= Nearly Hamiltonian system
{title2=$\dot z=J\nabla H(z)+\epsilon R(z)$}
A <nearly Hamiltonian system> is a weak perturbation of a <Hamiltonian system>, written
$$
\dot z=J\nabla H(z)+\varepsilon R(z).
$$
Since $J$ is skew-symmetric, its energy changes by $\dot H=\varepsilon\nabla H\cdot R$. 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 is
$$
M(E)=\int_0^{T(E)}\nabla H(z_E(t))\cdot R(z_E(t))\,dt.
$$
Assume the family forms a regular annulus, the return section is transverse and $M(E_0)=0$, $M'(E_0)\ne0$. The return energy then has expansion $P_\varepsilon(E)-E=\varepsilon M(E)+O(\varepsilon^2)$. Dividing by $\varepsilon$ and applying the <implicit function theorem> produces a fixed return energy $E(\varepsilon)$ 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>.
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