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May  2014, 34(5): 2091-2104. doi: 10.3934/dcds.2014.34.2091

## On Hamiltonian flows whose orbits are straight lines

 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, United States

Received  February 2013 Revised  July 2013 Published  October 2013

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 $4$.
Citation: Hans Koch, Héctor E. Lomelí. On Hamiltonian flows whose orbits are straight lines. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2091-2104. doi: 10.3934/dcds.2014.34.2091
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