Mathematics Lecture Notes

ProofVolume Preservation by Hamiltonian Flows

We prove that Hamiltonian flows preserve the Liouville volume form $\Omega = \frac{\omega^n}{n!}$ .

Setup: Let $(M, \omega)$ be a $2n$ -dimensional symplectic manifold, $H: M \to \mathbb{R}$ a Hamiltonian function, and $\phi_t$ the flow of the Hamiltonian vector field $X_H$ defined by $\iota_{X_H} \omega = dH$ .

Step 1: Hamiltonian vector fields preserve $\omega$

By Cartan's formula:

$\mathcal{L}_{X_H} \omega = d(\iota_{X_H} \omega) + \iota_{X_H}(d\omega)$

Since $\iota_{X_H} \omega = dH$ and $d\omega = 0$ (closedness), we have:

$\mathcal{L}_{X_H} \omega = d(dH) + 0 = 0$

Thus $X_H$ preserves $\omega$ : $\mathcal{L}_{X_H} \omega = 0$ .

Step 2: Preservation extends to $\omega^n$

We need to show $\mathcal{L}_{X_H}(\omega^n) = 0$ . Using the derivation property of the Lie derivative:

$\mathcal{L}_{X_H}(\omega^n) = \mathcal{L}_{X_H}(\omega \wedge \omega^{n-1})$

Since $\mathcal{L}_{X_H}$ is a derivation on forms:

$= (\mathcal{L}_{X_H} \omega) \wedge \omega^{n-1} + \omega \wedge \mathcal{L}_{X_H}(\omega^{n-1})$

By Step 1, $\mathcal{L}_{X_H} \omega = 0$ , so:

$= 0 \wedge \omega^{n-1} + \omega \wedge \mathcal{L}_{X_H}(\omega^{n-1}) = \omega \wedge \mathcal{L}_{X_H}(\omega^{n-1})$

Proceeding inductively, if $\mathcal{L}_{X_H}(\omega^{n-1}) = 0$ , then $\mathcal{L}_{X_H}(\omega^n) = 0$ .

The base case is $\mathcal{L}_{X_H} \omega = 0$ from Step 1. Therefore $\mathcal{L}_{X_H}(\omega^n) = 0$ for all $n$ .

Step 3: Integration yields flow preservation

The Lie derivative $\mathcal{L}_{X_H}$ measures the infinitesimal change along the flow:

$\mathcal{L}_{X_H}(\omega^n) = \lim_{t \to 0} \frac{\phi_t^*(\omega^n) - \omega^n}{t}$

If $\mathcal{L}_{X_H}(\omega^n) = 0$ , then:

$\frac{d}{dt}\phi_t^*(\omega^n) = 0$

Integrating from $0$ to $t$ :

$\phi_t^*(\omega^n) = \phi_0^*(\omega^n) = \omega^n$

Since $\Omega = \frac{\omega^n}{n!}$ , we have $\phi_t^* \Omega = \Omega$ , completing the proof.

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Remark

The key insight is that preservation of $\omega$ by $X_H$ automatically implies preservation of $\omega^n$ . This is a special feature of Hamiltonian vector fields: they preserve not just volume, but the entire symplectic structure.

DefinitionDivergence-Free Condition

In local coordinates $(q_i, p_i)$ , the Hamiltonian vector field is:

$X_H = \sum_{i=1}^n \left(\frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial}{\partial p_i}\right)$

The divergence with respect to the volume form $\Omega = dq_1 \wedge \cdots \wedge dq_n \wedge dp_1 \wedge \cdots \wedge dp_n$ is:

$\text{div}(X_H) = \sum_{i=1}^n \left(\frac{\partial^2 H}{\partial q_i \partial p_i} - \frac{\partial^2 H}{\partial p_i \partial q_i}\right) = 0$

The mixed partials cancel by equality of mixed partials, so $X_H$ is divergence-free.

ExampleExplicit Computation for Harmonic Oscillator

For $H = \frac{1}{2}(p^2 + q^2)$ on $\mathbb{R}^2$ :

$X_H = p \frac{\partial}{\partial q} - q \frac{\partial}{\partial p}$

The flow is $\phi_t(q, p) = (q\cos t + p\sin t, -q\sin t + p\cos t)$ , a rotation.

For any region $A$ :

$\int_{\phi_t(A)} dq \wedge dp = \int_A \phi_t^*(dq \wedge dp)$

Computing the pullback:

$\phi_t^*(dq \wedge dp) = (\cos t \, dq + \sin t \, dp) \wedge (-\sin t \, dq + \cos t \, dp)$ $= (\cos^2 t + \sin^2 t) dq \wedge dp = dq \wedge dp$

Thus area is preserved.

Remark

Alternative proof via Jacobian: If $\phi_t$ is the Hamiltonian flow, its Jacobian matrix $J_{\phi_t}$ satisfies:

$\det(J_{\phi_t}) = 1$

for all $t$ . This follows from the fact that $\phi_t \in \text{Sp}(2n, \mathbb{R})$ at each point, and symplectic matrices have determinant 1.

The volume element transforms as:

$\phi_t^*(\Omega) = \det(J_{\phi_t}) \cdot \Omega = \Omega$

DefinitionConservation of Phase Space Density

In statistical mechanics, the phase space density $\rho(q, p, t)$ evolves according to Liouville's equation:

$\frac{\partial \rho}{\partial t} + \{H, \rho\} = 0$

This is equivalent to $\frac{d}{dt}(\rho \circ \phi_t) = 0$ , i.e., $\rho$ is constant along Hamiltonian flow. Combined with volume preservation, this gives the conservation of the measure $\rho \, \Omega$ .

Math Notes

Proof of Liouville's Theorem