ON HAMILTONIAN FLOWS WHOSE ORBITS ARE STRAIGHT LINES

We consider real analytic Hamiltonians on R-n X R-n whose flow depends linearly on time. Trivial examples are Hamiltonians H(q,p) that do not depend on the coordinate q is an element of 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.