On Hamiltonian flows whose orbits are straight lines

Hans Koch; Héctor E. Lomelí

doi:10.3934/dcds.2014.34.2091

Article Contents

2014, Volume 34, Issue 5: 2091-2104. Doi: 10.3934/dcds.2014.34.2091

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On Hamiltonian flows whose orbits are straight lines

Hans Koch^1, and
Héctor E. Lomelí^2,

1.
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712
2.
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526

Received: January 31, 2013

Revised: June 30, 2013

Published: April 2014

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Abstract

We consider real analytic Hamiltonians on $\mathbb{R}^n \times \mathbb{R}^n$ whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q\in \mathbb{R}^n$. By a theorem of Moser [11], 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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Mathematics Subject Classification: Primary: 26C05; Secondary: 37J10, 37M15, 14E07, 58C25.

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