What is Poincare's Integral Invariant

[Greg Bernhardt]

Definition/Summary

Poincare's integral invariant is the most fundamental invariant in Hamiltonian Dynamics. For any phase space set, the sum of the areas of all of its orthogonal projections onto all the non-intersection canonically conjugate planes is invariant under Hamiltonian evolution.

Equations

$$\sum {\int_{{\Delta _k}} {d{q^k}d{p_k}} }$$

$${\Delta _k}$$ is the projection of a phase-space set onto the kth conjugate plane (usually taken to be the kth position-momentum plane),

Extended explanation

Poincare's integral invariant states that the differential form $$\sum {d{q^k}d{p_k}}$$ (or equivalently $$\sum {d{q^k} \wedge d{p_k}}$$ considering any manifold structure on configuration space) is preserved under Hamiltonian evolution.

All Hamiltonian systems are Poincare integral invariant. Similarly, it can also be shown that Poincare's integral invariant holds only for Hamiltonian systems.

In statistical mechanics, Poincare's integral invariant may replace the Liouville integral invariant when defining entropy.

Mathematicians sometimes use the term symplectic capacity when referring to symplectic spaces in general. However, the most (or only) natural symplectic capacity in dynamics is the Poincare integral invariant.

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[APhysicist]

Thanks for the overview of Poincare's integral invariant states