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Since $N_i=N_i^{\ast }{e}^{x_i}$, division of each population equation by $N_i$ givesDefine the Hamiltonian functionand the antisymmetric matrixThen

The functional derivative of the action isFor the Hamiltonian functionand a time-independent Lagrangian,Therefore

Hamiltonian Monte Carlo augments the position $x$ by an independent momentum $p\sim N(0,M)$ and uses the Hamiltonian functionFrom the current $x$, draw a fresh $p$, apply a fixed number of leapfrog steps to approximate Hamiltonian flow, and obtain $(x^{\prime },p^{\prime })$. The Metropolis–Hastings acceptance probability isotherwise retain $x$. Momentum negation may be included to make the proposal explicitly reversible. The leapfrog map is volume preserving and reversible, while the acceptance step corrects its discretization error, leaving $\pi$ invariant after the momentum is discarded.

Again take a symplectic two-sphere, but let $L$ be a small latitude bounding a cap of area strictly below half the total area. A rotation carrying that cap to a disjoint cap carries $L$ to a disjoint Lagrangian circle. Rotations of the symplectic sphere are flows of Hamiltonian vector fields; for rotation about an axis, a height function is a Hamiltonian function. Thus this rotation is a Hamiltonian isotopy, and $L$ is Hamiltonian displaceable.

For identical point vortices on the plane, use $\omega ={\sum }_{a}d{x}_a\wedge d{y}_a$, ${\iota }_{X_H}\omega =dH$, and $L=\frac{1}{2}{\sum }_a(x_a^2+y_a^2)$. The generator $Y$ of simultaneous counterclockwise rotations is $-X_L$, and $L$ is its moment map with the convention ${\iota }_Y\omega =-dL$. In coordinates rotating with angular velocity $\Omega$, the Hamiltonian vector field is $X_H-\Omega Y=X_{H+\Omega L}$. Hence a collision-free configuration rigidly rotating at that velocity is exactly a critical point of the augmented Hamiltonian function $H+\Omega L$. For $H=-{\sum }_{a<b}\log |{\mathbf{r}}_a-{\mathbf{r}}_b{|}^2$, scaling all positions shows ${\sum }_a{\mathbf{r}}_a\cdot {\nabla }_{a}H=-N(N-1)$; a rotating relative equilibrium therefore obeys $2\Omega L=N(N-1)$. For $N\ge 2$ this implies positive $\Omega$ in the stated circulation convention. Reversing the contraction convention requires consistent changes to the Hamiltonian and rotation signs.