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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.

For a nearly Hamiltonian system, the unperturbed Hamiltonian is conserved and its level curves typically contain a family of periodic orbits. A weak perturbation changes the energy slowly. The energy balance method integrates $\dot{H}$ over one unperturbed period to obtain the leading energy change $\Delta H(H)$. A zero of this change selects a possible persistent periodic orbit. A simple zero persists for sufficiently small perturbation by a Poincaré return map and the implicit function theorem. A negative slope of the averaged energy drift gives attraction; a positive slope gives repulsion. A multiple zero requires higher-order analysis, and a zero-free leading drift rules out periodic motion in the corresponding perturbative energy range.