Equivalent Integrable Metrics on the Sphere with Quartic Invariants

We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we can construct new integrable systems on the sphere with quartic invariants.


Introduction
In the study of metric spaces, there are various notions of two metrics on the same underlying space Q being "the same", or equivalent.For instance, there are topologically equivalent metrics and strong equivalent metrics [3].In the Riemannian geometry two metrics are projectively equivalent if their geodesics coincide, in Kähler geometry two metrics are c-projectively equivalent if their J-planar curves coincide and so on, see [9,10,12] and references within.
In symplectic geometry, it is natural to say that two metrics g and g ′ on the configuration space Q are equivalent if the corresponding geodesic Hamiltonians g ij (q)p i p j and T ′ = n i,j=1 g ′ ij (q)p i p j (1.1) are related by some transformation of the phase space preserving canonical symplectic form ω = dp ∧ dq.Here q = q 1 , . . ., q n are coordinates on Q and p = p 1 , . . ., p n are are fibrewise coordinates with respect to the cotangent vectors dq 1 , . . ., dq n .Well-known examples of such canonical transformations are point transformations and non-point transformations in T * R n ρ : q i → p i and p i → −q i , i = 1, . . ., n, relating two geodesic Hamiltonians (1.1) when both metrics are the homogeneous polynomials of second order in coordinates.We aim to construct and classify other non-point canonical transformations relating to two polynomials of the second order in momenta (1.1).Canonical transformation preserves the form of canonical Poisson brackets, which allows us to obtain new integrable geodesic flows by using the following algorithm: take some known integrable geodesic flow with Hamiltonian T = T 1 and independent integrals of motion T 2 , . . ., T n in the involution {T i , T j } = 0, i, j = 1, . . ., n; take non-point canonical transformation ρ (1.2), which maps geodesic Hamiltonian T to geodesic Hamiltonian T ′ , and calculate a set of independent functions ρ(T k ) in the involution on T * Q with respect to the same canonical Poisson brackets {ρ(T i ), ρ(T j )} = 0, i, j = 1, . . ., n; compute n − 1 functions K m on integrals of motion ρ(T k ), so that functions K m are polynomials in momenta, which simplifies all further calculations; find potential V (q) solving equations {H i , H j } = 0, i, j = 1, . . ., n, where H 1 = T ′ + V (q) and H m = K m + W m (p, q), with respect to V and polynomials in momenta W m ; calculate new integrable metric g on Q by using Maupertuis principle gij (q)p i p j . (1. 3) The main unsolved problems in this method are the construction of the non-point canonical transformations ρ (1.2) relating a given quadratic polynomial T with other quadratic polynomial T ′ and computation of the applicable to the Maupertuis principle polynomials in momenta K m .Several canonical transformations ρ were obtained in the framework of algebraic geometry for the 2D Euclidean space in [16,22,24,25], for the 2D sphere in [17,18,20,21] and for the 2D ellipsoid in [19].
In this note, we present canonical transformation ρ (1.2) on the cotangent bundle to (n − 1)dimensional sphere S (n−1) using globally defined coordinates on the ambient space R n .At n = 3 this transformation was obtained in [20,21] in terms of the locally defined coordinates on the sphere.Because we only want to prove the existence of such non-point canonical transformations ρ (1.2) and their applicability to the construction of new integrable metrics and so-called magnetic Hamiltonians H 1 = T ′ + V with generalized potential V depending on velocities we do not discuss the properties of obtained integrable systems, the curvature of the metrics, etc.

Non-point canonical transformations
Let us consider Cartesian coordinates x = (x 1 , . . ., x n ) in Euclidean space R n and the conjugated momenta p x i on T * R n , so that The unit (n − 1)-dimensional sphere S (n−1) ⊂ R n and its cotangent bundle T * S (n−1) ⊂ T * R n are defined via constraints Induced symplectic structure on T * S n−1 is given by the Dirac-Poisson bracket which reads as Images of these variables (x, p x ) we denote as y = (y 1 , y 2 , y 3 ) and p y = (p y 1 , p y 2 , p y 3 ) Proposition 2.1.Consider the following mapping of the cotangent bundle T * S (n−1) defined by equations where This mapping preserves constraints and the form of induced Poisson brackets (2.2).
The proof is a straightforward verification of the Poisson bracket, the forms of constraints, and the form of Hamiltonian.
Below we also consider composition of ρ b (2.3) and similar map ρ c ρ c : which is the canonical transformation depending on 2n parameters b i , c i , i = 1, . . ., n.This composition also preserves canonical Poisson brackets (2.2) and the form of Hamiltonian Here px i are momenta corresponding to coordinates xi .We can construct a family of equivalent integrable metrics on the sphere using these canonical transformations ρ b and σ bc .For instance, applying mapping (2.3) to the geodesic Hamiltonian on T * S (n−1) we obtain geodesic Hamiltonian of the similar form When b i = 1, we have a simple permutation of parameters a i ↔ c i in the original Hamiltonian (2.6).This permutation of parameters is not as trivial as it seems.Let us take Hamiltonian T (2.6) and polynomial of the second order in momenta where w(y) is a polynomial of second order in squares y 2 j , which is not a full square.If we substitute T (2.6) and K (2.8) into {T, K} = 0 and solve the resulting system of algebraic equations for b i , c i , and d i , we obtain a geodesic flow with two integrals of motion which are polynomials of second order in momenta.
Mapping ρ b (2.3) relates second order polynomial in momenta T to the second order polynomial in momenta ρ b (T ) (2.7) commuting with ρ b (K), and with its square ρ 2 (K), which is a polynomial of the fourth order in momenta.
For instance, when n = 3 and b i = 1, the following Hamiltonian commutes with the polynomial of second order in momenta where After transformation (2.3), we obtain geodesic Hamiltonian (2.7) commuting with a square root ρ b (K) (2.9) and with its square ρ 2 (K) which is the quartic polynomial in momenta.
So, on the two-dimensional sphere, we have at least one non-trivial example of equivalent geodesic flows with quadratic and quartic polynomial invariants T , K and ρ b (T ), ρ 2 (K), respectively.An application of the Maupertuis principle to the construction of the corresponding nonequivalent metrics (1.3) is discussed in Section 3.
In the next subsection, we rewrite canonical transformations ρ b (2.3) and σ bc (2.5) in other variables on cotangent bundle T * S 2 to the two-dimensional sphere S 2 and study properties of these transformations.It allows us to construct other examples of equivalent metrics and understand how to construct similar ones on the (n − 1)-dimensional sphere.
For brevity, below we will drop ρ b and σ bc which do not affect understanding, and simply write H instead of ρ b (H) or σ bc (H).

Euler flow on two-dimensional sphere
The three-dimensional Euler top on the phase space so(3) is defined by Hamiltonian commuting with any component M 1 , M 2 and M 3 of the angular momentum vector Many implicit and explicit maps preserve a form of this Hamiltonian, see [23] and references within.
We consider another Hamiltonian system defined by the same Hamiltonian (2.11) but on the six-dimensional phase space T * S 2 , when vector M = x × p x is a cross product of two vectors x and p x so that We denote the similar cross product of the vectors y and p y from (2.3) as L = y × p y .By definition and the symplectic structure on T * S 2 is given by the bracket where ε ijk is the skew-symmetric tensor.
Let us rewrite map (2.3) in these variables on 3) on T * S 2 has the following form

14)
This map preserves the angular momentum vector and the Poisson brackets (2.13).
Proof: From (2.3) and (2.12), we have and Solving these equations for x i and M i we obtain where is the square of the angular momentum vector.After that, we can directly verify that when constraints (2.12) hold.Using (2.15), we can also directly check the Poisson brackets (2.13).

Magnetic flow on the sphere
Let us take geodesic Hamiltonian on the sphere (2.4) which in (x, M ) coordinates reads as After the shift of momenta we obtain the magnetic flow on the sphere defined by the Hamiltonian with linear terms in momenta [2,8,11].An integrable map preserving this flow is given by mapping (2.15) after the shift of momenta.
In contrast with the previous transformation (2.14), this transformation changes the angular momentum vector so that Thus, we rewrite map ρ b (2.3) using the entries of the angular momentum vector (2.15) and construct its trivial generalisation ρ β (2.17).Below we apply these maps to construct equivalent geodesic and non-equivalent potential flows on the two-dimensional sphere.
3 Main example of equivalent metrics on the sphere Elliptic coordinate system u 1 , . . ., u n−1 on the sphere with parameters that implies x 2 i = 1.Elliptic coordinates are orthogonal and locally defined, they take values in the intervals The Poisson bracket between elliptic coordinates u k and their conjugated momenta p u k is the canonical Poisson bracket When n = 3 six variables x i and M i are expressed via four variables u 1,2 and p u 1,2 in the following way Similar second pair of elliptic coordinates v 1,2 on the sphere together with the conjugated momenta determine the second set of variables on T * S 2 In elliptic coordinates, the square of the angular momentum is equal to and mapping (2.15) can be rewritten using a pair of the so-called Abel polynomials on auxiliary variable z defining intersection divisor on the hyperelliptic curve C [20,21]: and where These equations (3.4) and (3.5) should be interpreted as an identity for z and each set of elliptic variables u, p u and v, p v .Transformation ρ b (2.14) is a partial case of the integrable maps associated with the nonholonomic Chaplygin and Veselova systems on the sphere [20,21].

Integration of the original integrable flow
Let us come back to the geodesic Hamiltonian (2.10), which becomes additive separable Hamiltonian in elliptic variables commuting with linear integrals of motion quadratic integral of motion J = S 1 − S 2 (2.8) and any other functions f (S 1 , S 2 ) on S 1,2 .
The corresponding diagonal metric has a non-trivial isometry group.Integrals of motion f (S 1 , S 2 ) are in involution with respect to the Poisson brackets associated with the canonical Poisson bivector P , and second compatible Poisson bivector This pair of compatible Poisson bivectors determines the bi-Hamiltonian vector field where The Hamilton-Jacobi equation H = E (3.6) admits additive separation Because and we have the following separate equations Standard substitution reduces equations (3.10) to equations for the elliptic Weierstrass function dw dt where Thus, we can express variables v 1 , v 2 (3.13) and p v 1 , p v 2 (3.11) via two elliptic ℘-functions on time.
Below we apply transformation ρ b (2.14) to this simple geodesic flow (3.12).

Some properties of the equivalent metrics
Let us introduce the diagonal matrix and its spectral characteristics It allows us to rewrite Hamiltonian H (3.6) using the angular momentum vector After transformation ρ b (2.14) this Hamiltonian has the following form so in elliptic coordinates (3.2) we have which allows us to calculate the corresponding diagonal metric on S 2 For metric space S 2 , g we can define a vector space of symmetric (m, 0) Killing tensors K, which are solutions of the Killing equations where [[•, •]] is a Schouten bracket.When m = 1, solutions K are said to be infinitesimal isometries that form an isometry group.According to [7]: "everybody knows that isometry group Isom(M, g) = Id for generic Riemannian or pseudo-Riemannian metrics g for dim M ≥ 2".Our metric is no exception.Proposition 3.1.Metric g (3.17) on a two-dimensional sphere S 2 has the trivial isometry group and trivial vector spaces of Killing tensors of valency two and three when m = 2, 3.
The proof is a straightforward solution of the Killing equation (3.18) at m = 1, 2, 3.
Transformation ρ b (2.14) maps integral of motion to the following polynomial of fourth order in momenta In elliptic coordinates (3.2) these factors are equal to and changes the form of the second Poisson bivector P ′ (3.9) where and φ k = φ(u k ) for brevity.The structure of this Poisson bivector P ′ is completely different from the structure of the so-called natural Poisson bivectors on the sphere [15,26].Both bivectors P (3.8) and P ′ (3.21) are invertible, which allows us to introduce a hereditary or recursion operator defined as The spectral curve of N has the form where H and K are given by (3.16)- (3.19).Thus, the following equation holds where P and P ′ are given by (3.20) and (3.21) and .
So, we have a geodesic flow on the bi-Hamiltonian manifold T * S 2 , P, P ′ defined by the coefficients of the Casimir functions of the Poisson pencil P λ = P + λP ′ .
The proof is a straightforward calculation.

Potential motion
Let us discuss potentials V (x) which can be added to the geodesic Hamiltonian H (3.15).For instance, starting with the following separable Hamiltonian so that we obtain Hamiltonian commuting with the second integral of motion Other potentials may be obtained using the following substitution The equation {H 1 , H 2 } = 0 has several solutions from which we single out the following polynomial "cubic" potential where b = a 1 + a 2 + a 3 , and In this case polynomial H 1 (3.22) becomes a rational function in v-variables and we know nothing about the separation of variables and the bi-Hamiltonian structure of the corresponding vector field when α ̸ = 0. Following [6,13,14], we can use the Maupertuis principle to construct a new metric on the sphere associated with the Hamiltonian The corresponding additional integral of motion is the polynomial of the fourth order in momenta.

One family of equivalent metrics
As an example, we take another separable Hamiltonian which in (y, L) variables reads as up to the factor 1/4.In Cartesian coordinates, it has the form Ĥ = a 1 p 2 y 1 + a 2 p 2 y 2 + a 3 p 2 y 3 .
Below we apply transformations ρ b (2.3) with different values of b i to this Hamiltonian and present equivalent metrics on the sphere related to canonical transformation σ (2.5).Case 1.Using transformation ρ b (2.3) with we obtain the following integrals of motion where and φ(z) is given by (3.7).
Case 2. Using transformation ρ b (2.3) with we obtain the following integrals of motion we obtain the following integrals of motion where b = a −1 1 + a −1 2 + a −1 3 .
Thus, we have three equivalent metrics on the two-dimensional sphere.

Potential motion
We can try to destroy this equivalence of the geodesic flows by adding potentials to the geodesic Hamiltonian, for instance, changing Hamiltonians in the following way Let us present some potentials explicitly: In the third case, we have Using relations we can rewrite these potentials in terms of Cartesian coordinates.
As above, the Maupertuis principle allows the construction of a new metric on the sphere associated with the Hamiltonian where g and V are metrics and potentials associated with H (1) (4.1), H (2) (4.2) and H (3) (4.3).

Hamiltonians with linear in momenta terms
Let us come back to the separable Hamiltonian (3.6) Using transformation ρ b (2.3) with we obtain where In elliptic coordinates, this Hamiltonian has the following form where the metric is , and b, c and d are combinations of a 1 , a 2 and a 3 (3.14).The corresponding quartic invariant is a product of two polynomials in momenta The main difference is that canonical transformation p k → p k + β k acts trivially when b i = 1 according to constraints (2.1).When b i = a i , this transformation adds nontrivial term to the Hamiltonian H (2.4), which is linear polynomial in momenta As a result, applying a transformation (2.17) to the Hamiltonian we obtain Hamiltonian on the T * S 2 involving linear terms in velocity.Here H is given by (4.4) and the corresponding second integral of motion is equal to where K is given by (4.5) and According [2,8,11] this Hamiltonian defines magnetic flow on the sphere.

Conclusion
We discuss a relatively simple map ρ b (2.3) preserving the form of Hamiltonian and the Dirac-Poisson bracket (2.2) on cotangent bundle T * S (n−1) to the sphere.Applying this map to the following Hamiltonian, which in terms of elliptic coordinates (3.1) has the form we obtain polynomials of the second order in momenta at b i = a ℓ i , ℓ = 0, 1, . . ., Because these polynomials commute with n independent, non-polynomial functions ρ b (S k ), they determine a set of equivalent metrics g b (x) on the sphere.By adding various potentials V b to these equivalent geodesic Hamiltonians ρ b (T ) we can construct different integrable flows and different metrics (1.3) on the sphere.
The main problem is how to get a set of functions on ρ b (S k ) which are polynomials in momenta.In this note, we study two-dimensional sphere when n = 3 and prove that secondorder polynomial ρ b (S 1 + S 2 ) commutes with a polynomial of fourth order in momenta ρ b (S 1 S 2 ).In further publications, we will present a similar result for equivalent metrics on the threedimensional 3D sphere.
Another interesting problem is to consider canonical transformations preserving Hamiltonian of the form which were obtained for different partial cases in [17,18,19,20,21].

A The Maupertuis principle
In modern invariant, coordinate-free Hamiltonian mechanics [1,27], an integrable system is defined as a Lagrangian submanifold in which n parameters are considered as functions on 2n-dimensional symplectic manifold.In a generic case, the Lagrangian submanifold depends on m > n parameters and gives rise to a family of C n m integrable systems with common trajectories.In traditional Hamiltonian mechanics, there are several coordinate-dependent descriptions of the integrable system with common trajectories [4], and the Maupertuis principle is the oldest of them.Roughly speaking, the Maupertuis or Jacobi-Maupertuis principle says that trajectories of the natural Hamiltonian systems are geodesics for the suitable metrics on configuration space, see [5,6,13,14] and references within.
Below we present known technical construction of the geodesic Hamiltonians in a suitable to our purpose form.Let us take the Hamilton function in the so-called natural form g ij (q)p i p j , where potential V (q) is a function on coordinates q and c.Suppose that H commutes with a sum of the homogeneous polynomials of m-order in momenta where N is an arbitrary integer number, all terms in the polynomial K have the same parity.
From {H, K} = 0 follows that geodesic Hamiltonian T = i,j gij (q)p i p j = T h − V , g(q) = g(q) h − V , where h is a constant, commutes T , K = 0 with a sum of the homogeneous polynomials of m-order in momenta Indeed, we can rewrite equation {H, K} = 0 as a set of equations {T, K j } + {V, K j+2 } = 0, j = m, m − 2, . . ., K m+2 = K −1 = K −2 = 0 by using Euler's homogeneous function theorem.Substituting these equations into and grouping terms of the same order in momenta we directly verify that T commutes with K.

. 5 )
These integrals of motion H and K coincide with integrals H (3.16) and K (3.19) after canonical transformation σ bc (2.5) depending on parameters b i = 1 and c k = a k .
19))which allows us to determine the Killing tensor K of valency m = 4 on the sphere S 2 , which satisfies the Killing equation(3.18).Transformation ρ b (2.14) preserves the form of the canonical Poisson bivector P (3.8)