Finiteness of integrable n-dimensional homogeneous polynomial potentials

Finiteness of integrable n-dimensional homogeneous polynomial potentials

We consider natural Hamiltonian systems of n > 1 degrees of freedom with polynomial homogeneous potentials of degree k. We show that under a genericity assumption, for a fixed k, at most only a finite number of such systems is integrable. We also explain how to find explicit forms of these integrable potentials for small k.

New Integrable Family in the n-Dimensional Homogeneous Lotka–Volterra Systems with Abelian Lie Algebra

where aij , i, j = 1, . . . , n are complex parameters. Integrability of the HLV system (1) has been fairly investigated in some particular cases, e.g. soliton systems, 3 the 2-dimensional systems or the 3-dimensional ABC system. 8, 9, 10 Almost all the above studies dealt with first integrals in the criterions of integrability. However it is known that a system of ordinary differential equatio...

A list of all integrable 2D homogeneous polynomial potentials with a polynomial integral of order at most 4 in the momenta

We searched integrable 2D homogeneous polynomial potential with a polynomial first integral by using the so-called direct method of searching for first integrals. We proved that there exist no polynomial first integrals which are genuinely cubic or quartic in the momenta if the degree of homogeneous polynomial potentials is greater than 4. PACS numbers: 45.50.-j, 45.20.Jj, 02.30.Jr

Integrable Systems in n-dimensional Riemannian Geometry

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an n-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair...

Integrable systems in n-dimensional conformal geometry