On Hamiltonian flows whose orbits are straight lines

Hans Koch; Héctor E. Lomelí

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arXiv · 1304.3377

On Hamiltonian flows whose orbits are straight lines

Hans Koch·Héctor E. Lomelí

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Abstract

We consider real analytic Hamiltonians whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q$. By a theorem of Moser, every polynomial Hamiltonian of degree 3 reduces to such a $q$-independent Hamiltonian via a linear symplectic change of variables. We show that such a reduction is impossible, in general, for polynomials of degree 4 or higher. But we give a condition that implies linear-symplectic conjugacy to another simple class of Hamiltonians. The condition is shown to hold for all nondegenerate Hamiltonians that are homogeneous of degree 4.

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Hans Koch, Héctor E. Lomelí. 2013-04-11. On Hamiltonian flows whose orbits are straight lines. https://arxiv.org/abs/1304.3377

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