ(X̄_{1}, X̄_{2})_{q}, X_{i} ∈ T_{ρ(q)}(Q/G), X̄_{i} ∈ H_{q}, X_{i} = dρ(X̄_{i}).

Since (T^{∗}Q)_{0} = Φ^{−1}(0) = ⋃_{q} ann V_{q}, ann V_{q} = {p ∈ T_{q}^{∗}Q | (p, ξ_{q}) = 0, ξ ∈ 𝔤}, after passing to the quotient spaces H/G ≈ T (Q/G) and (T^{∗}Q)_{0}/G ≈ T^{∗}(G/H) we see that the reduced Hamiltonian is the Hamiltonian of the natural mechanical system (Q/G, K, V).

Theorem 3.1. The trajectories q(t) of the G-invariant, natural mechanical system (Q, κ, v) with velocities q(t) that belong to H project to the trajectories ρ(q(t)) of the reduced natural mechanical system (Q/G, K, V)

For Abelian groups this is the classical method of Routh for eliminating cyclic coordinates [73]. The non-Abelian construction for the zero level-set of and for the other values of the momentum mapping is given in [2] and [52], respectively. The above formulation is taken from [2]. In the case of geodesic flows the horizontal geodesic lines projects to the geodesic lines of the submersion metric.

4. Integrable Systems Related to Hamiltonian Actions

4.1. Collective Motions. Consider the following two natural classes of functions on M. Let F_{1} be the set of functions in C^{∞}(M) obtained by pulling-back the algebra C^{∞}(𝔤^{∗}) by the moment map F_{1} = Φ^{∗}C^{∞}(𝔤^{∗}). Let F_{2} be the set of G-invariant functions in C^{∞}(M). The mapping f ↦ f ∘Φ is a morphism of Poisson structures:

(4.1)  {f_{1} ∘ Φ, f_{2} ∘ Φ}(x) = {f_{1}, f_{2}}_{𝔤^{∗}}(η), η = Φ(x),

where {·, ·}_{𝔤^{∗}} is the Lie–Poisson bracket on 𝔤^{∗}:

(4.2)  {f_{1}, f_{2}}_{𝔤^{∗}}(η) = (η, [df_{1}(η), df_{2}(η)]), f_{1}, f_{2} : 𝔤^{∗} → ℝ.

Thus, F_{1} is closed under the Poisson bracket. Since G acts in a Hamiltonian way, F_{2} is closed under the Poisson bracket as well. The second essential fact is that h ∘ Φ commute with any G-invariant function (the Noether theorem). In other words {F_{1}, F_{2}} = 0.

Assumption 4.1 (Separation of generic orbits by invariant functions). Let a general orbit of the action have dimension m. We shall suppose that

(4.3)  span {df(x), f ∈ F_{2}} = ann(T_{x}(G · x)),

for general x ∈ M, dim G · x = m. Whence ddim F_{2} = 2n − m. By reg F_{2} denote the open dense set where (4.3) holds.

Assumption 4.1 holds for any proper group action because all orbits are separated by invariant functions. Moreover, for the action of a compact group G the algebra F_{2} is generated by a finite number of functions. To be more precise, let the G action have a finite number of orbit types. Then there exist functions f_{1}, …, f_{r} ∈ F_{2}, such that every function f ∈ F_{2} is of the form: f = F (f_{1}, …, f_{r}). This theorem was proved by Schwarz [76]. If M is compact then M has a finite number of orbit types. Furthermore, Mann proved that if M is an orientable manifold whose homology groups H_{i}(M, ℤ) are finitely generated, then the number of orbit types of any action of a compact Lie group on M is finite. A review of results concerning invariant functions of G actions can be found in [66].

The following theorem, although it is a reformulation of some well known facts about the momentum mapping (e.g., see [38]), is fundamental in the considerations below.

Let A ⊂ C^{∞}(𝔤) be a Lie subalgebra and Φ^{∗}A = {h ∘ Φ, h ∈ A} the pull-back of A by the momentum mapping. Then we have:

Theorem 4.1 (Bolsinov and Jovanović [12]). Let a connected Lie group G act on 2n-dimensional connected symplectic manifold (M, ω). Suppose the action is Hamiltonian and Assumption 4.1 holds. Then:

(i) The algebra of functions F_{1} + F_{2} is complete:

ddim (F_{1} + F_{2}) + dind (F_{1} + F_{2}) = dim M.

(ii) Φ^{∗}A+F_{2} is a complete algebra on M if and only if A is a complete algebra on a generic adjoint orbit O(μ) ⊂ Φ(M).

(iii) If B is complete (commutative) subalgebra of F_{2} and A is complete (commutative) algebra on the orbit O(μ), for generic μ ∈ Φ(M) then Φ^{∗}A+B is complete (commutative) algebra on M.

Proof. It is enough to prove item (i). We shall need a well known fact that ker dΦ(x) is symplectically orthogonal to the tangent space at x to the orbit of x (see [38, 46]):

(4.4)  ker dΦ(x) = (T_{x}(G · x))^{ω}.

Let U = reg (F_{1} + F_{2}) = reg F_{2} be the open dense set in M, such that for x ∈ U the moment map (3.1) has maximal rank (by (4.4) it is equivalent to the fact that the orbit G·x has maximal dimension) and that (4.3) holds. Let x belong to reg (F_{1} + F_{2}). Consider the tangent spaces to the level sets of integrals F_{1} and F_{2}:

D_{1} = D_{1}(x) = span {X | (df_{x}, X) = 0, f ∈ F_{1}} ⊂ T_{x}M
D_{2} = D_{2}(x) = span {X | (df_{x}, X) = 0, f ∈ F_{2}} ⊂ T_{x}M

We shall prove that D_{1} ∩D_{2} is isotropic: D_{1} ∩D_{2} ⊂ (D_{1} ∩D_{2})^{ω} which is equivalent to the completeness of the algebra F_{1} + F_{2} (see remark 2.1). For x ∈ reg (F_{1} + F_{2}) we have D_{1} = ker dΦ(x) and D_{2} = T_{x}(G · x). Therefore by (4.4)

(D_{1} ∩ D_{2})^{ω} = D_{1}^{ω} + D_{2}^{ω} = D_{2} + D_{1} ⊃ D_{1} ∩ D_{2}.  □

Following Arnold [3], define ad^{∗} by

(ad_{ω}^{∗}μ, ξ) = (μ, ad_{ω}ξ) = (μ, [ω, ξ]), μ ∈ 𝔤^{∗}, ω, ξ ∈ 𝔤.

Let f_{h} = h ∘ Φ ∈ F_{1} and let μ_{0} = Φ(x_{0}). If

ad_{dh(μ_{0})}^{∗}μ_{0} = 0 ⟺ dh(μ_{0}) ∈ ann T_{μ_{0}}O(μ_{0})
(4.5)  ⟺ {f, h}_{𝔤^{∗}}(μ_{0}) = 0, f ∈ C^{∞}(𝔤^{∗}),

then by (4.1) and (4.2) we get that f_{h} commutes with all functions from F_{2} at x_{0}.

By the proof of Theorem 4.1, for x_{0} ∈ reg (F_{1} +F_{2}), the condition (4.5) implies that df_{h}(x_{0}) belongs to the span {df(x_{0}), f ∈ F_{2}}. Thus we have:

ddim (F_{1} + F_{2}) = dim M − dim G · x + dim O(μ)
= dim M + dim G_{x} − dim G_{μ}
dind (F_{1} + F_{2}) = dim G_{μ} − dim G_{x},

for general x ∈ M, μ = Φ(x) (G_{μ} and G_{x} denotes the isotropy groups of G action at μ and x).

Note that if h : 𝔤^{∗} → ℝ is an invariant of the co-adjoint action, then (4.5) hold on 𝔤^{∗} and h ∘ Φ will be a G-invariant function on M.

Remark 4.1. Similar construction can be performed on dual pairs (see Panasyuk [69]). Recall, the dual pair is given by two Poisson mappings

N_{1} ⟵^{Φ_{1}} M ⟶^{Φ_{2}} N_{2}

such that all level-sets of Φ_{1} and Φ_{2} are regular and symplectically orthogonal in the intersection points [82]. Then the algebra F_{1} + F_{2} is complete on M, where F_{i} = Φ_{i}^{∗}(C^{∞}(N_{i})), i = 1, 2.

Definition 4.1. [38] The Hamiltonian H : M → ℝ is said to be collective if H is of the form H = h ∘ Φ.

The solutions of the Hamiltonian equation ẋ = X_{h∘Φ} are of the form

x(t) = g(t) · x_{0},

where g(t) are solutions of the kinematic equation ġ = dh(μ(t)) · g(t) and μ(t) are solutions of the Euler equations (4.6) (see [38]).

For example, if M = T^{∗}G, the geodesic flow of the right-invariant metric on G is described by the collective Hamiltonian with respect to the natural left G-action.

Corollary 4.1. Suppose that assumption 4.1 holds. Let h : 𝔤^{∗} → ℝ be a Hamiltonian function such that the Euler equations:

(4.6)  μ̇ = −ad_{dh(μ)}^{∗}μ ⟺ ḟ = {f, h}_{𝔤^{∗}}, f ∈ C^{∞}(𝔤^{∗}),

are completely integrable on general co-adjoint orbits O(μ) ⊂ Φ(M) with a set of Lie–Poisson commuting integrals f_{i}, i = 1, …, \frac{1}{2} dim O(μ). Then the Hamiltonian equations on M with the collective Hamiltonian function h ∘ Φ are completely integrable (in the non-commutative sense). The complete set of first integrals is

\{f_{i} ∘ Φ | i = 1, …, \frac{1}{2} dim O(μ)\} + F_{2}.

The above theorem for the multiplicity-free actions is given by Guillemin and Sternberg (see [36, 37, 38]).

If G is a compact group, then the connected components of regular invariant submanifolds are isotropic tori of dimension \frac{1}{2}(dim G + dim G_{μ}) − dim G_{x}.

4.2. Geodesic Flows on Homogeneous Spaces of Compact Lie Groups. The classical example is the geodesic flow of a left-invariant metric on the Lie group SO(3). The geodesic flow of such a metric describes the motion of a rigid body about a fixed point under its own inertia. This problem was solved by Euler (e.g., see [3, 2].

In general, the geodesic flow of a left-invariant metrics on a Lie group G after G-reduction reduces to the 𝔤^{∗} = (T^{∗}G)/G. The reduced bracket of the canonical Poisson bracket on T^{∗}G are the Lie–Poisson bracket (4.2) (multiplied by −1) [3]. Therefore the reduced system is described by Euler equations (4.6) (multiplied by −1), where now h is the reduced Hamiltonian.

A multidimensional generalization of the Euler case to so(n) has been suggested by Manakov [53]. Using his idea, Mishchenko and Fomenko proposed the argument shift method (see below) and constructed integrable examples of Euler equations for all compact groups [60] and proved the integrability of the original geodesic flows [61, 62]. There are many other important constructions on various Lie algebras (e.g., see [79, 8, 72]).

Thimm [78] and Mishchenko [59] proved the complete integrability of geodesic flows of normal metrics on compact symmetric spaces G/H. By a normal Ginvariant Riemannian metric on the homogeneous space G/H of a compact group G, we mean the submersion metric induced from a bi-invatiant metric on G.

These results are generalized to the class of homogeneous spaces G/H on which all G-invariant Hamiltonian systems are integrable by means of Noether integrals (see [36, 37, 55]). In those cases (G, H) is a spherical pair and G/H is a weekly symmetric space (see [80] and references therein). Examples of homogeneous, non (weekly) symmetric spaces G/H with integrable geodesic flows are given in [78, 71, 57]. It appears that many of those examples can be considered together within the framework of non-commutative integrability.

Let G be a compact connected Lie group with the Lie algebra 𝔤 = T_{e}G. Let us fix some bi-invariant metric ds_{0}^{2} on G, i.e., Ad_{G}-invariant scalar product ⟨·, ·⟩ on 𝔤. We can identify 𝔤^{∗} and 𝔤 by ⟨·, ·⟩.

Consider an arbitrary homogeneous space G/H of the Lie group G. The metric ds_{0}^{2} induces so called normal metric (or standard metric) on G/H. We shall denote the normal metric also by ds_{0}^{2}. By the use of ds_{0}^{2} we identify T^{∗}G ≅ T G and T^{∗}(G/H) ≅ T (G/H). Let 𝔥 be the Lie algebra of H and 𝔤 = 𝔥 + 𝔳 the orthogonal decomposition. Then 𝔳 can be naturally identified with T_{ρ(e)}(G/H) and T_{ρ(e)}^{∗}(G/H), where ρ : G → G/H is the canonical projection.

The momentum mapping Φ : T^{∗}(G/H) → 𝔤 of the natural G-action on T^{∗}(G/H) is given by Φ(g · η) = Ad_{g}(η). where η ∈ 𝔳, g · η ∈ T_{ρ(g)}^{∗}(G/H).

The Hamiltonian function of the geodesic flow of the normal metric ds_{0}^{2} is simply given by

(4.7)  h_{0} = \frac{1}{2}⟨Φ, Φ⟩.

Since ⟨ξ, ξ⟩ is Ad_{G}-invariant, h_{0} Poisson commute with F_{1} and F_{2}. Note that for F_{1} and F_{2} we can take analytic functions, polynomial in momenta. Thus we get

Theorem 4.2. [11] Let G be a compact Lie group. The geodesic flows of the normal metrics on the homogeneous spaces G/H are completely integrable in the non-commutative sense by means of analytic functions, polynomial in momenta.

Let ℝ[𝔤]^{G} be the algebra of Ad_{G}-invariant polynomials on 𝔤. The polynomials

A_{a} = {p_{a}^{λ} = p(· + λa) | λ ∈ ℝ, p ∈ ℝ[𝔤]^{G}}

obtained from the invariants by shifting the argument are all in involution [60]. Furthermore, for every adjoint orbit in 𝔤, one can find a ∈ 𝔤, such that A_{a} is a complete involutive set of functions on this orbit. For regular orbits it is proved in [60]. For singular orbits there are several different proofs, see [54, 19, 8]. (Note that Φ(T^{∗}(G/H)) can be often a subset of singular set in 𝔤^{∗}.)

According item (ii) of theorem 4.1, Φ^{∗}A_{a}+F_{2} is a complete algebra on T^{∗}(G/H). Let h_{C}(ξ) = \frac{1}{2}⟨C(ξ), ξ⟩ be a quadratic positive definite polynomial in A_{a} (operators C : 𝔤 → 𝔤 are described in [60]). Then h_{C} ∘ Φ is the Hamiltonian of the geodesic flow of a certain metric that we shall denote by ds_{C}^{2}. The metric ds_{C}^{2} has the following nice geometrical meaning. This is the submersion metric of the right-invariant Riemannian metric on G whose Hamiltonian function is obtained from h_{C}(ξ) by right translations. Thus some integrable (in non-commutative sense) deformations of the normal metric always exist [11, 12]. This construction for symmetric spaces is done by Brailov [18, 19].

Whence, to construct a complete commutative algebra of analytic integrals of the geodesic flow of the normal metric we need to solve the following nontrivial problem: to find a complete commutative subalgebra B of G-invariant functions on T^{∗}(G/H).

For symmetric and weekly symmetric spaces (spherical pairs) the algebra F_{2} is already commutative and for the integrability we need only Noether’s integrals F_{1}. Spherical pairs for G simple and semisimple are classified in [44] and [55], respectively.

The examples of Thimm [78] ((SO(n), SO(n−2))) and Paternain and Spatzier [71] ((SU(3), 𝕋^{2})) are almost spherical pairs, that are the homogeneous spaces where for a complete commutative set of functions on T^{∗}(G/H) we can take an arbitrary complete commutative subalgebra A ⊂ F_{1} and one G-invariant function functionally independent of F_{1} [57, 56]. Almost spherical pairs are classified, for G compact and semisimple in [70, 57].

There are two natural methods for constructing commutative families of Ginvariant functions, namely the shift-argument method and chain of subalgebras method, see [11, 13]. In many examples (Stiefel manifolds, flag manifolds, orbits of the adjoint actions of compact Lie groups etc.) those methods lead to complete commutative algebras (see Bolsinov and Jovanović [11, 13], Buldaeva [20], Mykytyuk and Panasyuk [58]).

Remark 4.2. We also note that integrable potential systems on symmetric spaces and coadjoint orbits can be found, e.g., in [75] and [15], respectively. Then one can easily construct, by means of the classical Maupertuis principle, corresponding integrable geodesic flows that are not G invariant (e.g., see [9]).

4.3. Magnetic Geodesic Flows. Similar construction is used for the magnetic geodesic flows of the normal metrics on a class of homogeneous spaces, in particular (co)adjoint orbits of compact Lie groups [27, 28, 14, 15].

The geodesic flow on (Q, κ) can be interpreted as the inertial motion of a particle on Q with the kinetic energy given by (1.4) (we take v ≡ 0). The motion of the particle under the influence of the additional magnetic field given by a closed 2-form

Ω = ∑_{1⩽i<j⩽n} F_{ij}(q)dq^{i} ∧ dq^{j},

is described by the following equations:

(4.8)  \frac{dq^{i}}{dt} = \frac{∂h}{∂p_{i}} ,  \frac{dp_{i}}{dt} = −\frac{∂h}{∂q^{i}} + ∑_{j=1}^{n} F_{ij} \frac{∂h}{∂p_{j}} .

The equations (4.8) are Hamiltonian with respect to the “twisted” symplectic form ω + σ^{∗}Ω, where σ : T^{∗}Q → Q is the natural projection. Namely, the new Poisson bracket is given by

(4.9)  \{f, g\}_{Ω} = \{f, g\} + ∑_{i,j=1}^{n} F_{ij} \frac{∂f}{∂p_{i}} \frac{∂g}{∂p_{j}} ,

and the Hamiltonian equations ḟ = {f, h}_{Ω} read (4.8). Here {·, ·} are canonical Poisson bracket (1.5).

The flow (4.8) is called magnetic geodesic flow on the Riemannian manifold (Q, κ) with respect to the magnetic field Ω. For simplicity, we shall refer to (4.9) as a magnetic Poisson bracket and to (T^{∗}Q, ω +σ^{∗}Ω) as a magnetic cotangent bundle.

We introduce a class of homogeneous spaces G/H having a natural construction of the magnetic term, consisting of pairs (G, H), where H have one-point adjoint orbits. Let a ∈ 𝔥 be the H-adjoint invariant. Then H is a subgroup of the Gadjoint isotropy group G_{a} of a. The adjoint orbit O(a) through a carries the Kirillov–Konstant symplectic form Ω_{KK} (e.g., see [38]). Then we have canonical submersion of homogeneous spaces τ : G/H → G/G_{a} ≅ O(a) and the closed twoform Ω = τ^{∗}Ω_{KK} gives us the required magnetic field on G/H.

The form Ω is G-invariant. From the definition of Ω_{KK} one can easily prove that at the point ρ(e), Ω is given by

(4.10)  Ω(ξ_{1}, ξ_{2})|_{ρ(e)} = −⟨a, [ξ_{1}, ξ_{2}]⟩, ξ_{1}, ξ_{2} ∈ 𝔳 ≅ T_{ρ(e)}(G/H)

Lemma 4.1. The momentum mapping Φ_{ϵ} : T^{∗}(G/H) → 𝔤 of the natural Gaction on T^{∗}(G/H) with respect to the symplectic form ω + ϵσ^{∗}Ω is given by

Φ_{ϵ}(g · η) = Φ_{0}(g · η) + ϵ Ad_{g}(a),

where η ∈ 𝔳, g · η ∈ T_{π(g)}^{∗}(G/H).

Now, let F_{1}^{ϵ} be the algebra of all analytic, polynomial in momenta, functions of the form F_{1}^{ϵ} = {p ∘ Φ_{ϵ}, p ∈ ℝ[𝔤]} and F_{2} be the algebra of all analytic, polynomial in momenta, G-invariant functions on T^{∗}(G/H). Then {F_{1}^{ϵ}, F_{2}}_{ϵ} = 0, where {·, ·}_{ϵ} are magnetic Poisson bracket with respect to ω + ϵσ^{∗}Ω.

Consider the Hamiltonian h_{ϵ} = \frac{1}{2}⟨Φ_{ϵ}, Φ_{ϵ}⟩ ∈ F_{1}^{ϵ}. We have

h_{ϵ}(g · η) = \frac{1}{2}⟨Ad_{g} η, Ad_{g} η⟩ + ϵ⟨Ad_{g} η, Ad_{g} a⟩ + \frac{ϵ^{2}}{2} ⟨Ad_{g} a, Ad_{g} a⟩ = h_{0}(g · η) + const,

where we used that η ∈ 𝔳 is orthogonal to a ∈ 𝔥 and h_{0} is the Hamiltonian of the normal metric (4.7). Thus, the Hamiltonian flows of h_{0} and h_{ϵ} coincides. Since h_{ϵ} belongs to F_{1}^{ϵ} its commutes with F_{2}. On the other side, as a composition of the momentum mapping with an invariant polynomial, the function h_{ϵ} is also G-invariant and commutes with F_{1}^{ϵ}.

Theorem 4.3 (Bolsinov and Jovanović [15]). The magnetic geodesic flow of the normal metrics ds_{0}^{2} on the homogeneous space G/H with respect to the Ginvariant closed 2-form ϵΩ given at ρ(e) by (4.10) is completely integrable in the non-commutative sense. The complete algebra of first integrals is F_{1}^{ϵ} + F_{2}.

5. Reductions and Integrability

5.1. Reconstruction. Suppose the reduced flow (3.7) is completely integrable (in the non-commutative sense) with a complete algebra of first integrals F. Then F can be considered as a complete algebra

B = π^{∗}F

of F_{2} = C^{∞}(M)^{G}. According Theorem 4.1 we get the following theorem, explicitly given in [83].

Theorem 5.1 (Zung [83]). The integrability of the reduced flow (3.7) implies the integrability of the original system (1.1).

In the case of geodesic flows of left-invariant metric, Theorem 5.1 is proved by Mishchenko and Fomenko [61, 62].

5.2. Poisson Reduction. We have the following simple general observation (see e.g. [40]).

Lemma 5.1. Suppose a connected Lie group G acts effectively on a symplectic manifold (M, ω). Let h be a G-invariant Hamiltonian function.

(i) If the Hamiltonian system (1.1) is completely integrable by means of Ginvariant first integrals, then G is commutative. Moreover, if the action is free, the reduced Hamiltonian system on N_{η} is completely integrable for a generic value η of the moment map, i.e., the reduced flow (3.7) on the Poisson manifold M/G is completely integrable.

(ii) If the system (1.1) is 𝕋^{n}-dense completely integrable, then the Lie group G is commutative.

Proof. (i) Let F be a complete G-invariant algebra of integrals and let D_{x} and W_{x} be the subspaces of T_{x}M defined in Remark 2.1. Since all functions from F are G-invariant we have that T_{x}(G · x) ⊂ D_{x} is isotropic, for generic x ∈ M. The Hamiltonian vector fields of the functions (3.2) generate one-parameter groups of symplectomorphisms. Therefore ω_{x}(X_{ϕ_{ξ_{1}}} (x), X_{ϕ_{ξ_{2}}} (x)) vanishes in an open dense set of M. Thus, according to (3.3)

ϕ_{[ξ_{1},ξ_{2}]} = {ϕ_{ξ_{1}}, ϕ_{ξ_{2}}} = 0

for every ξ_{1}, ξ_{2} ∈ 𝔤. Since the action is effective, we get that G is commutative.

Let the Abelian group G act freely on M. Take x ∈ M such that F is complete at x. Since all functions from F are G-invariant, the coisotropic space W_{x} = {X_{f}(x) | f ∈ F} belongs to T_{x}Φ^{−1}(η), η = Φ(x).

Let F_{η} = {F_{η}, f ∈ F} be the induced algebra of integrals on N_{η}. From the definition of the reduced symplectic structure, one can easily see that the space

W_{x}^{η} = {X_{F_{η}}(π_{η}(x)) | F ∈ F_{η}} = dπ_{η}(W_{x})

is also coisotropic, i.e., the induced algebra F_{η} is complete at π_{η}(x).

(ii) If the system (1.1) is 𝕋^{n}-dense completely integrable, then in every toroidal domain the functions ϕ_{ξ}(x), ξ ∈ 𝔤 depend only on the action variables and so {ϕ_{ξ_{1}}, ϕ_{ξ_{2}}} = ϕ_{[ξ_{1},ξ_{2}]} vanishes on reg M. Therefore [𝔤, 𝔤] = 0 and the Lie group G is commutative. □

From Lemma 5.1, the condition on a G-invariant algebra F on (M, ω) to be complete forces G to be abelian, which is too restrictive. However, we can consider the case when we have integrability of (1.1) with non-invariant integrals. Also, note that by item (ii) the existence of the non-Abelian group of symmetries is closely related to the non-commutative integrability of the Hamiltonian flow.

Definition 5.1. [83] We shall say that the Hamiltonian system (1.1) is completely integrable by means of an algebra of all integrals if the family of all integrals of the system form a complete algebra.

It is clear that when F is a complete algebra of integrals, dind F = r, the connected components of regular invariant manifolds are compact and the system is 𝕋^{r}-dense within reg M, then the system is integrable by means of an algebra of all integrals.

Theorem 5.2 (Zung [83]). If G is compact and the Hamiltonian system (1.1) is completely integrable by means of an algebra of all integrals, then the reduced system (3.7) on M/G will be integrable by means of an algebra of all integrals.

Proof. For the completeness in the exposition, we shall present the proof given in [83] slightly modified and adopted for our notation.

Let F_{h} be the algebra of all integrals of the system (1.1). Without loss of generality it can be assumed that there exist functions f_{1}, …, f_{l}, h_{1}, …, h_{r} ∈ F_{h} (l = dind F_{h}, r = dind F_{h}) such that {F_{h}, h_{i}} = 0, i = 1, …, r and df_{1}∧⋯∧df_{l} ≠ 0 and dh_{1} ∧ ⋯ ∧ dh_{r} ≠ 0 at a generic point in M.

As in Section 3, by capital letters we denote the functions on M/G and by small letters corresponding G-invariant functions on M (f = π^{∗}(F), where π : M → M/G is the natural projection).

Let s be the generic dimension of the intersection of a common level set of f_{i}, i = 1, …, l with an orbit of G in M. The set of all integrals F_{H} of the flow of the reduced Hamiltonian H can be obtained from the functions in F_{h} by averaging with respect to the G-action. Therefore

(5.1)  l − ddim F_{H} = ddim F_{h} − ddim F_{H} ⩽ dim G − s.

An important observation is that the functions h_{i} are G-invariant. Since h is G invariant, the functions (3.2) are first integrals, i.e., ϕ_{ξ} ∈ F_{h}, ξ ∈ 𝔤. This implies {ϕ_{ξ}, h_{i}} = 0, i = 1, …, r, ξ ∈ 𝔤, which means that h_{i} are G-invariant. Whence

(5.2) r − dim span {X_{H_{1}}, …, X_{H_{r}}}|_{π(x)}

= dim span {X_{h_{1}}, …, X_{h_{r}}}|_{x} − dim span {X_{H_{1}}, …, X_{H_{r}}}|_{π(x)} = s,

for generic x ∈ M. Here X_{h_{i}} and X_{H_{i}} are Hamiltonian vector fields on M and M/G, respectively.

Further, the functions H_{1}, …, H_{r} belongs to the kernel of F_{H}. Indeed if F ∈ F_{H}, then f = π^{∗}(F) ∈ F_{h} and {h_{i}, f} = {H_{i}, F }^{G} = 0.

Taking into account rank G-Casimir functions of the Poisson bracket {·, ·}^{G}, from (5.2) we get

(5.3)  dind F_{H} ⩾ r − s + rank G.

Finally, from (5.1) and (5.3) it follows

ddim F_{H} + dind F_{H} ⩾ l + s − dim G + r − s + rank G = dim M/G + corank {·, ·}^{G}

implying the completeness of F_{H}. □

5.3. Symplectic Reduction. Suppose we are given a G-invariant natural mechanical system (Q, κ, v) with a completely integrable flow. The natural question arises: will the reduced natural system on Q/G be integrable?

This approach is used in the construction of new manifolds with completely integrable geodesic flows starting from the integrable geodesic flows with symmetries. Paternain and Spatzier proved that if the manifold Q has geodesic flow integrable by means of S^{1}-invariant integrals and if N is a surface of revolution, then the submersion geodesic flow on Q×_{S^{1}} N = (Q×N)/S^{1} will be completely integrable [71]. Combining submersions and Thimm’s method (see [78]), Paternain and Spatzier [71] and Bazaikin [5] proved integrability of geodesic flows on certain interesting bi-quotients of Lie groups. We shall explain these submersion examples in the framework of a general construction.

Note that, if G is not Abelian, T^{∗}(Q/G) is the singular symplectic leaf in (T^{∗}Q)/G so we can not directly apply Theorem 5.1.

Let G be a compact connected Lie group. Suppose we are given an integrable G-invariant Hamiltonian system ẋ = X_{H}(x) with compact iso-energy levels M_{h} = H^{−1}(h). Then M is foliated by invariant tori in an open dense set that we shall denote by reg M. Suppose 0 is the regular value of the momentum mapping.

Theorem 5.3. If reg M intersects the submanifold Φ^{−1}(0) in a dense set, then the reduced Hamiltonian system (3.5) on (N_{0}, ω_{0}) will be completely integrable.

Proof. Take some toroidal domain O = 𝕋^{r}{φ} × B_{σ}{I, p, q} ⊂ reg M such that O_{1} = O ∩ Φ^{−1}(0) ≠ ∅ and that the Hamiltonian flow (1.1) is 𝕋^{r}-dense in O. Consider the G-invariant sets U_{1} = U ∩Φ^{−1}(0), U = G·O = {g ·x | x ∈ O, g ∈ G}.

Note that the functions (3.2) are first integrals of the Hamiltonian flow and so do not depend on φ in O. In other words, the action of G preserves the foliation by r-dimensional invariant isotropic tori of the sets U and U_{1} as well.

The foliation T of U_{1} by tori induces a foliation L of U_{1} by (dim G + r − s)- dimensional G-invariant submanifolds, with tangent spaces of the form

T_{x}L = T_{x}(G · x) + T_{x}(T ) ⊂ T_{x}Φ^{−1}(0),

where s = dim S_{x}, S_{x} = T_{x}(G · x) ∩ T_{x}(T ), x ∈ U_{1} (for x ∈ O_{1} we have that

T_{x}L = {X_{ϕ_{a}}(x) | X_{I_{i}}(x)}, S_{x} = {X_{ϕ_{a}}(x)} ∩ {X_{I_{i}}(x)}

and the foliation L|_{O_{1}} does not depend on φ).

Let π_{0} : Φ^{−1}(0) → Φ^{−1}(0)/G = N_{0} be the natural projection and let Ũ = π_{0}(U_{1}) ⊂ N_{0}. The foliation L induces a foliation T̃ = π_{0}(L) of Ũ by an (r − s)- dimensional invariant manifolds of the geodesic flow of the submersion metric. We shall see below that T̃ is a foliation of Ũ by an invariant tori with respect to certain complete algebra of first integrals F_{0}.

The foliation L can be seen as the level sets of G-invariant integrals f_{1}, …, f_{ρ} on U_{1}, ρ = dim M − dim G + s − r. Indeed, this is always true locally: for σ small enough there are functions f_{i}^{O_{1}} = f_{i}^{O_{1}}(I, p, q) on O_{1} such that the tangent spaces T_{x}L (recall that L|_{O_{1}} does not depend on φ) are given by the equations

T_{x}L = {ξ ∈ T_{x}O_{1} | df_{i}^{O_{1}}(x)(ξ) = 0, i = 1, …, ρ}.

Then functions f_{i} are G-invariant extensions of f_{i}^{O_{1}} to U_{1}, i = 1, …, ρ. Let F_{0} be the induced algebra of first integrals in Ũ. Since

D_{π_{0}(x)} = {ξ ∈ T_{π_{0}(x)}Ũ | df_{0}(π_{0}(x))(ξ) = 0, f_{0} ∈ F_{0}} = T_{π_{0}(x)}T̃ = dπ_{0}(x)(T_{x}L),

x ∈ U_{1} are isotropic spaces, by Remark 1 we get that F_{0} is complete in Ũ. Note that ddim F_{0} = dim N_{0} + s − r, dind F_{0} = r − s.

Now fill up N_{0} with countably many disjoint torodial domains

Õ_{α} = 𝕋^{r(α)}{φ̃} × B_{σ(α)}{Ĩ, p̃, q̃}

(with possible different dimensions of tori). As in Theorem 2.2, in every domain Õ_{α}, one can construct complete involutive set of integrals that can be then “glued” in order to obtain a complete involutive set of integrals globally defined.

We have the following m = \frac{1}{2} dim N_{0} commuting integrals in Õ_{α}:

h_{1} = Ĩ_{1}^{2}, …, h_{r} = Ĩ_{r}^{2}, h_{r+1} = p̃_{1}^{2} + q̃_{1}^{2}, …, h_{m} = p̃_{k}^{2} + q̃_{k}^{2}.

Let g_{α} : ℝ → ℝ be a smooth nonnegative function such that g_{α}(x) is equal to zero for |x| > σ(α), g_{α} monotonically increases on [−σ(α), 0] and monotonically decreases on [0, σ(α)]. Let h_{α}(y) = g_{α}(h_{1}(y) + ⋯ + h_{m}(y)). This function can be extended by zero to the whole manifold N_{0}. Then f_{i}^{α} = h_{α} · h_{i}, i = 1, …, n will be commuting functions, independent on an open dense subset of Õ_{α}. With a “good” choice of g_{α}’s, a complete commutative set of smooth integrals is given by f_{i}(y) = f_{i}^{α}(y) for y ∈ Õ_{α} ⊂ ⋃_{β} Õ_{β} and zero otherwise, i = 1, …, m. □

Note that we do not suppose that integrals of the original system are Ginvariant. Also, the general construction used in the proof of the theorem leads to smooth commuting integrals on M_{0}.

Remark 5.1. It is clear that a similar statement holds for an arbitrary value of the momentum mapping: if reg M intersects the submanifold Φ^{−1}(η) in a dense set, then the reduced Hamiltonian system on (N_{η}, ω_{η}) will be completely integrable. Also one can get the following modification of Theorem 5.2: Suppose we are given an integrable Hamiltonian system (1.1) invariant with respect to the action of a compact group G and with compact iso-energy levels M_{h} = H^{−1}(h). Then the reduced system (3.7) on M/G will be completely integrable.

Corollary 5.1. [40] Let a compact connected Lie group G act freely by isometries on a compact Riemannian manifold (Q, κ). Suppose that the geodesic flow of κ is completely integrable. If reg T^{∗}Q intersects the space of horizontal vectors H ≅ Φ^{−1}(0) in a dense set then the geodesic flow on Q/G endowed with the submersion metric K is completely integrable.

Example 5.1. Eschenburg constructed bi-quotients M_{k,l,p,q}^{7} = SU(3)//T_{k,l,p,q} endowed with the submersion metrics ds_{t,sub}^{2} with strictly positive sectional curvature. Here T_{k,l,p,q} ≅ T^{1} ⊂ T^{2} × T^{2}, where T^{2} is a maximal torus and ds_{t}^{2} is a one-parameter family of left-invariant metrics on SU(3) (see [29]). One can prove that the geodesic flows of the metrics ds_{t}^{2} are completely integrable and that we can apply Theorem 5.1 to get the integrability of the geodesic flows of the submersion metrics on M_{k,l,p,q}^{7}.

Suppose we are given Hamiltonian G-actions on two symplectic manifolds (M_{1}, ω_{1}) and (M_{2}, ω_{2}) with moment maps Φ_{M_{1}} and Φ_{M_{2}}. Then we have the natural diagonal action of G on the product (M_{1} × M_{2}, ω_{1} ⊕ ω_{2}), with moment map

(5.4)  Φ_{M_{1}×M_{2}} = Φ_{M_{1}} + Φ_{M_{2}}.

If (Q_{1}, κ_{1}) and (Q_{2}, κ_{2}) have integrable geodesic flows, then (Q_{1} × Q_{2}, κ_{1} ⊕ κ_{2}) also has integrable geodesic flow. Using (5.4), one can easily see that if the Gactions on Q_{1} and Q_{2} are almost everywhere locally free, and free on the product Q_{1} × Q_{2}, then a generic horizontal vector of the submersion

Q_{1} × Q_{2} → Q_{1} ×_{G} Q_{2} = (Q_{1} × Q_{2})/G

belongs to reg T^{∗}(Q_{1}×Q_{2}) = reg T^{∗}Q_{1}×reg T^{∗}Q_{2}. Thus, with the above notation, we get

Theorem 5.4. The geodesic flow on Q_{1} ×_{G} Q_{2}, endowed with the submersion metric, is completely integrable.

Example 5.2. Let a compact Lie group G acts freely by isometries on (Q, κ) and let G_{1} be an arbitrary compact Lie group that contains G as a subgroup. Let ds_{1}^{2} be some left-invariant Riemannian metric on G_{1} with integrable geodesic flow. Then G acts in the natural way by isometries on (G_{1}, ds_{1}^{2}). Therefore, if the geodesic flow on Q is completely integrable, then the geodesic flows on Q ×_{G} Q and Q ×_{G} G_{1} endowed with the submersion metrics will be also completely integrable.

6. Partial Integrability

In this section we study reductions of the Hamiltonian flows restricted to their invariant submanifolds (see [41]). Apparently, the lowering of order in Hamiltonian systems having invariant relations was firstly studied by Levi-Civita (e.g., see [45, ch. X]).

6.1. Hess–Appel’rot System. The classical example of the system having an invariant relation is a celebrated Hess–Appel’rot case of a heavy rigid body motion [39, 1]. Recall that the motion of a heavy rigid body around a fixed point, in the moving frame, is represented by the Euler–Poisson equations

(6.1)  \frac{d}{dt}m⃗ = m⃗ × ω⃗ + 𝔊𝔐 γ⃗ × r⃗, \frac{d}{dt}γ⃗ = γ⃗ × ω⃗, ω⃗ = Am⃗,

where ω⃗ is the angular velocity, m⃗ the angular momentum, I = A^{−1} the inertial tensor, 𝔐 mass and r⃗ the vector of the mass center of a rigid body; γ⃗ is the direction of the homogeneous gravitational field and 𝔊 is the gravitational constant.

The equations (6.1) always have three integrals, the energy, geometric integral and the projection of angular momentum:

(6.2)  F_{1} = \frac{1}{2}(m⃗, ω⃗) + 𝔐𝔊(r⃗, γ⃗), F_{2} = (γ⃗, γ⃗) = 1, F_{3} = (m⃗, γ⃗).

For the integrability we need a forth integral. There are three famous integrable cases: Euler, Lagrange and Kowalevskaya [2, 35].

Apart of these cases, there are various particular solutions (e.g., see [35]). The celebrated is partially integrable Hess–Appel’rot case [39, 1]. Under the conditions:

(6.3)  r_{2} = 0, r_{1}\sqrt{a_{3} − a_{2}} ± r_{3}\sqrt{a_{2} − a_{1}} = 0, A = diag(a_{1}, a_{2}, a_{3}),

a_{3} > a_{2} > a_{1} > 0, system (6.1) has an invariant relation given by

(6.4)  F_{4} = (m⃗, r⃗) = r_{1}m_{1} + r_{3}m_{3} = 0.

The system is integrable up to one quadrature: the compact connected components of the regular invariant sets

(6.5)  F_{1} = c_{1}, F_{2} = 1, F_{3} = c_{3}, F_{4} = 0

are tori, but not with quasi-periodic dynamics. The classical and algebro-geometric integration can be found in [35] and [25], respectively.

There is a nice geometrical interpretation of the conditions (6.3) given by Zhukovski: the intersection of the plane orthogonal to r⃗ with the ellipsoid

(Am⃗, m⃗) = const

is a circle [84, 16]. If instead of the moving base given by the main axes of the inertia, we take the moving base f⃗_{1}, f⃗_{2}, f⃗_{3} such that the mass center of a rigid body r⃗ is proportional to f⃗_{3},

r⃗ = ρf⃗_{3}, ρ = \sqrt{r_{1}^{2} + r_{3}^{2}},

then the inverse of the inertial operator reads

A = (\begin{matrix}a_{2} & 0 & a_{13} \\ 0 & a_{2} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{matrix})

and the invariant relation (6.4) is simply given by

(6.6)  m_{3} = 0.

Note that, considered on the whole phase space T^{∗}SO(3) of the rigid body motion, the function m_{3} is the momentum mapping of the right SO(2)-action-rotations of the body around the line directed to the center of the mass.

Further, let R⃗ and Γ⃗ be the vectors of the mass center of a rigid body and the direction of the gravitational field considered in the space frame. The motion of R⃗(t) is described by the spherical pendulum equations (see [84, 16])

(6.7)  \frac{d^{2}}{dt^{2}} R⃗ = −a_{2}ρ^{2}𝔊𝔐Γ⃗ + μR⃗

Here the Lagrange multiplier μ is determined from the condition (R⃗, R⃗) = ρ^{2}.

Historical overview, with an application of Levi–Civita ideas to the Hess– Appel’rot system and other classical rigid body problems can be found in Borisov and Mamaev [16].

Recently, an interesting construction of a n-dimensional variant of the Hess– Appel’rot system is studied by Dragović and Gajić [26]. Besides, by generalizing analytical and algebraic properties of the classical system, they introduced a class of systems with invariant relations, so called Hess–Appel’rot type systems with remarkable property: there exist a pair of compatible Poisson structures, such that the system is Hamiltonian with respect to the first structure and invariant relations are Casimir functions with respect to the second structure (for more details see [26]). On the other side, considering invariant relation (6.6), we are interested in the following problem: suppose that h is not a G-invariant function, but M_{0} is still an invariant manifold of the Hamiltonian system (1.1). As a modification of the regular Marsden–Weinstein reduction, we study (partial) reduction of the Hamiltonian system (1.1) from M_{0} to N_{0} and relationship between the integrability of the reduced and nonreduced system.

6.2. Invariant Relations and Reduction. Let G be a connected Lie group with a free proper Hamiltonian action on a symplectic manifold (M, ω) with the momentum map (3.1). Assume that 0 is a regular value of Φ.

Let ξ_{1}, …, ξ_{p} be the base of 𝔤. Then the zero level set of the momentum mapping (3.1) is given by the equations

(6.8)  M_{0} :  ϕ_{α} = (Φ, ξ_{α}) = 0,  α = 1, …, p.

Theorem 6.1. (i) Suppose that the restriction of h to (6.8) is a G-invariant function. Then M_{0} is an invariant manifold of the Hamiltonian system (1.1) and X_{h}|_{M_{0}} projects to the Hamiltonian vector field X_{H_{0}}

(6.9)  dπ_{0}(X_{h})|_{x} = X_{H_{0}}|_{y=π_{0}(x)},

where H_{0} is the induced function on N_{0} defined by (3.6).

(ii) The inverse statement also holds: if (6.8) is an invariant submanifold of the Hamiltonian system (1.1), then the restriction of h to M_{0} is a G-invariant function and X_{h}|_{M_{0}} projects to the Hamiltonian vector field X_{H_{0}} on N_{0}, where H_{0} is defined by (3.6).

In both cases, the Hamiltonian vector field X_{h} is not assumed to be G-invariant on M. Moreover X_{h}|_{M_{0}} may not be G-invariant as well. It is invariant modulo the kernel of dπ_{0}, which is sufficient the tools of symplectic reduction are still applicable.

Definition 6.1. We shall refer to the passing from ẋ = X_{h}|_{M_{0}} to

(6.10)  ẏ = X_{H_{0}}

as a partial reduction.

The partial reduction can be seen as a special case of the symplectic reductions studied in [6, 47] (see also [46], Ch. III). There, the symplectic reduction of a symplectic manifold (M, ω) is any surjective submersion p : N → P of a submanifold N ⊂ M onto another symplectic manifold (P, Ω), which satisfies p^{∗}Ω = ω|_{N}.

Proof of theorem 6.1. (i) The action of G is generated by Hamiltonian vector fields X_{ϕ_{α}}. Since h|_{M_{0}} is G-invariant, for x ∈ M_{0} we have

(6.11)  (dh, X_{ϕ_{α}}) = {h, ϕ_{α}} = −(dϕ_{α}, X_{h}) = 0, α = 1, …, p.

Thus (6.8) is an invariant submanifold.

Let h^{∗} be an arbitrary G-invariant function that coincides with h on M_{0}. Then X_{h^{∗}} is a G-invariant vector field on M_{0} which project to X_{H_{0}} [51]:

(6.12)  dπ_{0}(X_{h^{∗}})|_{x} = X_{H_{0}}|_{y=π_{0}(x)}.

Let δ = h − h^{∗}. From the condition δ|_{M_{0}} = 0 we can express δ, in a an open neighborhood of (6.8), as δ(x) = ∑_{α=1}^{p} δ_{α}(x)ϕ_{α}(x).

Now, let f be a G-invariant function on M. Since {f, ϕ_{α}} = 0 (Noether’s theorem), we get

{δ, f}|_{M_{0}} = (∑_{α}{δ_{α}, f}(ϕ_{α} − η_{α}) + ∑_{α} δ_{α}{ϕ_{α}, f})|_{M_{0}} = 0.

Thus the Poisson bracket {δ, f}|_{M_{0}} = −(df, X_{δ})|_{M_{0}} vanish for an arbitrary Ginvariant function f. In other words

(6.13)  X_{δ}|_{x} ∈ T_{x}(G · x) ∩ T_{x}M_{0}.

Combining (6.12), X_{h} = X_{h^{∗}} + X_{δ} and (6.13) with the well known identity (e.g., see [67])

T_{x}(G · x) ∩ T_{x}M_{0} = T_{x}(G_{η} · x) = ker dπ_{0}|_{x},

we prove the relation (6.9).

(ii) Suppose that (6.8) is an invariant submanifold of (1.1). Then (6.11) holds for x ∈ M_{0}. Since the action of G is generated by Hamiltonian vector fields X_{ϕ_{α}}, from (6.11) we get that h is invariant with respect to the infinitesimal action of G. Therefore we have well defined reduced Hamiltonian function H_{0} on N_{0}.

Since M_{0} is a closed submanifold of M and the action is proper, we can find a G-invariant function h^{∗} on M which coincides with h on M_{0}. Now, the relation (6.9) follows from the proof of item (i). □

An immediate corollary of theorem 6.1 is

Corollary 6.1. Suppose that the partially reduced system (6.10) is completely integrable in the non-commutative sense and N_{0} is almost everywhere foliated on r-dimensional isotropic invariant manifolds, level sets of integrals F_{i}, i = 1, …, dim N_{0} − r. Then M_{0} is almost everywhere foliated on (r + dim G)-invariant isotropic manifolds

(6.14)  M_{c} = {f_{i} = F_{i} ∘ π_{0} = c_{i} | i = 1, …, dim N_{0} − r}

of the system (1.1).

Remark 6.1. If the partially reduced system (6.10) is integrable, then we need (dim G)-additional quadratures for the solving of ẋ = X_{h}|_{M_{0}} (the reconstruction equations). If r = \frac{1}{2} dim N_{0}, the invariant manifolds (6.14) are Lagrangian.

Definition 6.2. We shall say that the Hamiltonian system (1.1) is partially integrable if it has invariant relations of the form (6.8) and that the reduced system (6.10) is completely integrable in the non-commutative sense.

Example 6.1. Let G be a torus 𝕋^{p} and let the reduced flow be completely integrable in the commutative sense by integrals F_{1}, …, F_{m}, m = \frac{1}{2} dim N_{0}. Consider the regular compact connected component level set of F_{1}, …, F_{m}. By Liouville’s theorem, it is diffeomorphic to a m-dimensional torus 𝕋^{m} with quasi-periodic flow of (6.10). Thus the compact connected component M̂_{c} = π_{0}^{−1}(𝕋^{m}) of (6.14) is a torus bundle over 𝕋^{m}:

(6.15)  \begin{aligned}𝕋^{p} ⟶ & M̂_{c} \\ ↓ & π_{0} \\ & 𝕋^{m} \end{aligned}

Suppose that f_{1}, …, f_{m} can be extended to commuting 𝕋^{p}-invariant functions in some 𝕋^{p}-invariant neighborhood V of M̂_{c}. Then, within V, M̂_{c} is given by the equations

f_{1} = c_{1}, …, f_{m} = c_{m}, ϕ_{1} = 0, …, ϕ_{p} = 0.

From the Noether theorem the functions ϕ_{α} commute with all 𝕋^{n}-invariant functions on M and the following commuting relations hold on V:

{f_{a}, f_{b}} = {f_{a}, ϕ_{α}} = {ϕ_{α}, ϕ_{β}} = 0,
a, b = 1, …, m, α, β = 1, …, p.

Now, as in the case of commutative integrability of Hamiltonian systems M̂_{c} is a Lagrangian torus with tangent space spanned by X_{f_{a}}, X_{ϕ_{α}}, i.e., the bundle (6.15) is trivial.

In general, the flow of ẋ = X_{h} over the torus M̂_{c} is not quasi-periodic: the vector field X_{h} do not commute with vector fields X_{f_{1}}, …, X_{f_{m}}, X_{ϕ_{1}}, …, X_{ϕ_{p}} (although Poisson brackets {h, f_{a}}, {h, ϕ_{α}} vanish on M̂_{c}).

Example 6.2. To clear up the difference between partial and usual integrability, let us write down the above problem for the simplest case of 𝕋^{p}-action, when M = 𝕋^{m} × ℝ^{m} × 𝕋^{p} × ℝ^{p} = {(q, p, φ, ϕ)},

ω = ∑_{i=1}^{m} dp_{i} ∧ dq_{i} + ∑_{α=1}^{p} dϕ_{α} ∧ dφ_{α},

and the 𝕋^{p}-action is given by translations in φ coordinates. Then the momentum mapping is Φ = (ϕ_{1}, …, ϕ_{p}), (M)_{0} = {(q, p, φ, 0)} and the reduced space is

(N_{0}, ω_{0}) = (𝕋^{m} × ℝ^{m}, ∑_{i=1}^{m} dp_{i} ∧ dq_{i}).

The Hamiltonian equations with Hamiltonian h(q, p, φ, ϕ) are

(6.16)  q̇_{i} = \frac{∂h}{∂p_{i}} , ṗ_{i} = −\frac{∂h}{∂q_{i}} , i = 1, …, m,
(6.17)  φ̇_{α} = \frac{∂h}{∂ϕ_{α}} , ϕ̇_{i} = −\frac{∂h}{∂φ_{α}} , α = 1, …, p.

From (6.17) it is clear that M_{0} is an invariant manifold of the flow if and only if the Hamiltonian h is 𝕋^{p} invariant on M_{0}. Let H_{0}(q, p) = h(q, p, φ, ϕ)|_{Φ=0} be the reduced Hamiltonian. The partially reduced system

q̇_{i} = \frac{∂H_{0}}{∂p_{i}} , ṗ_{i} = −\frac{∂H_{0}}{∂q_{i}} , i = 1, …, m

coincides with non-reduced equation (6.16) restricted to M_{0}.

If H_{0} does not depend on action coordinates p_{i}, the system is quasi-periodic over the tori p_{i} = c_{i}. The reconstruction equations in variables φ_{α} are given by (6.17) for Φ = 0. In general, the partial derivatives \frac{∂h}{∂ϕ_{α}} restricted to M_{0} can depend on φ_{α}. Thus, in general, the system (6.17) is not solvable and the motion over invariant Lagrangian tori p_{i} = c_{i}, ϕ_{α} = 0 is not quasi-periodic.

If h, in addition does not depend on variables φ_{α} on the whole phase space M, equations (6.16), (6.17) are solvable on M_{0}, although the system on M may be non-integrable.

6.3. Reductions of Additional Symmetries. Suppose that equations (6.8) define invariant relations of the system (1.1) with Hamiltonian h and an additional free Hamiltonian action of a connected compact Lie group K is given. Let

Ψ : M → 𝔨^{∗}

be the corresponding momentum mapping. Let ζ_{1}, …, ζ_{q} be the base of 𝔨 and let

(6.18)  ψ_{j} = (Ψ, ζ_{j}),  j = 1, …, q.

Further, suppose that the actions of G and K commute, that is we have

{ϕ_{α}, ψ_{j}} = 0, α = 1, …, p, j = 1, …, q.

The induced K-action on (N_{0}, ω_{0}) is also Hamiltonian, with the momentum mapping Ψ_{0} satisfying Ψ|_{M_{0}} = Ψ_{0} ∘ π_{0}.

Denote the projection M → M/K by σ.

If h is K-invariant, we can reduce the system to the Poisson manifold M/K as well:

(6.19)  Ḟ = {F, H}^{K},  F ∈ C^{∞}(M/K),

where H is the reduced Hamiltonian H (h = H ∘σ) and {·, ·}^{K} are reduced Poisson bracket. Since ϕ_{α} are K-invariant, we obtain reduced invariant relations

(6.20)  Φ_{α} = 0, ϕ_{α} = Φ_{α} ∘ σ, α = 1, …, p

defining the invariant manifold M_{0}/K of the reduced flow:

M ⊃ M_{0} ⟶^{π_{0}} N_{0} = M_{0}/G
σ↓  ↓
M/K ⊃ M_{0}/K

On the other side, since h is K-invariant, we have that the reduced Hamiltonian H_{0} on N_{0} is also K-invariant, and therefore the momentum mapping Ψ_{0} is conserved along the flow of the partially reduced system (6.10).

Suppose we have additional integrals F implying partial integrability with respect to G action. Then M_{0} is almost everywhere foliated on invariant isotropic manifolds M_{c}, level sets of (6.18) and integrals π_{0}^{∗}F (see Corollary 6.1).

Now, as in the proof of Theorem 5.3 we obtain invariant foliation of M_{0}/K on manifolds of the form P_{c} = (K · M_{c})/K. It is clear that P_{c} are isotropic submanifolds of the symplectic leafs in (M/K, {·, ·}^{K}).

Example 6.3. In the case of Hess–Appelrot system we have M = T^{∗}SO(3), G = SO(2) (rotations of the body around the vector r⃗) and K = SO(2) (rotations of the body around the vector γ⃗).

The system is partially integrable with respect to G-action: the reduced system is the spherical pendulum (6.7) and the reduced phase phase N_{0} = T^{∗}S^{2} is foliated on two-dimensional tori, level sets of the reduced Hamiltonian H_{0} and momentum mapping of the G-action (rotation of the sphere around vector Γ⃗). Here the sphere S^{2} is identified with the positions of the vector R⃗.

Therefore, the invariant manifold M_{0} ⊂ T^{∗}SO(3) defined by (6.6) is foliated on 3-dimensional invariant Lagrangian tori.

The reduction of K = SO(2)-symmetry leads to the Euler–Poisson equations (6.1), restricted on the invariant set F_{1} = (γ⃗, γ⃗) = 1 (note that T^{∗}SO(3))/SO(2) ≈ ℝ^{3} × S^{2}). The integral F_{3} = (m⃗, γ⃗) is the casimir function. The invariant 2-tori (6.5) can be also seen as a reduction on invariant 3-tori of M_{0}.

Note that the system (6.1), (6.3), (6.4) has no invariant measure. Namely, by the Euler–Jacobi theorem (e.g., see [2, p. 131]), the invariant measure together with the integrals (6.2), (6.4) would imply solvability of the system by quadratures.

6.4. Invariant Measure. By the Liouville theorem, the Hamiltonian flow (1.1) preserves the canonical measure Ω = ω^{dim M/2}. Let L_{X_{h}} denotes the Lie derivative with respect to the flow (1.1). Then

(6.21)  L_{X_{h}}Ω = 0.

We define the restriction of Ω to M_{0} as follows (e.g., [2, p. 28]). Let Θ be an arbitrary volume form on M_{η}. We can cover an open neighborhood of M_{η} with charts U_{i} having local coordinate systems of the form (z^{i}, ϕ) = (z_{1}^{i}, …, z_{m}^{i}, ϕ_{1}, …, ϕ_{p}), where ϕ_{α} are given by (6.8). On U_{i} we have

Ω^{i}(z^{i}, ϕ) = μ^{i}(z^{i}, ϕ) dϕ_{1} ∧ dϕ_{2} ∧ ⋯ ∧ dϕ_{n} ∧ Θ^{i}(z^{i}),

for some smooth nonvanishing function μ^{i}. Then the restriction of Ω to M_{η} is the form μ̂ Θ locally given by μ^{i}(z^{i}, ϕ) Θ^{i}(z^{i})|_{ϕ_{α}=0}.

The functions ϕ_{α} are particular integrals of the equations (1.1). Therefore the time derivative of ϕ_{α}, in an open neighborhood of M_{η}, is of the form

(6.22)  ϕ̇_{α}(x) = {ϕ_{α}, h} = ∑_{β} ψ_{αβ}(x)ϕ_{α},

where ψ_{αβ} are smooth functions. From (6.22) we get

L_{X_{h}}dϕ_{α} = ∑_{β} (ψ_{αβ}dϕ_{α} + ϕ_{α}dψ_{αβ}) .

Whence, at the points of M_{0} ∩ U_{i} we have

(6.23) L_{X_{h}}Ω^{i} = dϕ_{1} ∧ ⋯ ∧ dϕ_{p} ∧ L_{X_{h}}(μ^{i} Θ^{i}) + tr ψ μ^{i} dϕ_{1} ∧ dϕ_{2} ∧ ⋯ ∧ dϕ_{p} ∧ Θ^{i},

where tr ψ is the trace of the matrix ψ_{αβ}. The relations (6.21), (6.23) lead to the following proposition:

Proposition 6.1. (i) The flow (1.1) restricted to the invariant manifold (6.8) preserve the restriction of Ω to M_{0} if and only if tr ψ(x) = 0, for x ∈ M_{0}. In particular, if the Hamiltonian h is a G-invariant function, then μ̂ Θ is an invariant volume form.

(ii) If in addition we have the Hamiltonian action of the compact Lie group K as described above and the flow (1.1) restricted to the invariant manifold (6.8) preserve the restriction of Ω to M_{0}, then the reduced flow (6.19) restricted to the invariant manifold (6.20) also has the invariant measure.

Item (ii) follows from the following general statement (e.g., see [31]):

Lemma 6.1. Suppose a compact group K acts freely on a manifold N with local coordinates z, and there is a K-invariant dynamical system ż = Z(z) on N. If this system has an invariant measure (which is not necessarily K-invariant), then the reduced system on the quotient manifold N/K also has an invariant measure.

6.5. Partial Lagrange–Routh Reductions. Let (Q, κ, v) be a natural mechanical system. Let G be a connected Lie group acting freely and properly on Q and ρ : Q → Q/G be the canonical projection. As above we define vertical and horizontal distribution by (3.8) and (3.9), respectively (now we do not suppose that κ is G-invariant).

Since κ_{q}(H_{q}) = ann V_{q}, we see that H is invariant with respect to the “twisted” G-action

(6.24)  g ⋄ (q, X) = (g · q, κ_{g·q}^{−1} ∘ (dg^{−1})^{∗} ∘ κ_{q}(X)), X ∈ T_{q}Q,

that is the pull-back of canonical symplectic G-action on T^{∗}Q via metric κ:

T Q ⟶^{κ} T^{∗}Q
g⋄↓  ↓^{g·}
T Q ⟶_{κ} T^{∗}Q

From Theorem 6.1 we obtain (see [41])

Theorem 6.2. (i) (Partial Noether theorem) The horizontal distribution (3.9) is an invariant submanifold of the Euler–Lagrange equations (1.2) if and only if the potential v and the restriction κ_{H} of the metric κ to H are G-invariant with respect to the action (6.24).

(ii) (Partial Lagrange–Routh reduction) If H is an invariant submanifold of the system (Q, κ, v), then the trajectories q(t) with velocities q̇(t) that belong to H project to the trajectories b(t) = π(q(t)) of the natural mechanical system (Q/G, K, V) with the potential V (π(q)) = v(q) and the metric K obtained from κ_{H} via identification H/G ≈ T (Q/G).

Note that when κ is G-invariant, the twisted G-action (6.24) coincides with usual G-action: g·(q, X) = (g·q, dg(X)) and the induced metric K is the submersion metric. In this case we have theorem 3.1.

The partial reduction of the classical Hess–Appel’rot problem is the spherical pendulum on S^{2} (6.7). The partial reduction of the n-dimensional variant of the system given in [41] is a spherical pendulum on S^{n−1}. It appears that the 4dimensional variant of the Hess–Appel’rot system given by Dragović and Gajić [26] can be also considered within the framework of partial reductions, where the reduced system is a pendulum system on oriented Grassmannian variety G^{+}(4, 2) of oriented two-dimensional planes in ℝ^{4} (see [41]).

The spherical pendulum as well as the pendulum system on Grassmannian variety G^{+}(4, 2) are completely integrable (the complete integrability of the later system follows from complete integrability of the pendulum type systems on adjoint orbits of compact Lie groups established in [15]).

Therefore it is natural to define

Definition 6.3. We shall say that a natural mechanical system (Q, κ, v) satisfies geometrical Hess–Appel’rot conditions if it is partially integrable, i.e, it has an invariant relation (3.9) and the partially reduced system (Q/G, K, V) is completely integrable.

Example 6.4. Let (Q, κ_{∗}) be a compact Riemannian manifold with a Ginvariant metric. Let h^{∗} = \frac{1}{2}(p, κ_{∗}^{−1}p). The reduction of the geodesic flow X_{h^{∗}} to T^{∗}(Q/G) ≈ (T^{∗}Q)_{0}/G is the geodesic flow of the submersion metric K on Q/G. Let pr_{ann V} be the orthogonal projection to ann V_{q} in T_{q}^{∗}Q with respect to the metric κ_{∗}^{−1}, and let A be an arbitrary metric on the cotangent bundle. Then h = h^{∗} + \frac{1}{2}(pr_{ann V} p, A_{q}p) will be the Hamiltonian function of the certain metric that we shall denote by κ. Since h = h^{∗}|_{(T^{∗}Q)_{0}} we can perform the partial Lagrange–Routh reduction. Note that the metrics κ and κ_{∗} induce the same metric K on Q/G, but their horizontal distributions H and H^{∗} are different:

H_{q} = κ^{−1} ∘ κ_{∗}|_{q}(H_{q}^{∗}).

Further, let us suppose that the geodesic flow of the submersion metric is completely integrable. Then the geodesic flow of the metric κ is, in general, nonintegrable but has the following interesting property: (T^{∗}Q)_{0} is almost everywhere foliated by compact invariant Lagrangian submanifolds.

6.6. Complete Integrability on Invariant Manifolds. The notion of partial integrability introduced here differs from those based on the Poincare-Lyapo-unov-Nekhoroshev theorem [64, 33, 65] (see also [34, 50]), where a Hamiltonian system (1.1) restricted to an invariant submanifold N ⊂ M of lower dimension is completely integrable, i.e., N is filled with periodic or quasi-periodic trajectories of (1.1).

Let (M, ω) be a 2n-dimensional symplectic manifold. Let f_{1}, …, f_{k} be real, Poisson commuting functions on M, where k < n. Assume that there exists a compact connected k-dimensional manifold T^{k} invariant under all Hamiltonian flows X_{f_{i}}, and that dF has rank k on all points of T^{k}, where

F : M → ℝ^{k}, F = (f_{1}, …, f_{k}).

Then, as in the Liouville theorem [3], T^{k} is diffeomorphic to k-dimensional torus and T^{k} is a submanifold of F^{−1}(β_{0}) for some β_{0} ∈ ℝ^{k}.

Take a k-parameter family of local (2n − k)-manifolds Σ_{m}, passing through m ∈ T^{k} and transversal to both T^{k} and F^{−1}(β_{0}) at m. In a neighborhood of m ∈ T^{k} consider also the local foliation L^{m} generated by vector fields X_{f_{i}}.

Any homotopy class γ in π_{1}(T^{k}) can be realized as the orbit of some vector field obtained as a linear combination of X_{f_{i}}:

X_{c} = ∑_{i=1}^{k} c_{i}X_{f_{i}}

such that X_{c} has closed trajectories with period 1 on T^{k}.

If we consider the time-one flow of points p ∈ Σ_{m} under X_{c}, this defines a map Θ_{c}^{m} from Σ_{m} to M. By construction Θ_{c}^{m}(m) = m, while in general Θ_{c}^{m}(p) can fail to be in Σ_{m}. However, in an appropriate neighborhood of m, the leaf of the foliation L^{m} through Θ_{c}(p) intersects Σ_{m} in a unique point p′. In this way we have well defined map Ψ_{c}^{m} : Σ_{m} ⟶ Σ_{m}, Ψ_{c}^{m}(p) = p′ (for more details see [64, 33]). Following [33], we call this the Poincare–Nekhoroshev map. It is based at a point m and depends on the constants c_{1}, …, c_{k}, i.e., on the homotopy class γ.

With the above notations, we can state the following theorem:

Theorem 6.3 (Nekhoroshev [64]). Suppose that the spectrum of the linear part of the Poincare–Nekhoroshev map Ψ_{c}^{m} associated to X_{c} does not include the unity, for some m ∈ T^{k}. Then, in a neighborhood U of T^{k} in M, there is a symplectic submanifold N which is fibered over a domain B ⊂ ℝ^{k} with fibers been k-tori T_{β}^{k}N ∩ F^{−1}(β), β ∈ B invariant under the flows of X_{f_{i}}, i = 1, …, k.

The theorem extends the Poincare–Lyapounov theorem on persistence of periodic trajectories (invariant S^{1} orbits) to the case of higher dimensional tori. Also it extends the Liouville–Arnold theorem by showing that one can find partial action-angle coordinates on these invariant tori, i.e. one can define coordinates (I_{j}, φ_{j}; p_{a}, q_{a}), with j = 1, …, k,, a = 1, …, n−k, in a open neighborhood of T^{k} so that the invariant tori correspond to p = q = 0 and are parametrized by the value of the action coordinates I_{j} (see [64, 65]).

Acknowledgments. I am very grateful to Alexey V. Bolsinov (most of the survey is the result of our collaboration), Vladimir Dragović, Yuri N. Fedorov, Borislav Gajić and Milena Radnović for support and many interesting discussions on the subject. The survey is based on the talk given at the Conference MM-VII Symmetries and Mechanics, Novi Sad 2007. I would like to thank the Organizers for kind hospitality. Also, I would like to thank the referee for useful suggestions. The research is supported by the Serbian Ministry of Science, Project 144014, Geometry and Topology of Manifolds and Integrable Dynamical Systems.

References

[1] G. G. Appel’rot, The problem of motion of a rigid body about a fixed point, Uchenye Zap. Mosk. Univ. Otdel. Fiz. Mat. Nauk 11 (1894) 1–112 (in Russian).

[2] V. I. Arnol’d, V. V. Kozlov and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Itogi Nauki i Tekhniki. Sovr. Probl. Mat. Fundamental’nye Napravleniya, Vol. 3, VINITI, Moscow 1985. English transl.: Encyclopadia of Math. Sciences, Vol.3, Springer-Verlag, Berlin 1989.

[3] V. I. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, 1978.

[4] M. Audin, Spinning tops. A course on integrable systems, Cambridge Studies in Advanced Mathematics 51, Cambridge University Press, 1966.

[5] Ya. V. Bazaikin, Double quotients of Lie groups with an integrable geodesic flows, Sibirsk. Mat. Zh. 41 (2000) No.3, 513–530 (Russian); English translation: Siberian Math. J. 41 (2000) No. 3, 419–432.

[6] S. Benenti and W. M. Tulczyjew, Remarques sur les réductions symplectiques, C. R. Acad. Sci. Paris, 294 Série I (1982) 561–564.

[7] O. I. Bogoyavlenkij, Theory of tensor invariants of integrable Hamiltonian systems. II. Theorem on symmetries and its applications, Comm. Math. Phys. 184 (1997) 301–365.

[8] A. V. Bolsinov, Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Izv. Acad. Nauk SSSR, Ser. matem. 55 (1991), no.1, 68–92 (Russian); English translation: Math. USSR-Izv. 38 (1992), no.1, 69–90

[9] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology and Classification I,II, Izhevsk, Izd. dom ”Udmurtskii universitet”, 1999 (Russian).

[10] A. V. Bolsinov and I. A. Taimanov, Integrable geodesic flow with positive topological entropy, Invent. Math. 140 (2000) 639–650.

[11] A. V. Bolsinov and B. Jovanović, Integrable geodesic flows on homogeneous spaces, Matem. Sbornik 192 (2001), no. 7, 21–40 (Russian); English translation: Sb. Mat. 192 (2001), no. 7–8, 951–969 [12] A. V. Bolsinov and B. Jovanović, Non-commutative integrability, moment map and geodesic

flows, Annals of Global Analysis and Geometry 23, no. 4, 305–322 (2003), arXiv: mathph/0109031

[13] A. V. Bolsinov and B. Jovanović, Complete involutive algebras of functions on cotangent bundles of homogeneous spaces, Mathematische Zeitschrift 246 (2004), no. 1–2, 213–236

[14] A. V. Bolsinov and B. Jovanović, Magnetic Geodesic Flows on Coadjoint Orbits, J. Phys. A: Math. Gen. 39 (2006), L247–L252, arXiv: math-ph/0602016.

[15] A. V. Bolsinov and B. Jovanović, Magnetic Flows on Homogeneous Spaces, Com. Mat. Helv., 83 (2008), no. 3, 679700, arXiv: math-ph/0609005.

[16] A. V. Borisov and I. S. Mamaev, The Hess case in the dynamics of a rigid body, Prikl. Mat. Mekh. 67 (2003) no. 2, 256–265 (Russian); English transl. J. Appl. Math. Mech. 67 (2003) no. 2, 227–235.

[17] A. V. Brailov, Complete integrability of some geodesic flows and integrable systems with noncommuting integrals, Dokl. Akad. Nauk SSSR 271(2) (1983), 273–276 (Russian); English translation: Soviet Math. Dokl. 28(1) (1983), 61–65 .

[18] A. V. Brailov, Some cases of complete integrability of the Euler equations and applications, Dokl. Akad. Nauk SSSR 268 (1983), 1043–1046 (Russian); English translation: Soviet Math. Dokl. 27 (1983), 209–212.

[19] A. V. Brailov, Construction of complete integrable geodesic flows on compact symmetric spaces, Izv. Acad. Nauk SSSR, Ser. Mat. 50(2) (1986), 661–674 (Russian); English translation: Math. USSR-Izv. 50(4) (1986), 19–31.

[20] E. A. Buldaeva, On integrable geodesic flows on the adjoint orbits of orthogonal groups, Diploma work, Dept. of Mathematics and Mechanics, Moscow State University, May 2002.

[21] L. Butler, Integrable geodesic flows on n-step nilmanifolds, J. Geom. Phys. 36 (2000), 315– 323.

[22] L. Butler, The Toda Lattice and Positive-Entropy Integrable Systems, Invent. Math. 158(3) (2004), 515–549.

[23] P. Dazord and T. Delzant, La probleme general des variables actions-angles, J. Diff. Geom. 26 (1987), 223–251.

[24] J. J. Duistermaat, On global action-angle variables, Comm. Pure Appl. Math. 33 (1980), 687–706.

[25] V. Dragović and B. Gajić, An L-A pair for the Hess–Appel’rot system and a new integrable case for the Euler-Poisson equations on so(4) × so(4), Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 845–855; arXiv math-ph/9911047.

[26] V. Dragović and B. Gajić, Systems of Hess–Appel’rot type, Comm. Math. Phys. 256 (2006), 397–435; arXiv: math-ph/0602022.

[27] D. I. Efimov, The magnetic geodesic flows in a homogeneous field on the complex projective space, Siberian Math. J. 45(3) (2004), 465–474.

[28] D. I. Efimov, The magnetic geodesic flows on a homogeneous symplectic manifold, Siberian Math. J. 46(1) (2005), 83–93.

[29] J. H. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982), 469–480.

[30] F. Fasso, The Euler-Poisson top: A non-commutatively integrable system without global action angle coordinates, ZAMP 47 (1996), 953–976.

[31] Yu. N. Fedorov and B. Jovanović, Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces, J. Non. Sci. 14 (2004), 341–381; arXiv: math-ph/0307016.

[32] E. Fiorani and G. Sardanashvily, Noncommutative integrability on noncompact invariant manifolds, J. Phys. A 39(45) (2006), 14035–14022; arXiv: math.DS/0604104

[33] G. Gaeta, The Poincare-Lyapounov-Nekhoroshev theorem, Ann. Phys. 297 (2002), 157–173.

[34] G. Giachetta, L. Mangiarotti and G. Sardanashvily, Bi-Hamiltonian partially integrable systems, J. Math. Phys. 44(5) (2003), 1984–1997, arXiv: math.DS/0211463.

[35] V. V. Golubev, Lectures on integration of the equations of motion of a rigid body about a fixed point, Moskva, Gostenhizdat, 1953 (in Russian); English translation: Transl. Philadelphia, PA: Coronet Books, 1953.

[36] V. Guillemin and S. Sternberg, On collective complete integrability according to the method of Thimm, Ergod. Th. Dynam. Sys. 3 (1983), 219–230.

[37] V. Guillemin and S. Sternberg, Multiplicity-free spaces, J. Diff. Geom. 19 (1984), 31–56.

[38] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambrige University press, 1984.

[39] W. Hess, Über die Euler’schen Bewegungsgleichungen und über eine neue particuläre Lösung des Problems der Bewegung eines starren Körpers un einen festen Punkt, Math. Ann. 37(2) (1890), 178–180.

[40] B. Jovanović, On the Integrability of Geodesic Flows of Submersion Metrics, Lett. Math. Phys. 61 (2002), 29–39; arXiv: math-ph/0204048

[41] B. Jovanović, Partial Reduction of Hamiltonian Flows and Hess–Appelrot Systems on SO(n), Nonlinearity 20 (2007), 221–240; arXiv:math-ph/0611062

[42] V. V. Kozlov and N. N. Kolesnikov, On theorems of dynamics, Prikl. Mat. Mekh. 42(1) (1978), 28–33 (Russian); English transl.: J. Appl. Math. Mech. 42(1) (1979), 26–31.

[43] V. V. Kozlov, Remarks on a Lie theorem on the integrability of differential equations in closed form, Diff. Uravn. 41(4) (2005),553–555 (Russian); English translation: Differ. Equ. 41(4) (2005), 588–590.

[44] M. Kramer, Spharische untrgruppen in kompakten zusammenhangenden Liegruppen, Compositio Math. 38 (1979), 129–153.

[45] T. Levi-Civita and V. Amaldi, Lezioni di Meccanica Razionale, Volume Secondo, Parte Seconda, Bologna, 1927.

[46] P. Libermann and C. M. Marle, Symplectic geometry and analytic mechanics, Math. Appl. 35, Reidel, Boston, 1987.

[47] A. Lichnerowicz, Variété symplectique et dynamique associée à une sous-variété, C. R. Acad. Sci. Paris, Sér. A 280 (1975) 523–527.

[48] J. Liouville, Note sur lintégration des équations differentielles de la dynamique, présentée au bureau des longitudes le 29 juin 1853, J. Math Pures Appl. 20 (1855), 137–138.

[49] A. J. Maciejewski and M. Przybylska, Differential Galois obstructions for non-commutative integrability; arXiv: nlin.SI/0701061.

[50] A. J. Maciejewski, M. Przybylska and H. Yoshida, Necessery conditions for partial and superintegrability of Hamiltonian systems with homogeneous potential; arXiv: nlin.SI/0701057.

[51] J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974) 121–130.

[52] J. E. Marseden, T. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange–Routh reduction, J. Math. Phys. 41(6) (2000), 3379–3429.

[53] S. V. Manakov, Note on the integrability of the Euler equations of n-dimensional rigid body dynamics, Funkc. Anal. Pril. 10(4) (1976), 93–94 (Russian).

[54] I. V. Mikityuk, Homogeneous spaces with integrable G-invariant Hamiltonian flows, Izv. Akad. Nauk SSSR, Ser. Mat. 47(6) (1983), 1248–1262 (Russian)

[55] I. V. Mikityuk, On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Mat. Sbornik 129[171](4) (1986), 514–534 (Russian); English translation: Math. USSR Sbornik 57(2) (1987), 527–547.

[56] I. V. Mykytyuk, Actions of Borel subgroups on homogeneous spaces of reductive complex Lie groups and integrability, Composito Math. 127 (2001), 55–67.

[57] I. V. Mykytyuk and A. M. Stepin, Classification of almost spherical pairs of compact simple Lie groups, In: Poisson Geometry, Banach Center Publ., 51, Polish Acad. Sci., Warsaw, 2000, pp. 231–241.

[58] I. V. Mykytyuk and A. Panasyuk, Bi-Poisson structures and integrability of geodesic flows on homogeneous spaces, Transformation Groups 9(3) (2004), 289–308.

[59] A. S. Mishchenko, Integration of geodesic flows on symmetric spaces, Mat. Zametki 31(2) (1982), 257–262 (Russian); English translation: Math. Notes. 31(1–2) (1982), 132–134.

[60] A. S. Mishchenko and A. T. Fomenko, Euler equations on finite-dimensional Lie groups, Izv. Akad. Nauk SSSR, Ser. Mat. 42(2) (1978), 396–415 (Russian); English translation: Math. USSR-Izv. 12(2) (1978), 371–389.

[61] A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funkts. Anal. Prilozh. 12(2) (1978), 46–56 (Russian); English translation: Funct. Anal. Appl. 12 (1978), 113–121.

[62] A. S. Mishchenko and A. T. Fomenko, Integration of Hamiltonian systems with noncommutative symmetries, Tr. Semin. Vekt. Tenz. Anal. Prilozh. Geom. Mekh. Fiz. 20 (1981), 5–54 (Russian).

[63] N. N. Nekhoroshev, Action-angle variables and their generalization, Tr. Mosk. Mat. O.-va. 26 (1972), 181–198, (Russian); English translation: Trans. Mosc. Math. Soc. 26 (1972), 180–198.

[64] N. N. Nekhoroshev, The Poincare–Lyapounov–Liouville–Arnold theorem, Funct. Anal. Appl. 28 (1994), 128–129.

[65] N. N. Nekhoroshev, Types of integrability on a submanifold and generalizations of Gordon’s theorem, Tr. Mosk. Mat. Obs. 66 (2005), 184–262 (Russian); English translation: Trans. Moscow Math. Soc. (2005), 169–241.

[66] A. L. Onishchik, E. B. Vinberg and V. V. Gorbatsevich, Foundations of Lie Theory and Lie Transformation Groups, Springer-Verlag, 1993.

[67] J. P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progress in Math. 222, Birkhäuser, 2004.

[68] A. A. Oshemkov, On the topological structure of the set of singularities of integrable Hamiltonian systems, In: Topological classification of integrable systems, Adv. in Soviet Math. 6 AMS, Providence, 1991, pp. 67–146.

[69] A. Panasyuk, Projection of Jordan bi-Poisson structures that are Kronecker, diagonal actions and the classical Gaudin systems, J. Geom. Phys. 47 (2003) 379–397; Erratum: J. Geom. Phys. 49 (2004), 116–117.

[70] D. I. Panyushev, Complexity of quasiaffine homogeneous varieties, t-decompositions, and affine homogeneous spaces of complexity 1, Adv. in Soviet Math. 8 (1992), 151–166.

[71] G. P. Paternain and R. J. Spatzier, New examples of manifolds with completely integrable geodesic flows, Adv. in Math. 108 (1994), 346–366; arXiv: math.DS/9201276

[72] A. G. Reyman and M. A. Semenov-Tian-Shanski, Group theoretical methods in the theory of finite dimensional integrable systems, In: V. I. Arnold, S. P. Novikov (eds.), Dynamical Systems VII, Springer-Verlag 1994, pp. 116–225

[73] E. J. Routh, Treatise on the Dynamics of a System of Rigid Bodies, MacMillan, London 1860.

[74] S. T. Sadetov, A proof of the Mishchenko–Fomenko conjecture (1981), Dokl. Akad. Nauk 397(6) (2004), 751–754 (Russian)

[75] P. Saksida, Nahm’s Equations and Generalizations of the Neumann System, Proc. London Math. Soc. 78(3) (1999), 701–720.

[76] G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14(1) (1975), 63–68.

[77] I. A. Taimanov, The topology of Riemannian manifolds with integrable geodesic flows, Tr. Mat. Inst. Steklova 205 (1994), 150–164 (Russian); English translation: Proc. Steklov. Inst. Math. 205(4) (1995), 139–150.

[78] A. Thimm, Integrable geodesic flows on homogeneous spaces, Ergod. Th. Dynam. Sys. 1 (1981), 495–517.

[79] V. V. Trofimov and A. T. Fomenko, Algebra and Geometry of Integrable Hamiltonian Differential Equations, Moskva, Faktorial, 1995 (Russian).

[80] E. B. Vinberg, Commutative homogeneous spaces and co-isotropic symplectic actions, Uspekhi Mat. Nauk 561 (2001), 3–62 (Russian); English translation: Russian Math. Surveys 56(1) (2001), 1–60.

[81] E. B. Vinberg and O. S. Yakimova, Complete families of commuting functions for coisotropic Hamiltonian actions; arXiv: math.SG/0511498.

[82] A. Weinstein, The local tructure of Poisson manifolds, J. Diff. Geom. 18 (1983) 523–557.

[83] N. T. Zung, Torus actions and integrable systems, In: A. V. Bolsinov, A. T. Fomenko, A. A. Oshemkov (eds.), Topological Methods in the Theory of Integrable Systems, Cambridge Scientific Publ., 2006; arXive: math.DS/0407455

[84] N. E. Zhukovski, Geometrische Interpretation des Hess’schen Falles der Bewegung eines schweren starren Körpers um einen festen Punkt, Jber. Deutchen Math. Verein. 3 (1894) 62–70.

Mathematical Institute Serbian Academy of Science and Art Kneza Mihaila 36, 11000 Belgrade Serbia

bozaj@mi.sanu.ac.rs