A new class of separable Lagrangian systems generalizing Sawada-Kotera

Some characteristics of the Sawada-Kotera Lagrangian system lend themselves to generalization, producing a large class of separable Lagrangian systems with two degrees of freedom.

1 Main result and related literature Theorem 1.The two smooth Lagrangians in two degrees of freedom L = 1 2 ( q2 1 + q2 2 ) − U(q 1 , q 2 ), L = q1 q2 − Ũ(q 1 , q 2 ) (1.1) have the same Lagrange equations and, hence, share the two energy first integrals if and only if there exist two smooth functions f, g of one variable such that U(q 1 , q 2 ) = f (q 1 +q 2 )+g(q 1 −q 2 ), Ũ(q 1 , q 2 ) = f (q 1 +q 2 )−g(q 1 −q 2 ).(1.3)If this happens, the change of variables (x, y) = (q 1 + q 2 , q 1 − q 2 ) separates the Lagrange equations into Our motivation was the study of the Lagrangian which we call the Sawada-Kotera system.It is integrable, too, by means of the energy and the supplementary first integral This case is also separable in the coordinates (q 1 + q 2 , q 1 − q 2 ) and its solution can be expressed through elliptic functions.This result was obtained by Aizawa and Saito [1], but their names did not stick, and the system is found associated with Sawada and Kotera, and is connected with soliton theory [2,6,12].The Sawada-Kotera system is special case of the following family of Lagrangian functions, depending on the parameter b ∈ R: See for instance the larger family of Hamiltonian functions (1) in [15] (not the one in the Abstract) which we restricted choosing ω 1 = ω 2 = a = 1.Since these Lagrangians are autonomous, the associated Lagrange equations have the first integral of energy (1.8) The special case of b = 1 is the influential Hénon-Heiles system that was introduced in [9] in order to model a Newtonian axially-symmetric galactic system.Such model has no analytic first integral independent of energy, as proved by Ito [10], and its chaotic dynamic behaviour has been extensively studied [4,14,3].(Note that Wojciechowski calls after Hénon-Heiles the whole family in his formula (1) quoted above).
By contrast, two other members are known to be integrable.One is the case b = −6 studied by Dorizzi, Grammaticos and Ramani [5,8], and independently integrated by Wojciechowski [15].
The case b = −1 is precisely the Sawada-Kotera system, which is integrable, too, as we mentioned above.In the same spirit of solitons the case b = −6 is called Korteweg-De Vries.
Let us mention a recent paper by Sottocornola [13], which deals with separation of variables for seven integrable systems related to Hénon-Heiles, and presents some open questions.

Two Lagrangians for the Sawada-Kotera equations
Consider the Sawada-Kotera cubic Lagrangian The associated Sawada-Kotera Lagrange equations are The energy first integral is and there is another first integral quadratic in the velocities (2.4) Let us consider also the Lagrangian function (2.5) It happens that so that the Euler-Lagrange equation for the two system are exactly the same: The first integral of energy for L happens to coincide with (2.4) above (the central dot is the scalar product).

Proof of the main theorem
Suppose we have a Lagrangian of the form where the potential U is smooth but it isn't necessarily a polynomial function.
The associated Euler-Lagrange equations are Inspired by the Sawada-Kotera results of the previous Section, let us consider a second Lagrangian of the form whose equations of motions are Let us impose that the equations (3.2) coincide with those in (3.4): We claim that this occurs if and only if for some smooth functions f, g.
To prove this, let us make a change of dependent variables through a 45 • rotation in the (q 1 , q 2 ) plane, introducing new variables r 1 , r 2 , V, Ṽ : that is to say, In terms of the new variables the differential equations (3.5) become which mean that there exist one-variable functions ϕ, ψ such that that is, which are equivalent to (3.6).
The Euler-Lagrange equations (3.2) have the first integral of energy while the first integral of energy for (3.4) is Whenever conditions (3.5) hold, the Euler-Lagrange equations coincide and E and Ẽ are first integrals for both.In this case, in terms of the new variables (x, y) = (q 1 + q 2 , q 1 − q 2 ) (just a little simpler than r 1 , r 2 above) the Lagrangian function L becomes for which the Euler-Lagrange equations are separated: Similarly, the Lagrangian L becomes Given a Lagrangian function L(t, q, q), q, q ∈ R n , it is easy to build other Lagrangians which have the same Lagrange equations: simply take an arbitrary smooth scalar function G(t, q), and define L(t, q, q) := L(t, q, q) + ∂G ∂t (t, q) + ∂G ∂q (t, q) • q . (3.20) We say L and L are related by a gauge transform.Our Lagrangian functions L in (3.1), and L in (3.3), are not related by a gauge transform because their difference is quadratic in the velocities.

Examples
Here are some some sample systems that are covered by our results.
Example 4.1.(Recovering Sawada-Kotera) The functions give the Sawada-Kotera U, Ũ: Ũ(q 1 , q 2 ) = q 1 q 2 + 1 3 Hence, from (3.2), the separated Euler-Lagrange equations for Example 4.2.The class of system to which our result applies is not limited to polynomial potential.Let us consider the following Lagrangian, that is a variation on Calogero's potential for n-bodies constrained on a line with inversely quadratic pair potentials: (4.5) The equations of motion are The functions give the potential in (4.5) and satisfy the hypotheses of Theorem 1.Hence, from (3.2), the separated Euler-Lagrange equations are that have simple solutions.