Hierarchy of integrable geodesic flows

Peter Topalov

November 5, 1998

Abstract

A family of integrable geodesic flows is obtained. Any such a family corresponds to a pair of geodesically equivalent metrics.

Contents

1 Introduction 1

2 Geodesically equivalent metrics and the corresponding 1-parameter family 4

3 Hierarchy of integrable flows 7

4 S-hierarchy for the ellipsoid 12

5 A family of integrable Hamiltonians in e(3)^{∗} 17

1 Introduction

In papers [1, 2] a simple approach was suggested for obtaining first integrals of Hamiltonian systems if a trajectorial diffeomorphism is given. This approach is closely related to the ideas presented in [3].

In paper [2] the construction was applied to a classical example where such a diffeomorphism exists – the geodesic flows corresponding to a pair of geodesically equivalent metrics.

Let g and ḡ be Riemannian metrics on the manifold M^{n}.

Definition 1 . The pseudo-Riemannian metrics g and ḡ are called geodesically equivalent iff they have the same geodesics (considered as unparameterized curves).

Denote by M^{q}(g, ḡ) (1 ≤ q ≤ n) the set of points y ∈ M^{n} such that the Riemannian metrics g and ḡ have exactly q distinct eigenvalues in an open neighborhood of the point y. If the set M^{n} is everywhere dense in M^{n} we say that the Riemannian metrics g and ḡ are strictly non-proportional on M^{n}.

In the paper [2] the following was proved.

Theorem 1 . Suppose the Riemannian metrics g and ḡ are geodesically equivalent; then

a) the geodesic flow of the metric g admits n integrals (k ≥ 0)

I_{k} =^{def} (−1)^{k} (\frac{det(g)}{det(ḡ)})^{\frac{k+2}{n+1}} ḡ(S_{k}v, v),  (1)

where

S_{k} =^{def} Ḡ^{k} − σ_{1}(Ḡ)Ḡ^{k−1} + … + (−1)^{k}σ_{k}(Ḡ)  (2)

and Ḡ =^{def} (g^{ij}ḡ_{jk}), σ_{i}(Ḡ) are the elementary symmetric polynomials of degree i;

b) the integrals I_{k} (k ≥ 0) are in involution^{1};

c) if M^{q} ≠ ∅, then the rank of the differentials dI_{0}, …, dI_{n−1} is equal to q almost everywhere in T M^{q}. Moreover, on T M^{q} we have

rk(dI_{0}, …, dI_{n−1}) = rk(dI_{0}, …, dI_{q−1}) ≤ q.  (3)

This theorem is closely related to some results proved by U.Dini, P.Painlevé, T.Levi-Civita and R.Liouville (see [4, 5, 6]).

In the present paper we assign to any pair of geodesically equivalent metrics a hierarchy of integrable geodesic flows (Theorem 5). The paper is organized as follows.

In Section 2 we present some important facts from the theory of geodesic mappings needed for the sequel.

1i.e., making the Legendre transformation we obtain n functions commuting with respect to the canonical symplectic structure on T^{∗}M

In Section 3 we assign to any pair of strictly non-proportional geodesically equivalent metrics g and ḡ a family of completely integrable Riemannian metrics S(g, ḡ) (see Theorem 5). Some simple properties of these families are obtained.

In Section 4 the results obtained in Section 3 are applied to the metrics

dg^{2} =^{def} ∑_{i=1}^{n} dx^{i}^{2}  (4)

and

dḡ^{2} =^{def} \frac{1}{∑{n \atop i=1} (\frac{x^{i}}{a^{i}} )2} ∑_{i=1}^{n} \frac{dx^{i}^{2}}{a^{i}}  (5)

which are geodesically equivalent on the standard ellipsoid E_{h} =^{def} \{∑_{i=1}^{n} \frac{x^{i}^{2}}{a^{i}} = h\}, h > 0. We give explicit formulae for the family S(dg^{2}|_{E_{h}}, dḡ^{2}|_{E_{h}}) in terms of some metrics on the whole R^{n} (Theorem 6). It is interesting that this family contains the metric on the Poisson sphere, well-known in classical mechanics. It immediately permits us to prove that the geodesic flow on the standard ellipsoid and the geodesic flow on the Poisson sphere have the same Liouville foliation (Corollary 3). This theorem is a multidimensional generalization of the well-known one in the two-dimensional case. In addition, we give an explicit formula for a metric geodesically equivalent to the metric on the Poisson sphere (Corollary 2).

In Section 5 we obtain a family of completely integrable Hamiltonians with respect to the standard Lie-Poisson bracket on e(3)^{∗} (the dual space to the Lie algebra e(3)). In particular, we obtain an algebraic Hamiltonian H^{(1)} such that the corresponding Hamiltonian system sgrad H^{(1)} is orbitally equivalent to the Euler case of the free motion of the rigid body (see Corollary 6).

Throughout the paper the standard agreement holds, i.e. Riemannian metric means a positive-definite symmetric form and pseudo-Riemannian metric means a non-degenerate symmetric form. We consider mainly Riemannian metrics; nevertheless many results hold in the pseudo-Riemannian case.

As we said above the theory of geodesically equivalent metrics (or more generally – the theory of projective mappings) is a classical object. It is not possible to mention here all schools and mathematicians who have worked in this area. We are able to mention only a few of them – U.Dini, T.Levi-Civita, P.Painlevé, H.Weyl, E.Cartan, P.A.Shirokov, S.Kobayashi, N.S.Sinyukov, A.Petrov, P.Venzi, J.Mikeš, A.Aminova and many others. More information may be found in the paper [7]. We would like to mention here that we present a ’symplectic’ point of view on this field.

The author is grateful to A.V.Bolsinov, A.T.Fomenko, V.V.Kozlov, V.S.Matveev, S.Tabachnikov and I.A.Taimanov for useful discussions. This paper was written during my stay at the Max-Planck-Institut für Mathematik in Bonn. It is pleasure to thank the Institut for its hospitality and financial support.

2 Geodesically equivalent metrics and the corresponding 1-parameter family

The following lemma is needed for the sequel.

Lemma 1 (see [8]) .

1) Suppose the pseudo-Riemannian metrics g and ḡ are geodesically equivalent; then the tensors a_{ij} and λ_{i}

a_{ij} =^{def} A_{i}^{α}g_{αj},  (6)
λ_{i} =^{def} −A_{i}^{α}ψ_{α},  (7)

where (n + 1)ψ_{i} = \frac{1}{2}∂_{i} ln |\frac{ḡ}{g}|, g = det(g_{ij}), and the operator A is given by formula

A_{j}^{i}(g, ḡ) =^{def} |\frac{ḡ}{g}|^{\frac{1}{n+1}} ḡ^{iα}g_{αj},  (8)

satisfying the equation

a_{ij,k} = λ_{i}g_{jk} + λ_{j}g_{ik}.  (9)

Here a_{ij,k} denotes the covariant derivative ∇_{k}a_{ij}, where ∇ is the Levi-Civita connection coressponding to the metric g.

2) Conversely, if a non-degenerate symmetric tensor field a_{ij} and an 1-form λ_{i} satisfy equation (9), then the metric

ḡ_{ij} =^{def} (\frac{ĝ}{g}) ĝ_{ij},  (10)

where ĝ_{ij} =^{def} g_{iα}a^{αβ}g_{βj}, is geodesically equivalent to g.

It is obvious that A is self-adjoint with respect the both metrics. Now, using Lemma 1 we can prove

Proposition 1 . Suppose the pseudo-Riemannian metrics g and ḡ are geodesically equivalent; then if for some parameters α and β the operator (λA + β) is invertible on M^{n}, then the pseudo-Riemannian metrics g and

ḡ_{α,β}(X, Y) =^{def} \frac{1}{det(αA + β)}g((αA + β)^{−1}X, Y),  (11)

are geodesically equivalent.

Remark 1 . Remark here that ḡ_{0,1} = g and ḡ_{1,0} = ḡ.

Remark 2 . If the manifold M^{n} is compact, then formula (11) gives a 1-parameter family of geodesically equivalent metrics. Moreover, in a small neighborhood of each point of M^{n} we also obtain 1-parameter family.

Remark 3 . Let us take two representatives ḡ_{α,β} and ḡ_{λ,μ}. Now, we are able to apply Proposition 1 again. The corresponding family of geodesically equivalent metrics is

ḡ_{a,b} = \frac{1}{det((aλ + bα)A + (aμ + bβ))}g((aλ+bα)A+(aμ+bβ))^{−1}. (12)

Therefore, if (α : β) ≠ (λ : μ), then we obtain the same family. Remark here also that

A(ḡ_{α,β}, ḡ_{λ,μ}) = \frac{λA + μ}{αA + β} .  (13)

Proof of Proposition 1. Let g and ḡ be geodesically equivalent metrics. Using (6), we obtain that the symmetric form a = gA satisfies equation (9) for some λ. Hence, the form a_{c} =^{def} gA + cg (c = const) also satisfied this equation. We use here that ∇ is the Levi-Civita connection corresponding to the metric g. Assume that a_{c} is nondegenerate on M^{n}. We get ĝ =^{def} g(gA + cg)^{−1}g = g(A + c)^{−1} and using the converse formula (10) obtain that the pseudo-Riemannian metric

ḡ_{c} = \frac{1}{det(A + c)}g(A + c)^{−1}  (14)

is geodesically equivalent to g, q.e.d.

Now, using Proposition 1 we are able to present another ’nonsymplectic’ proof of the first item of Theorem 1.

Corollary 1 . If the Riemannian metrics g and ḡ are geodesically equivalent, then the geodesic flow of the metric g admits a 1-parameter family of first integrals

I_{α,β}(g, ḡ) =^{def} det(αA + β)g(αA + β)^{−1}.  (15)

Proof of Corollary 1. The corollary easily follows from the next theorem (see [6]).

Theorem 2 (Painlevé) . If the metrics g and ḡ are geodesically equivalent, then the function

I_{0}(g) =^{def} (\frac{det(g)}{det(ḡ)})^{\frac{2}{n+1}} ḡ  (16)

is an integral of the the geodesic flow of the metric g.

In a small neighborhood of each point of M^{n} we can find a 1-parameter family ḡ_{1,β} of geodesically equivalent Riemannian metrics. Therefore, locally the Riemannian metrics g and ḡ_{1,β} are geodesically equivalent, where β ∈ (a, b), a < b. It permits us to apply Theorem 2. We have

I_{1,β}(g) =^{def} (\frac{det(g)}{det(ḡ_{1,β})})^{\frac{2}{n+1}} ḡ_{1,β}  (17)
= det(A + β)g(A + β)^{−1}  (18)
= I_{0} + I_{1}β + … + I_{n−1}β^{n−1},  (19)

where the functions I_{k} are the same as in Theorem 1. The last equality in this chain can easily be proved. Therefore, the functions I_{k} are integrals in a small neighborhood of each point of M^{n}. But they are globally defined on M^{n}. Thus, they are integrals, q.e.d.

Remark 4 . If we apply Corollary 1 to the geodesically equivalent metrics g and ḡ_{α,β} ((α : β) ≠ (0 : 1)), then we’ll obtain the same family of integrals.

Remark 5 . Let us consider the map Φ : ξ ↦ \frac{||ξ||_{g}}{||ξ||ḡ} ξ ^{2},

Φ : (T M)_{0} → (T M)_{0}.  (20)

It is obvious that Φ maps each geodesic trajectory of the metric g into a geodesic trajectory of the metric ḡ. Therefore, the pull-back Φ^{∗}(I_{α,β}(ḡ, g)) gives a family of integrals for the geodesic flow of the metric g. It can easily be checked that

Φ^{∗}(I_{α,β}(ḡ, g)) = \frac{g(ξ, ξ)}{I_{1,0}(g, ḡ)}I_{β,α}(g, ḡ).  (21)

Thus, we won’t be able to obtain new family of integrals.

3 Hierarchy of integrable flows

As we have seen if the Riemannian metrics g and ḡ are geodesically equivalent, then they are contained in a family of geodesically equivalent metrics

ḡ_{α,β} = \frac{1}{det(αA + β)}g(αA + β)^{−1},  (22)

where α and β are parameters.

Remark 6 . In the following sections we consider only Riemannian metrics although the most of the constructions pass in pseudo-Riemannian case.

It is interesting that using geodesically equivalent Riemannian metrics g and ḡ we can produce another families of geodesically equivalent metrics. We need

Theorem 3 (see [8]) . If the Riemannian metrics g and ḡ are geodesically equivalent, then for each integer k the Riemannian metrics g^{(k)} =^{def} gA^{k} and ḡ^{(k)} =^{def} ḡA^{k} are also geodesically equivalent.

2Sometimes we will denote this mapping by Φ(g, ḡ).

This theorem may be proved by direct calculations. For details see paper [8]. We will slightly generalize this result later using some other arguments. At first we need some notations.

Let B be a self-adjoint operator on the connected Riemannian manifold (M^{n}, g). By definition, put

r(B) =^{def} inf_{x∈M} min{spec B(x)}  (23)

and

R(B) =^{def} sup_{x∈M} max{spec B(x)}.  (24)

Denote by I the set obtained by adding to the interval (r, R) its endpoints iff they are achieved for some x ∈ M^{n}. Let us consider the set of all real Laurent series L_{a}(x) = ∑_{k} c_{k}(x − a)^{k} which are convergent on some open neighborhood of I. Denote by ω^{+}(B) the cone of all finite linear combinations of such series which give positive functions on I. Of course, we are able to consider more general set of functions but it will only complicate our construction.

Theorem 4 . Suppose the Riemannian metrics g and ḡ are geodesically equivalent and F (x) ∈ ω^{+}(A); then the metrics g^{F} =^{def} gF (A) and ḡ^{F} =^{def} ḡF (A) are also geodesically equivalent. These metrics are contained in a family of geodesically equivalent Riemannian metrics

ḡ_{α,β}^{F} =^{def} \frac{1}{det(αA + β)}gF (A)(αA + β)^{−1},  (25)

where α and β are parameters such that the operator (αA + β) is non-degenerate.

Proof of Theorem 4. Let g and ḡ be geodesically equivalent Riemannian metrics on a manifold M^{n}. Denote by ρ^{1}, …, ρ^{m} (1 ≤ m ≤ n) the common eigenvalues of the metrics g and ḡ. Suppose the functions ρ^{1}, …, ρ^{m} are different at every point of an open domain D ⊂ M^{n}. In the paper [6], T.Levi-Civita proved that for every point P ∈ D there is an open neighborhood U(P) ⊂ D and a coordinate system x̄ = (x̄_{1}, …, x̄_{m}) (in U(P)), where x̄_{i} = (x_{i}^{1}, …, x_{i}^{k_{i}}), (1 ≤ i ≤ m), such that the quadratic forms of the metrics g and ḡ have the following form:

g( ˙x̄, ˙ x̄) = ∑_{i=1}^{m} Π_{i}(x̄)A_{i}(x̄_{i}, ˙ x̄_{i}),  (26)

ḡ( ˙x̄, ˙ x̄) = ∑_{i=1}^{m} ρ^{i}Π_{i}(x̄)A_{i}(x̄_{i}, ˙ x̄_{i}),  (27)

where A_{i}(x̄_{i}, ˙ x̄_{i}) are positive-definite quadratic forms in the velocities ˙x̄_{i} with coefficients depending on x̄_{i},

Π_{i} =^{def} (ϕ_{i} − ϕ_{1}) … (ϕ_{i} − ϕ_{i−1})(ϕ_{i+1} − ϕ_{i}) … (ϕ_{m} − ϕ_{i}),  (28)
ρ^{i} = \frac{1}{ϕ_{1}...ϕ_{m}} \frac{1}{ϕ_{i}}  (29)

and ϕ_{1}, ϕ_{2}, …, ϕ_{m} (0 < ϕ_{1} < ϕ_{2} < … < ϕ_{m}) are smooth functions such that

ϕ_{i} = \{ \begin{matrix}ϕ_{i}(x̄_{i}), if k_{i} = 1 \\ constant, else. \end{matrix}

Definition 2 . Let metrics g and ḡ be given by formulae (26) and (27) in a coordinate chart U. Then we say that the metrics g and ḡ have Levi-Civita local form of (type m), and the coordinate chart U is Levi-Civita coordinate chart (with respect to the metrics).

In the paper [6], Levi-Civita proved that the metrics g and ḡ given by formulae (26) and (27) are geodesically equivalent.

Denote by M the set of all point in M^{n} which are contained in some Levi-Civita chart, i.e. x ∈ M iff there is a Levi-Civita chart U which contains x. By definition, M is an open subset of M^{n}. In [2] was proved that M is everywhere dense in M^{n}.

In every Levi-Civita coordinate chart we have

(\frac{ḡ}{g})^{\frac{1}{n+1}} = [\frac{1}{ϕ_{1}^{k_{1}−1}...ϕ_{m}^{k_{m}−1}} ] \frac{1}{ϕ_{1}...ϕ_{m}} = const.\frac{1}{ϕ_{1}...ϕ_{m}} ,  (30)

(ḡ^{ik}g_{kj}) = diag(\underbrace{\frac{1}{ρ^{1}} , …, \frac{1}{ρ^{1}}}_{k_{1}}; …; \underbrace{\frac{1}{ρ^{m}} , …, \frac{1}{ρ^{m}}}_{k_{m}}).  (31)

Therefore,

A(g, ḡ) = const. diag(\underbrace{ϕ^{1}, …, ϕ^{1}}_{k_{1}}; …; \underbrace{ϕ^{m}, …, ϕ^{m}}_{k_{m}}).  (32)

Hence,

g(F (A) ˙x̄, ˙ x̄) = ∑_{i=1}^{m} Π_{i}(x̄)A′_{i}(x̄_{i}, ˙ x̄_{i}),  (33)

and

ḡ(F (A) ˙x̄, ˙ x̄) = ∑_{i=1}^{m} ρ^{i}Π_{i}(x̄)A′_{i}(x̄_{i}, ˙ x̄_{i}),  (34)

where A′_{i} = f_{i}(x̄_{i})A_{i} and f_{i}(x̄_{i}) is a smooth function depending of the variable x̄_{i}. Therefore, in any Levi-Civita chart the metrics gF (A) and ḡF (A) are geodesically equivalent. Let us consider the map Φ : ξ ↦ \frac{||ξ||_{g}}{||ξ||ḡ} ξ,

Φ : (T M)_{0} → (T M)_{0}.  (35)

Denote by v and v̄ the vector fields corresponding to the geodesic flows of the metrics gF (A) and ḡF (A). Therefore, the smooth vector fields v̄ ≠ 0 and Φ_{∗}(v) ≠ 0 are proportional on a dense subset of the iso-energetic surface U^{r}M, (r > 0). Thus, their directions coincide everywhere. It yields that the metrics gF (A) and ḡF (A) are geodesically equivalent.

Finally, using Proposition 1, we obtain a family of geodesically equivalent metrics

ḡ_{α,β}^{F} =^{def} \frac{1}{det(αA(g^{F} , ḡ^{F}) + β)}g^{F} (αA(g^{F} , ḡ^{F}) + β)^{−1} (36)
= \frac{1}{det(αA + β)}g^{F} (αA + β)^{−1}.  (37)

This completes the proof of Theorem 4.

If the Riemannian metrics g and ḡ are strictly non-proportional geodesically equivalent on M^{n}, then the metrics g^{F} and ḡ^{F} are also strictly non-proportional geodesically equivalent. Therefore, using Theorem 1, we obtain that all Riemannian metrics ḡ_{α,β}^{F} are completely integrable.

Theorem 5 . If g and ḡ are strictly non-proportional geodesically equivalent Riemannian metrics, then the family of Riemannian metrics

S(g, ḡ) =^{def} {ḡ_{α,β}^{F}|F (x) ∈ ω^{+}(A), α, β are parameters}  (38)

consists of completely integrable Riemannian metrics.

Remark 7 . An explicit formula for the integrals of the metric g^{F} is

I_{α,β}(g^{F} , ḡ^{F}) = det(αA + β)gF (A)(αA + β)^{−1}  (39)
= (\frac{g}{ḡ})^{\frac{2}{n+1}} I_{β,α}(ḡ^{F} , g^{F}).  (40)

Definition 3 . The family S(g, ḡ) is called S-hierarchy of integrable Riemannian metrics corresponding to the pair g and ḡ.

Remark 8 . The family S(g, ḡ) in some sense is ’complete’. It means that if we take two representatives ḡ_{α,β}^{F} and ḡ_{λ,μ}^{F} ((α : β) ≠ (λ : μ)), then S(ḡ_{α,β}^{F}, ḡ_{λ,μ}^{F}) = S(g, ḡ).

Now, we are going to establish some relations between the flows of the metrics of a given S-hierarchy.

Proposition 2 . The metrics from a given S-hierarchy have identical ’Liouville foliations’. It means that for every two metrics g_{1} and g_{2} we can find

(i) n independent almost everywhere on (T M)_{0} pairwise commuting integrals {I_{k}^{(1)}}_{k=0}^{n−1} for the geodesic flow of the metric g_{1};

(ii) n independent almost everywhere on (T M)_{0} pairwise commuting

integrals {I_{k}^{(2)}}_{k=0}^{n−1} for the geodesic flow of the metric g_{2}; (iii) a diffeomorphism ϕ : (T M)_{0} → (T M)_{0},

such that ϕ^{∗}I_{k}^{(2)} = I_{k}^{(1)}.

Proof of Proposition 2. Suppose g_{1}, g_{2} ∈ S(g, ḡ). Without loss of generality it can be assumed that g_{1} = g and g_{2} = ḡ_{α,β}^{G}. Let us consider the commutative diagram

\begin{aligned}(T M)_{0} & ←^{Φ} & (T M)_{0} \\ & ↖ & ↑ Ψ \\ & & (T M)_{0} \end{aligned} ,

where Φ =^{def} Φ(g^{G}, ḡ_{α,β}^{G}) and Ψ : ξ → (\sqrt{G(A)})^{−1}ξ. Let ϕ =^{def} Φ ∘ Ψ. Now, the proposition easily follows from Remark 5 and Remark 7, q.e.d.

Let g and ḡ be strictly non-proportional geodesically equivalent Riemannian metrics on the manifold M^{n}. Consider the diagram

\begin{matrix}↓ & & ↓ \\ g^{(k)} & ⟷^{g.e.} & ḡ^{(k)} \\ ↓ & & ↓ \\ g^{(k+1)} & ⟷^{g.e.} & ḡ^{(k+1)} \\ ↓ & & ↓ \\ g^{(k+2)} & ⟷^{g.e.} & ḡ^{(k+2)} \\ ↓ & & ↓ \end{matrix} ,

where the horizontal arrows mean that the corresponding Riemannian metrics are geodesically equivalent. Recall that g^{(k+1)} = g^{(k)}A, where A =^{def} A(g, ḡ). We proved that all metrics of the above sequence are completely integrable.

Proposition 3 . Let us consider the Riemannian metrics g^{(k)} and ḡ^{(k+2)}. Making the corresponding Legendre transformations we obtain two Hamiltonian systems on the cotangent bundle T^{∗}M which are completely integrable and the corresponding integrals, cannonicaly given by Theorem 1, coincide.

Proof of Proposition 3. Making the Legendre transformations we obtain two families of pairwise commuting functions on T^{∗}M

a) I_{α,β}(g^{(k)}, ḡ^{(k)}) = det(αA + β)(αA + β)^{−1}(gA^{k})^{−1};

b) I_{α,β}(ḡ^{(k)}, g^{(k)}) = det(αA^{−1} + β)(αA^{−1} + β)^{−1}(ḡA^{k})^{−1}.

We obviously have

I_{α,β}(ḡ^{(k)}, g^{(k)}) = \frac{|α + βA|}{|A|} [(αA^{−1} + β)^{−1}A^{−1}] A^{−k+2} [A^{−1}ḡ^{−1}]
= [\frac{1}{|A|} (\frac{g}{ḡ})^{\frac{1}{n+1}} ] (det(α + βA)(α + βA)^{−1} A^{−k+2}g^{−1}
= I_{β,α}(g^{(k−2)}, ḡ^{(k−2)}).

This completes the proof.

4 S-hierarchy for the ellipsoid

In [2] was proved that the restrictions of the metrics

dg^{2} =^{def} ∑_{i=1}^{n} dx^{i}^{2}  (41)

and

dḡ^{2} =^{def} \frac{1}{∑{n \atop i=1} (\frac{x^{i}}{a^{i}} )2} ∑_{i=1}^{n} \frac{dx^{i}^{2}}{a^{i}}  (42)

to the ellipsoid E_{h} =^{def} \{∑_{i=1}^{n} \frac{x^{i}^{2}}{a^{i}} = h\}, h > 0 are geodesically equivalent^{3}. Moreover, these metrics are strictly non-proportional on the ellipsoid. Without loss of generality we can assume that a^{1} < a^{2} < ... < a^{n}.

3the same result was independently obtained by S.Tabachnikov in [9]

Our aim in the present section is to find explicit formulae for Shierarchy S(g, ḡ). Moreover, we intend to extend these metrics on the whole R^{n} in a natural manner. By definition, put A =^{def} diag(a^{1}, ..., a^{n}). Denote by ⟨., .⟩ the natural pairing in R^{n}.

Let us consider two smooth operators on R^{n}\{0}

A(x) =^{def} A − \frac{1}{⟨A^{−1}x, x⟩}x ⊗ x + (A^{−1}x) ⊗ (A^{−1}x)  (43)

and

A′(x) =^{def} A^{−1} − \frac{(A^{−1}x) ⊗ (A^{−2}x) + (A^{−2}x) ⊗ (A^{−1}x)}{⟨A^{−1}x, A^{−1}x⟩}  (44)

where x = (x^{1}, ..., x^{n}).

Proposition 4 . The operators A and A′ satisfy the following conditions:

1) both of them are self-adjoint with respect to the metric g;

2) for each x ∈ R^{n} the tangent space T_{x}E_{h} is invariant subspace for these operators;

3) A(x)(A^{−1}x) = ⟨A^{−1}x, A^{−!}x⟩ A^{−1}x and

A′(x)(A^{−1}x) = −\frac{⟨A^{−2}x, A^{−1}x⟩}{⟨A^{−1}x, A^{−1}x⟩}A^{−1}x;

4) The restriction of the operator A(x) to each ellipsoid E_{h}, h > 0 coincides (up to multiplication on a positive constant) with the operator A(g|_{E_{h}}, ḡ|_{E_{h}}) and the restriction of A′ coincides (up to multiplication on a positive constant) with [A(g|_{E_{h}}, ḡ|_{E_{h}})]^{−1}.

These conditions determine the operators A and A′ uniquely.

Proof of Proposition 4. Items 1) and 3) are obvious.

Let us prove 2). Suppose that ξ ∈ T_{x}E_{h}, i.e. ⟨A^{−1}x, ξ⟩ = 0. We have

⟨A^{−1}x, A(x)(ξ)⟩ = ⟨A^{−1}x, Aξ − \frac{⟨x, ξ⟩}{⟨A^{−1}x, x⟩}x⟩ =
= ⟨x, ξ⟩ − ⟨x, ξ⟩ = 0

and

⟨A^{−1}x, A′(x)(ξ)⟩ = ⟨A^{−1}x, A^{−1}ξ − \frac{⟨A^{−2}x, ξ⟩}{⟨A^{−1}x, A^{−1}x⟩}A^{−1}x⟩ =
= ⟨A^{−1}x, A^{−1}ξ⟩ − ⟨A^{−2}x, ξ⟩ = 0.

Before proving 4) we need the next simple lemma.

Lemma 2 . Let V be an oriented real vector space and T ⊂ V be an oriented hyperplane. If g and ḡ are positive defined metrics, then

\frac{vol(ḡ|_{T})}{vol(g|_{T})} = \frac{vol(ḡ)}{vol(g)}g(n_{g}, nḡ),  (45)

where n_{g} and nḡ are positive directed unit normals to the subspace T calculated for the metrics g and ḡ, respectively.

The proof of Lemma 2 is trivial.

Now, using the lemma, we obtain

[\frac{det(ḡ|_{T_{x}})}{det(g|_{T_{x}})}]^{\frac{1}{(n−1)+1}} = \begin{matrix}⟨A^{−1}x, x⟩ & \frac{1}{n} \\ (Π_{i=1}^{n}a^{i}) & \frac{1}{n} \end{matrix} \frac{1}{⟨A^{−1}x, A^{−1}x⟩}.  (46)

If ξ, η ∈ T_{x}E_{h}, then we get

ḡ_{x}(A(x)(ξ), η) = \frac{1}{⟨A^{−1}x, A^{−1}x⟩} ⟨ξ − \frac{⟨x, ξ⟩}{⟨A^{−1}x, x⟩}A^{−1}x, η⟩ =
= \frac{1}{⟨A^{−1}x, A^{−1}x⟩} ⟨ξ, η⟩ = \frac{g_{x}(ξ, η)}{⟨A^{−1}x, A^{−1}x⟩}.

Therefore, we have proved that

A(x)|_{T_{x}E_{h}} = (\frac{Π_{i=1}^{n}a^{i}}{h} )^{\frac{1}{n}} A(g|_{E_{h}}, ḡ|_{E_{h}}).  (47)

Similarly, we have

g(A′(x)(ξ), η) = ⟨A^{−1}ξ − \frac{⟨A^{−2}x, ξ⟩}{⟨A^{−1}x, A^{−1}x⟩}A^{−1}x, η⟩ =
= ⟨A^{−1}ξ, η⟩ = ⟨A^{−1}x, A^{−1}x⟩ ḡ(ξ, η).

Hence,

A′(x)|_{T_{x}E_{h}} = (\frac{h}{Π_{i=1}^{n}a^{i}} )^{\frac{1}{n}} [A(g|_{E_{h}}, ḡ|_{E_{h}})]^{−1} .  (48)

Finally, it is obvious that if a self-adjoint operator is given on an invariant hyperplane, then the normal vector is an eigenvector and the corresponding eigenvalue determines the operator uniquely on the whole space, q.e.d.

Theorem 6 . Consider the family of Riemannian metrics g^{(k)} =^{def}

⟨A^{k}(x)(ξ), η⟩ and ḡ^{(k)} =^{def} \frac{1}{⟨A^{−1}x,A^{−1}x⟩} ⟨A^{k−1}(x)(ξ), η⟩ (k ∈ Z) on

R^{n}\{0}.

1) The restriction of g^{(k)} to any ellipsoid E_{h} gives a completely integrable Riemannian metric. The corresponding integrals are given by formula

I_{α,β}^{(k)} =^{def} \frac{det(αA(x) + β)}{α ⟨A^{−1}x, A^{−1}x⟩ + β} A^{k}(x)(αA(x) + β)^{−1},  (49)

where α and β are parameters.

2) The restriction of ḡ^{(k)} to any ellipsoid E_{h} gives a completely integrable Riemannian metric. The corresponding integrals are given by formula

Ī_{α,β}^{(k)} =^{def} \frac{I_{β,α}^{(k)}}{⟨A^{−1}x, A^{−1}x⟩^{2}} .  (50)

3) The metrics g^{(k)} and ḡ^{(k)} are contained in a 1-parameter family of metrics

ḡ_{α,β}^{(k)} =^{def} \frac{α ⟨A^{−1}x, A^{−1}x⟩ + β}{det(αA(x) + β)} A^{(k)}(x)(αA(x) + β)^{−1}. (51)

and the restrictions of any two metrics from this family to the ellipsoid E_{h} are geodesically equivalent.

The proof easily follows from Proposition 4, Theorem 4 and Theorem 5.

The Ellipsoid and the Poisson sphere. In the case k = 1 we obtain that the metrics

dg_{(1)}^{2} = ⟨A(x)dx, dx⟩  (52)
= ⟨Adx, dx⟩ − \frac{⟨x, dx⟩^{2}}{⟨A^{−1}x, x⟩} + ⟨A^{−1}x, dx⟩^{2}  (53)

and

dḡ_{(1)}^{2} = \frac{1}{⟨A^{−1}x, A^{−1}x⟩} ⟨dx, dx⟩  (54)

are geodesically equivalent on the ellipsoid E_{h}, h > 0.

The metric ḡ^{(1)}|E_{h} can be considered as a metric on the homogeneous space S^{n} = SO(n + 1)/SO(n), where the kinetic energy of the (n + 1)-dimensional rigid body is spread left-invariantly on the whole SO(n + 1). This construction generalize the well-known metric on the Poisson sphere considered in the classical mechanics (see [10]).

Corollary 2 . The restriction of the metric dg_{(1)}^{2} to the ellipsoid E_{h} has the same geodesic lines as the metric on the Poisson sphere.

Using Proposition 2, we obtain

Corollary 3 . The geodesic flow on the ellipsoid and the metric on the Poisson sphere have the same Liouville foliation.

If n = 2 the last result is well-known. Moreover, in [11] A.T.Fomenko and A.V.Bolsinov using the theory of orbital equivalence of the integrable Hamiltonian systems proved that the geodesic flow on 2dimensional sphere and the geodesic flow on the Poisson sphere are continuously orbitally equivalent. It is interesting to solve the same problem in multidimensional case.

The standard sphere and the Poisson sphere. In the case k = −1 we have

dg_{(−1)}^{2} = ⟨A′(x)dx, dx⟩ =  (55)
= ⟨A^{−1}dx, dx⟩ − 2\frac{⟨A^{−1}x, dx⟩ ⟨A^{−2}x, dx⟩}{⟨A^{−1}x, A^{−1}x⟩}.  (56)

The last term on the right side vanish on T E_{h}. Therefore, we can think that dg_{(−1)}^{2} = ⟨A^{−1}dx, dx⟩. Changing the variables x = \sqrt{A}y we see that g_{(−1)} is the standard metric on the sphere. Further,

dḡ_{(−1)}^{2} = \frac{\{⟨A^{−1}dx, A^{−1}dx⟩ − ^{\frac{⟨A^{−1}x,A^{−1}dx⟩^{2}}{⟨A^{−1}x,A^{−1}x⟩}} \}}{⟨A^{−1}x, A^{−1}x⟩}.  (57)

Corollary 4 . The restriction of the metric

dḡ_{(−1)}^{2} = \frac{⟨A^{−1}dy, dy⟩ − ^{\frac{⟨A^{−1}y,dy⟩^{2}}{⟨A^{−1}y,y⟩}}}{⟨A^{−1}y, y⟩}  (58)

to the sphere S^{n} = {⟨y, y⟩ = h > 0} has the same geodesic lines as the standard metric on the sphere.

Using Proposition 3, we obtain

Corollary 5 . The metric on the Poisson sphere dḡ_{(1)}^{2}|_{E_{h}} and the standard metric on the sphere dg_{(−1)}^{2}|_{E_{h}} are completely integrable on T^{∗}M such that the corresponding integrals are the same.

5 A family of integrable Hamiltonians in e(3)^{∗}

It is well-known that the Euler-Poisson equations which describes the motion of the rigid body can be written as a Euler equations on the dual space of the Lie algebra e(3)(see [12, 13]). More precisely, if the coordinates in e(3)^{∗} are denoted by (r_{1}, r_{2}, r_{3}, s_{1}, s_{2}, s_{3}), then the Lie-Poisson bracket can be given by relations {s_{i}, s_{j}} = ϵ_{ijk}s_{k}, {r_{i}, r_{j}} = 0, {s_{i}, r_{j}} = ϵ_{ijk}r_{k}. Therefore, if H = H(r, s) is a Hamiltonian function, then the Euler equations take the form of Kirchoff equations

ṡ = s × \frac{∂H}{∂s} + r × \frac{∂H}{∂r} ,  (59)
ṙ = r × \frac{∂H}{∂s} .  (60)

The Lie-Poisson bracket has two annulators F_{1} = r_{1}^{2} + r_{2}^{2} + r_{3}^{2} and F_{2} = r_{1}s_{1} + r_{2}s_{2} + r_{3}s_{3}. The Hamiltonian function for the rigid body is

H =^{def} \frac{1}{2} (\frac{s_{1}^{2}}{I_{1}} + \frac{s_{2}^{2}}{I_{2}} + \frac{s_{3}^{2}}{I_{3}} ) + mg ⟨l, r⟩ ,  (61)

where s = (s_{1}, s_{2}, s_{3}) denotes the angular momentum of the body, l = (l_{1}, l_{2}, l_{3}) denotes the coordinates of the center of gravity of the body, I = diag(I_{1}, I_{1}, I_{3}) is the tensor of inertia, r = (r_{1}, r_{2}, r_{3}) – the coordinates of the unit vertical vector in the space, and mg is the weight of the body.

The cotangent bundle T^{∗}S^{2} supplied with the canonical symplectic structure dp ∧ dq is symplectomorphic to the manifold O_{h} =^{def} {F_{1} = h, F_{2} = 0} ⊂ e(3)^{∗} (see [14]). Therefore, any integrable Hamiltonian on T^{∗}S^{2} gives an integrable Hamiltonian on O_{h}. Moreover, if the first Hamiltonian is polynomial in impulses, then the second one is also polynomial in momenta s = (s_{1}, s_{2}, s_{3}) and has the same degree. Using the results of [14] and Lemma 2 it is easy to prove

Lemma 3 . Suppose that the Riemannian metric dg^{2} = g_{ij}dx^{i}dx^{j}, x = (x^{1}, x^{2}, x^{3}) is smooth in a neighborhood of the sphere S_{h}^{2} = {⟨x, x⟩ = h} and the restriction dg^{2}|_{S_{h}^{2}} gives a metric whose geodesic flow is completely integrable; then the Hamiltonian function

H_{g}(r, s) =^{def} [\frac{det E}{det G} \frac{1}{e(n_{e}, n_{g})^{2}} ] ∑_{i,j} g_{ij}(r)s_{i}s_{j},  (62)

where de^{2} = (dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}, G = (g_{ij}), E = (e_{ij}) and n_{e} and n_{g} are the external unit normals to the sphere S_{h}^{2} calculated with respect to the metrics e and g at the point x = r ∈ S_{h}^{2}, is completely integrable on the submanifold O_{h}.

In Section 4 we found a family of metrics on R^{n} such that their restriction on the ellipsoid E_{h} give a family of integrable metrics. Now, using these metrics and Lemma 3 we are able to find a family of integrable Hamiltonians in e(3)^{∗}.

Theorem 7 . Consider the operators

A(r) =^{def} A − \frac{(\sqrt{A}r) ⊗ (\sqrt{A}r)}{⟨r, r⟩} + (A^{− \frac{1}{2}} r) ⊗ (A^{− \frac{1}{2}} r)  (63)

and

B^{(k)}(r) =^{def} \sqrt{A}A^{k}(r)\sqrt{A}.  (64)

For any fixed integer k the Hamiltonians

H^{(k)} =^{def} \frac{1}{⟨A^{−1}r, r⟩^{k+1}} ⟨B^{(k)}s, s⟩  (65)

and

H̄^{(k)} =^{def} \frac{1}{⟨A^{−1}r, r⟩^{k−1}} ⟨B^{(k−1)}s, s⟩  (66)

are Liouville integrable on any orbit O_{h}. Moreover, the corresponding Hamiltonian systems are orbitally equivalent.

Example. If k = 1, then we obtain that the Hamiltonians

H^{(1)} = \frac{1}{⟨A^{−1}r, r⟩^{2}} \{⟨A^{2}s, s⟩ − \frac{⟨Ar, s⟩^{2}}{⟨r, r⟩} + ⟨r, s⟩^{2}\}  (67)

and

H̄^{(1)} = ⟨As, s⟩  (68)

are orbitally equivalent.

Corollary 6 . The Hamiltonian system with Hamiltonian H^{(1)} and the Euler case of the free motion of the rigid body are orbitally equivalent on the orbit O_{h}, h > 0.

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