 Nearly Hamiltonian system

A nearly Hamiltonian system is a weak perturbation of a Hamiltonian system, writtenSince $J$ is skew-symmetric, its energy changes by $\dot{H}=\epsilon \nabla H\cdot R$. This identity is exact; it does not require replacing the perturbed trajectory by an unperturbed one.

For a smooth regular family of planar periodic Hamiltonian orbits labeled by energy $E$, the first-order change over one unperturbed period isAssume the family forms a regular annulus, the return section is transverse and $M(E_0)=0$, $M^{\prime }(E_0)\ne 0$. The return energy then has expansion $P_{\epsilon }(E)-E=\epsilon M(E)+O({\epsilon }^2)$. Dividing by $\epsilon$ and applying the implicit function theorem produces a fixed return energy $E(\epsilon )$ near $E_0$, hence a nearby periodic orbit for small perturbations.

These planar regular-orbit hypotheses are part of the persistence result. A single averaged energy equation is not automatically sufficient in a higher-dimensional phase space, and a multiple zero requires higher-order terms. Near an infinite-period separatrix, the Melnikov energy-balance method instead measures manifold splitting along a homoclinic orbit or a heteroclinic orbit.