On the case of Kovalevskaya and new examples of integrable conservative

systems on S^{2}

K.P. Hadeler

Mathematische Fakultät, Universität Tübingen Auf der Morgenstelle 10, 72076 Tübingen, Germany e-mail: hadeler@uni-tuebingen.de

E.N. Selivanova^{∗}

Department of Geometry, Nizhny Novgorod State Pedagogical University 603000 Russia, Nizhny Novgorod, ul. Ulyanova 1 e-mail: lena@moebius.mathematik.uni-tuebingen.de

Abstract. There is a well-known example of an integrable conservative system on S^{2}, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. The aim of this paper is to construct new families of examples of conservative systems on S^{2} possessing an integral of fourth degree in momenta.

1 Introduction

Let M be an n-dimensional Riemannian manifold, and U : M → R be a smooth function on M. For the Lagrangian L : T M → R we choose

L(η) = \frac{|η|^{2}}{2} + (U ∘ π)(η)

where π : T M → M is the canonical projection. In local coordinates q_{1}, ... , q_{n}, q̇_{1}, ... , q̇_{n} on T M we have

L(η) = \frac{1}{2} ∑ g_{ij}q̇^{i}q̇^{j} + U(q).

Identifying T M and T^{∗}M by means of the Riemannian metric, we get a Hamiltonian system with the Hamiltonian H : T^{∗}M → R which in local coordinates q_{1}, ... , q_{n}, p_{1}, ... , p_{n} on T^{∗}M has the form

H = \frac{1}{2} ∑ g^{ij}p_{i}p_{j} + U(q) = K + U.

∗Supported by DAAD.

We will call these Hamiltonian systems conservative systems on M.

A smooth function F : T^{∗}M → R which is an integral of the Hamiltonian system with the Hamiltonian H and which is independent of H we will call an integral of this system of degree m in momenta if in local coordinates F has the form

F = ∑_{k_{1}+ ... +k_{n}≤m} a_{k_{1} ... k_{n}}(q)p_{1}^{k_{1}} ... p_{n}^{k_{n}}.

We will say that two Hamiltonians H_{1} = K_{1}+U_{1} and H_{2} = K_{2}+U_{2} are equivalent if there exists a diffeomorphism ϕ of M and a diffeomorphism Φ of T^{∗}M such that the diagram

Φ : T^{∗}M → T^{∗}M
π′ ↓ ↓ π′
ϕ : M → M

is commutative, where Φ is linear for p ∈ M fixed, and if there are some nonzero constants κ, κ̃ such that Φ^{∗}(K_{1}) = κK_{2}, ϕ^{∗}(U_{1}) = κ̃U_{2}.

Clearly, if the Hamiltonians H_{1}, H_{2} are equivalent and one of the corresponding systems possesses an integral of degree m in momenta, then the other system has the same property.

There is a well-known example of an integrable conservative system on S^{2}, the case of Kovalevskaya [4] in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. The total energy (the Hamiltonian) of this system has the following form

H = \frac{du_{1}^{2} + du_{2}^{2} + 2du_{3}^{2}}{2u_{1}^{2} + 2u_{2}^{2} + u_{3}^{2}} − u_{1},  (1)

where S^{2} is given by u_{1}^{2} + u_{2}^{2} + u_{3}^{2} = 1, (see [1]).

Goryachev proposed in [2] a family of examples of conservative systems on S^{2} possessing an integral of fourth degree in momenta. The Hamiltonians of these systems have the following form

H = \frac{du_{1}^{2} + du_{2}^{2} + 2du_{3}^{2}}{2u_{1}^{2} + 2u_{2}^{2} + u_{3}^{2}} − 2B_{1}u_{1}u_{2} − B_{2}(u_{1}^{2} − u_{2}^{2}) − u_{1}  (2)

where S^{2} is given by u_{1}^{2} + u_{2}^{2} + u_{3}^{2} = 1, B_{1}, B_{2} are arbitrary constants. Clearly, this family reduces to the case of Kovalevskaya when B_{1} = B_{2} = 0.

The aim of this paper is to construct new families of examples of conservative systems on S^{2} possessing an integral F of fourth degree in momenta, i.e. examples which are not equivalent to the cases of Kovalevskaya or Goryachev.

We will show that these families include the integrable cases from [5] and the case of Kovalevskaya. In particular, we will obtain an explicit expression for the case of Kovalevskaya from a quite different point of view.

2 New examples

In [5] the following local criterion for integrability of the geodesic flow of a Riemannian metric ds^{2} by a polynomial of fourth degree in momenta has been obtained.

Theorem 2.1 Let ds^{2} = λ(z, z̄)dzdz̄ be a metric such that there exists a function f : R^{2} ↦ R, satisfying the following conditions

λ(z, z̄) = \frac{∂^{2}f}{∂z∂z̄}, Im (\frac{∂^{4}f}{∂z^{4}} \frac{∂^{2}f}{∂z∂z̄} + 3\frac{∂^{3}f}{∂z^{3}} \frac{∂^{3}f}{∂z^{2}∂z̄} + 2\frac{∂^{2}f}{∂z^{2}} \frac{∂^{4}f}{∂z^{3}∂z̄}) = 0. (3)

Then the geodesic flow of ds^{2} possesses an integral of fourth degree in momenta.

If the geodesic flow of a metric ds^{2} possesses an integral which is a polynomial of fourth degree in momenta and it does not depend on the Hamiltonian and an integral of smaller degree then there exist conformal coordinates x, y and a function f : R^{2} ↦ R such that ds^{2} = λ(z, z̄)dzdz̄ where z = φ + iy and (3) holds.

Equation (3) in some other form has been obtained also in [3].

As in [5] we consider now the solutions of (3) of the following form:

f(φ, y) = u(y) cos φ + ξ(y) + d(φ^{2} − y^{2})

where u(y), ξ(y) are some smooth functions and d is a constant. In [5] it has been shown that the geodesic flows of the metrics ds^{2} = λ(dφ^{2} + dy^{2}) where

λ = \frac{1}{4} ((u′′(y) − u(y)) cos φ + ξ′′(y)) ,

ξ′′ = \frac{d_{1}u(y) + c}{(u′(y))^{2}}, d_{1}, c − const, for d = 0,

or

ξ′′ = 2d\frac{u^{′2}(y) − u^{2}(y) + d_{1}(2d)^{−1}u(y) + p}{u′(y))^{2}}, d_{1}, p − const, for d ≠ 0,

and u(y) satisfies

2u^{′′2} − 3u^{2} + u′u′′′ = \frac{a}{2}, a − const,  (4)

possess an integral of fourth degree in momenta.

Using the well-known Maupertuis’s principle and taking into account that c and d are arbitrary constants we find that the Hamiltonian systems with the Hamiltonians

H = \frac{dφ^{2} + dy^{2}}{u^{′2}(y)} − (u′′(y) − u(y))u^{′2}(y) cos φ  (5)

and

H_{p} = \frac{u^{′2}(y) − u^{2}(y) + p}{u^{′2}(y)} (dφ^{2} + dy^{2}) − (u′′(y) − u(y))\frac{u^{′2}(y)}{u^{′2}(y) − u^{2}(y) + p} cos φ (6)

where u is a solution of (4), possess also an integral of fourth degree in momenta. Now we exploit the properties of the differential equation (4).

Proposition 2.2 The differential equation (4) is equivalent to the two-parameter family of first-order differential equations

u^{′4} = b + b_{1}u + au^{2} + u^{4}  (7)

where b, b_{1} are arbitrary constants.

Proof. Multiply equation (4) by u′

u′′′u^{′2} + 2u′′u′u′′ − 3u^{2}u′ = \frac{a}{2}u′

and integrate

u′′u^{′2} − u^{3} = \frac{a}{2}u + \frac{b_{1}}{4}

with some constant b_{1}. Multiply again with u′

u′′u^{′3} − u^{3}u′ = \frac{a}{2}uu′ + \frac{b_{1}}{4} u′

and integrate

\frac{1}{4}u^{′4} − \frac{1}{4}u^{4} = \frac{a}{4}u^{2} + \frac{b_{1}}{4} u + \frac{b}{4}

with some constant b. This is (7).

□

Thus the family (5) can be parametrized by a, b, b_{1}, u(0) and (6) by a, b, b_{1}, u(0), p.

We will consider the case b_{1} = 0 in (7). Denote A(u, a, b) = b + au^{2} + u^{4} where a and b are constants.

Thus, the construction of our examples is based on the properties of the following differential equation

u^{′4} = b + au^{2} + u^{4}.  (8)

Due to Proposition 2.2 we may obtain explicit expressions for the Hamiltonians (5) and (6) in the coordinates φ and u:

H = A^{− \frac{1}{2}} (u, a, b) (dφ^{2} + A^{− \frac{1}{2}} du^{2}) − \frac{u}{2}((a + 2u^{2}) − 2A^{\frac{1}{2}} ) cos φ  (9)

and

H_{p} = \frac{A^{\frac{1}{2}} − u^{2} + p}{A^{\frac{1}{2}}} (dφ^{2} + A^{− \frac{1}{2}} du^{2}) − \frac{u((a + 2u^{2}) − 2A^{\frac{1}{2}} )}{2(A^{\frac{1}{2}} − u^{2} + p)} cos φ  (10)

Thus, in (u, φ) coordinates, H depends only the parameters a, b, and H_{p} on a, b, p.

By a substitution u → α_{1}u, α_{1} − const in (8) we obtain

u^{′4} = \frac{b}{α_{1}^{4}} + \frac{a}{α_{1}^{2}} u^{2} + u^{4}.

So we can normalize b to 1, 0 or −1 and keep a.

In this section we will consider the case b = 1. In the next section we will consider the case b = 0, a = 1 and show that it is in fact the case of Kovalevskaya.

The case b = 1, a = 0 has been considered in [5]. It has been shown that in this case the systems given by (5), (6) (and therefore (9), (10)) define smooth conservative systems on S^{2} possessing an integral of fourth degree in momenta.

We will use the following proposition.

Proposition 2.3 If b = 1 in (8) and a > −2, then there is a solution u_{a}(y) : R → R of (8) such that the following holds

u′_{a}(y) = (exp y)ν_{a}(exp(−2y)) = exp(−y)ν_{a}(exp(2y)),
u_{a}^{′2}(y)(u′′_{a}(y) − u_{a}(y)) = exp(−y)μ_{a}(exp(−2y)) = −(exp y)μ_{a}(exp(2y))

where the functions μ_{a}, ν_{a} are of class C^{∞} and ν_{a} > 0 everywhere.

Proof. By a simple computation we get that if −2 < a < 0, then A(x, a, 1) ≥ 1 − \frac{a^{2}}{4} > 0 for all x ∈ R and if a ≥ 0, then A(x, a, 1) ≥ 1 for all x ∈ R.

Therefore, if b = 1 and a > −2, then all solutions u of (8) exist globally. All increasing (descreasing) solutions are translates of each other. Increasing solutions run from −∞ to +∞, similarly for descreasing solutions.

W.l.o.g. we may only consider increasing solutions of

u′ = (1 + au^{2} + u^{4})^{\frac{1}{4}} , u′ > 0.  (11)

Then we have for u ≥ 0

u′ ≤ u + c_{0}

with a constant c_{0} such that c_{0} ≥ 1 and 6c_{0}^{2} ≥ a and therefore

u(y) ≤ (u(0) + c_{0})(exp y) − c_{0}

and

\frac{u(y)}{exp y} ≤ u(0) + c_{0}.  (12)

Now consider any such solution. Put s = exp(−2y) and

g(s) = \sqrt{s}u (−\frac{1}{2} log s) .

Then u(y) = (exp y)g(exp(−2y)). The function g is of class C^{∞} on (0, ∞). We normalize u(0) = g(1) = 0.

Equation (11) can be rewritten as a differential equation for g(s), s = exp(−2y):

g′ = −\frac{s + ag^{2}}{2((s^{2} + ag^{2} + g^{4})^{\frac{1}{4}} + g)((s^{2} + ag^{2} + g^{4})^{\frac{1}{2}} + g^{2})} = Θ_{a}(s, g).

Thus, g is decreasing, g(0) is finite in view of (12), and g(0) > g(1) = 0. Therefore, there is a solution u_{a} of (8) which can be given as

u_{a}(y) = (exp y)g_{a}(exp(−2y)) = −(exp(−y))g_{a}(exp 2y)

where g_{a} is of class C^{∞} on [0, +∞) if a ≠ 0.

The case a = 0 has been considered in [5] but we can give here another proof. We can consider the function β(s^{2}) = g_{0}(s) and prove, with the same arguments as above, that β is smooth in zero, and therefore, g_{0} is smooth in zero, too.

Now the corresponding expressions for the functions ν_{a} and μ_{a} can be obtained in terms of g_{a}.

□

Further we will use some properties of the geodesic flows of metrics

ds^{2} = λ(r^{2})(r^{2}dφ^{2} + dr^{2})  (13)

on S^{2} which have been proved in [5]. These properties follow also from the results of Kolokol’tsov, published in his Ph.D. Dissertation, (Moscow State University, 1984).

Proposition 2.4 The geodesic flow of a Riemannian metric (13) on S^{2} does not possess a nontrivial integral quadratic in momenta (which does not depend on H and linear integrals).

It is known that a metric of constant positive curvature has the following form in polar coordinates

ds^{2} = \frac{C_{1}}{(1 + Dr^{2})^{2}} (r^{2}dφ^{2} + dr^{2}) ,  (14)

where C_{1}, D − const.

Proposition 2.5 The geodesic flow of a metric (13) on S^{2} possesses two independent linear integrals if and only if it is has the form (14), i.e. if it is a metric of constant positive curvature.

Corollary 2.6 Liouville coordinates φ, y = log r, related to polar coordinates φ, r of a metric (13) on S^{2} are unique up to shifts and the transform y → −y.

Now we will prove the main theorems.

Theorem 2.7 Assume b = 1. Then for any a > −2 the Hamiltonian (9) where φ ∈ [0, 2π) and u ∈ (−∞, +∞) defines a conservative system on S^{2} possessing an integral of fourth degree in momenta.

If a ≠ 2, then this integral is nontrivial, i.e. there is no quadratic or linear integral. If a = 2 it is a Hamiltonian of a metric of constant positive curvature.

The Hamiltonians (9) for different values of the parameter a are not equivalent.

Proof. The integrability of these systems follows immediately from Theorem 2.1, see above.

So, we have to prove only that the Hamiltonian (9), for b = 1 and a > −2, is a sum of a smooth Riemannian metric on S^{2} and a smooth function on S^{2}.

Using Proposition 2.3 we may rewrite the corresponding Hamiltonian in polar coordinates φ, r = exp y. By computation we obtain

H = \frac{1}{ν_{a}^{2}(r^{2})}(r^{2}dφ^{2} + dr^{2}) + μ_{a}(r^{2})r cos φ

= \frac{1}{ν_{a}^{2}(r̃^{2})}(r̃^{2}dφ̃^{2} + dr̃^{2}) − μ_{a}(r̃^{2})r̃cos φ̃

where r̃ = \frac{1}{r}, φ̃ = −φ. Since ν_{a}, μ_{a} are of class C^{∞} and ν_{a} ≠ 0, see Proposition 2.3, this system is a conservative system on S^{2}.

Now we prove that the systems with Hamiltonians (9) where b = 1 and a > −2 do not have linear or nontrivial quadratic integrals.

Write H = Ĥ + V (φ, y) where Ĥ is the Hamiltonian of the geodesic flow of the metric

ds_{1}^{2} = \frac{dφ^{2} + dy^{2}}{u^{′2}(y)}

and

V (φ, y) = −u^{′2}(y)(u′′(y) − u(y)) cos φ  (15)

where u(y) satisfies (8) with the parameters b = 1, a > −2. Note that ds_{1}^{2} has the form (13) in φ, r = exp y but it has the form (14) if and only if a = 2 (if b = 1). So, we can apply Propositions 2.4 and 2.5. We conclude that if a ≠ 2, then an integral quadratic in momenta of the geodesic flow of ds_{1}^{2} depends on the linear integral p_{φ} and the Hamiltonian Ĥ.

Let us assume that a system from our theorem has an integral which is independent of the energy H and which is quadratic in momenta (clearly, this assumption includes the case of linear integrals).

So, there is an integral F̃ of (9) which is quadratic in momenta. Thus, F̃ = D(p_{φ}, p_{y}, φ, y) + B(φ, y) where D(p_{φ}, p_{y}, φ, y) is a polynomial of second degree in momenta p_{φ}, p_{y}.

We write {F̃, H} = {D(p_{φ}, p_{y}, φ, y) + B(φ, y), Ĥ + V } ≡ 0 and, therefore, {D(p_{φ}, p_{y}, φ, y), Ĥ} ≡ 0.

Thus, the geodesic flow of ds_{1}^{2} has an integral quadratic in momenta. As mentioned above, this integral depends on p_{φ} and Ĥ. Since F̃ does not depend on H, we may put w.l.o.g. that

D(p_{φ}, p_{y}, φ, y) = p_{φ}^{2}.

We write now

{D, V } + {B, Ĥ} = {p_{φ}^{2}, V } + {B, Ĥ} ≡ 0.

By computation we obtain

\frac{∂B}{∂y} ≡ 0,

and

u^{′2}(y)\frac{∂B}{∂φ} = \frac{∂V}{∂φ} .

So, we get

V = (B(φ) + α(y))u^{′2}(y)

for a smooth function α(y). Comparing now this expression with (15), we get u′′(y) − u(y) ≡ const, that is not true if a ≠ 2. So, there is no nontrivial quadratic integral of the system given by (9) where b = 1 and −2 < a < 2 or a > 2.

In order to prove that the Hamiltonians (9) for different values of the parameter a are not equivalent we will apply Corollary 2.6.

Suppose that the Hamiltonians (9) for a = a_{1} and a = a_{2}, (a_{1} ≠ a_{2}) are equivalent. Therefore, from Corollary 2.6 the following holds identically

\frac{du}{dũ} = ± (\frac{1 + a_{1}u^{2} + u^{4}}{1 + a_{2}ũ^{2} + ũ^{4}} )^{\frac{1}{4}}

and

\frac{1}{\sqrt{1 + a_{1}u^{2} + u^{4}}} = κ\frac{1}{\sqrt{1 + a_{2}ũ^{2} + ũ^{4}}}

for some constant κ. Thus, we obtain u = ũ and, therefore, a_{1} = a_{2}.

With the same arguments as in [5], we may prove that no Hamiltonian from our theorem is equivalent to the Hamiltonian of the cases of Kovalevskaya or Goryachev.

From Corollary 2.6 it follows immediately that in the family (2) of Goryachev we need to consider only the case B_{1} = B_{2} = 0 which is in fact the case of Kovalevskaya, see the Introduction.

Let us write the Hamiltonian (1) of the case of Kovalevskaya in polar coordinates. We obtain

H = γ_{1}(r^{2})(r^{2}dφ^{2} + dr^{2}) − γ_{2}(r^{2}) cos φ

where γ_{2}(r^{2}) ≠ 0 for all 0 < r < +∞. Comparing this with (9) where u ∈ (−∞, +∞), we see that no Hamiltonian from our theorem is equivalent to the Hamiltonian of the case of Kovalevskaya.

□

Theorem 2.8 Assume b = 1. Then for any a > 2, p ∈ (−∞, −\frac{a}{2}) ∪ (−1, +∞) and −2 < a < 2, p ∈ (−∞, −1) ∪ (−\frac{a}{2}, +∞) the Hamiltonian H_{p} (or −H_{p}) where H_{p} has the form (10) and φ ∈ [0, 2π), u ∈ (−∞, +∞) defines a conservative system on S^{2} possessing an integral of fourth degree in momenta. This integral is nontrivial, i.e. there is no quadratic or linear integral.

These Hamiltonians for different values of parameters a and p are not equivalent. No Hamiltonian from this family is equivalent to the Hamiltonians of the cases of Kovalevskaya or Goryachev.

Proof. The integrability follows immediately from Theorem 2.1.

We will use Proposition 2.3 to rewrite the corresponding Hamiltonians in polar coordinates φ, r = exp y:

H_{p} = \frac{ξ_{a}(r^{2}) + p + 1}{ν_{a}^{2}(r^{2})}(r^{2}dφ^{2} + dr^{2}) + \frac{μ_{a}(r^{2})}{ξ_{a}(r^{2}) + p + 1}r cos φ

= \frac{ξ_{a}(r̃^{2}) + p + 1}{ν_{a}^{2}(r̃^{2})}(r̃^{2}dφ̃^{2} + dr̃^{2}) − \frac{μ_{a}(r̃^{2})}{ξ_{a}(r̃^{2}) + p + 1}r̃cos φ̃

where r̃ = \frac{1}{r}, φ̃ = −φ and

ξ_{a}(t) = ∫_{t}^{1} μ_{a}(s)ν_{a}^{−1}(s)ds.

Now we must find the admissible values of the parameter p. We have

p > max_{z≥0} f(z) or p < min_{z≥0} f(z)

where f(z) = z − \sqrt{1 + az + z^{2}}. By computation we obtain min f(x) = f(0) = −1, max f(x) = f(∞) = −\frac{a}{2} if −2 < a < 2 and min f(x) = f(∞) = −\frac{a}{2}, max f(x) = f(0) = −1 if a > 2.

In order to prove all other statements of the theorem one must only repeat the arguments from the proof of Theorem 2.7.

□

3 The case of Kovalevskaya

Theorem 3.1 If b = 0, a = 1, p = 0, then the Hamiltonian (10) where φ ∈ [0, 2π) and u ∈ [0, +∞) defines a conservative system on S^{2}, corresponding to the case of Kovalevskaya.

Proof. In this case the Hamiltonian (10) has the form

H = \frac{1}{\sqrt{1 + u^{2}}(\sqrt{1 + u^{2}} + u)} (dφ^{2} + \frac{du^{2}}{u\sqrt{1 + u^{2}}} ) + \frac{1}{2(\sqrt{1 + u^{2}} + u)} cos φ (16)

where u ∈ [0, +∞), φ ∈ [0, 2π).

Let us introduce new variables

x = Ψ(u) cos φ, y = Ψ(u) sin φ, z = ±\sqrt{1 − Ψ(u)^{2}}

where

Ψ(u) = \frac{1}{\sqrt{1 + u^{2}} + u} = \sqrt{1 + u^{2}} − u, 0 < Ψ(u) < 1.

We get

\sqrt{1 + u^{2}} = \frac{1}{2} (Ψ(u) + \frac{1}{Ψ(u)}) , u = \frac{1}{2} (−Ψ(u) + \frac{1}{Ψ(u)}) .

Let us compute

dx^{2} + dy^{2} + 2dz^{2} = Ψ^{′2}du^{2} + Ψ^{2}dφ^{2} + 2\frac{Ψ^{2}Ψ^{′2}}{1 − Ψ^{2}} du^{2} = Ψ^{2} (dφ^{2} + \frac{du^{2}}{u\sqrt{1 + u^{2}}} ) .

Then (16) can be rewritten in the form

H = 2\frac{Ψ^{2}}{Ψ^{2} + 1} (dφ^{2} + \frac{du^{2}}{u\sqrt{1 + u^{2}}} ) + \frac{1}{2}Ψ cos φ = 2\frac{dx^{2} + dy^{2} + 2dz^{2}}{2x^{2} + 2y^{2} + z^{2}} + \frac{1}{2}x

where

x^{2} + y^{2} + z^{2} = 1.
□

Acknowledgement. E. Selivanova would like to thank Professor Gerhard Huisken and the Arbeitsbereich ”Analysis” of the University of Tübingen for their hospitality.

References

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[2] Goryachev D.N.: New cases of integrability of Euler’s dynamical equations (Russian). Varshavskie Universitet’skie Izvestiya, 3, (1916) 3-15

[3] Hall, L.S.: A theory of exact and approximate configurational invariants. Physica 8D, (1983) 90-116

[4] Kovalevskaya S.V.: Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Mathematica, 12, (1889) 177-232

[5] Selivanova E.N.: New families of conservative systems on S^{2} possessing an integral of fourth degree in momenta, Preprint dg-ga/9712018 (1997)