Hamiltonian System - Sympletic Structure

Sympletic Structure

One important property of a Hamiltonian dynamical system is that it has a sympletic structure. Writing

$\nabla_{\boldsymbol{r}} H(\boldsymbol{r}) = \begin{bmatrix} \partial_\boldsymbol{q}H(\boldsymbol{q},\boldsymbol{p}) \\ \partial_\boldsymbol{p}H(\boldsymbol{q},\boldsymbol{p}) \\ \end{bmatrix}$

the evolution equation of the dynamical system can be written as

where

$S_N = \begin{bmatrix} 0 & I_N \\ -I_N & 0 \\ \end{bmatrix}$

and I_N the N×N identity matrix.

One important consequence of this property is that an infinitesimal phase-space volume is preserved. A corollary of this is the Liouville's theorem:

Liouville's theorem:

Liouville's theorem states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.

$\frac{d}{dt}\int_{S_t}d\boldsymbol{r} = \int_{S_t}\frac{d\boldsymbol{r}}{dt}\cdot d\boldsymbol{S} = \int_{S_t}\boldsymbol{F}\cdot d\boldsymbol{S} = \int_{S_t}\nabla\cdot\boldsymbol{F} d\boldsymbol{r} = 0$

where the third equality comes from the divergence theorem.

Read more about this topic: Hamiltonian System

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