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Polynomial in momenta invariants of Hamilton’s equations, divergent series and generalized functions. (English) Zbl 1276.37039
The author studies the existence of new first integrals for the system of Hamilton equations that governs the motion of two particles with a potential function of the form \(V(x^1, x^2)=U(x^1) + U(x^2) + W(x^1-x^2)\). The functions \(U\) and \(W\) are assumed to be \(2\pi\)-periodic and \(W\) is an even function. In the nontrivial case, it is well known that the system does not have linear or quadratic first integrals. Therefore, the author studies the existence of a first integral as a polynomial of degree four in moments, with periodic coefficients. The existence of such invariant is shown to be equivalent to a nonlinear functional-differential equation. In the smooth case, this equation has only trivial solutions. Therefore, the author studies and classifies the generalized solutions of this equation when the interacting potential \(W\) is a distribution, while the external potential \(U\) is a function.

MSC:
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70F05 Two-body problems
70H07 Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics
45J05 Integro-ordinary differential equations
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