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Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space. (English. Russian original) Zbl 1261.37023
Izv. Math. 76, No. 5, 907-921 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 5, 57-72 (2012).
This paper addresses an outstanding conjecture on the degrees of irreducible polynomial integrals of reversible Hamiltonian systems.
In particular, the authors study a special system arising in the analysis of irreducible polynomial integrals of degree 4. The broad form of the conjecture is that the degree of any such polynomial integral does not exceed 2.
The special case considered by the authors is the last step in proving the conjecture for degree 4 irreducible polynomials.
The special system considered here is described by the Hamiltonian equations \[ \dot x_k= {\partial H\over\partial y_k},\quad\dot y_k= -{\partial H\over\partial x_k}\quad (k= 1,2), \] where \(H= {1\over 2}(y^2_1+ y^2_2)- W(x_1, x_2)\) and where \(W\) is an infinitely differentiable function on the configuration space, namely the torus \(\mathbb{T}^2\). The authors ask whether the Hamiltonian equations admit non-trivial first integrals that are polynomial in the momenta \(y_1\) and \(y_2\). They conclude that there is a non-trivial polynomial integral of degree at most 2.

37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70F05 Two-body problems
70H07 Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics
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