29.4. Simplectic integrators

For Hamiltonian systems (see Classical Mechanics: Hamiltonian Mechanics), simplectic integrators preserve the volume in phase space.

\[\begin{split}\begin{aligned} \dot{\mathbf{p}} & =-\partial_{\mathbf{q}} H \\ \dot{\mathbf{q}} & = \partial_{\mathbf{p}} H \\ \end{aligned}\end{split}\]

Preservation of volume in phase space for Hamiltonian systems

Let \(V_t\) a material domain in phase space. The derivative of its volume reads

\[\begin{split}\begin{aligned} \frac{d}{dt} \int_{V_t} 1 & = \int_{V_t} \int_{V_t} \partial_t 1 + \oint_{\partial V_t} \mathbf{u} \cdot \hat{\mathbf{n}} = \\ & = \oint_{\partial V_t} \left( \partial_{\mathbf{p}} H, - \partial_{\mathbf{q}} H \right) \cdot \hat{\mathbf{n}} = \\ & = \int_{V_t} \nabla_{\mathbf{q}} \cdot \nabla_{\mathbf{p}} H - \nabla_{\mathbf{p}} \cdot \nabla_{\mathbf{q}} H = \\ & = 0 \ . \end{aligned}\end{split}\]

having applied Reynolds’ transport theroem in the phase space, with \(\mathbf{u} = \frac{d \mathbf{r}}{dt} = \left( \partial_{\mathbf{p}} H, -\partial_{\mathbf{q}} H \right)\) the “velocity” in the phase space, with the state \(\mathbf{r} = ( \mathbf{q}, \mathbf{p} )\), and \(\hat{\mathbf{n}}\) the unit normal vector pointing outwards. In the last steps, Hamilton equations are used to write velocity components as a function of partial derivatives of the Hamiltonian function, and divergence theorem is used in the phase space to transform the surface integral to a volume integral. The last step reads in components reads

\[\nabla_{\mathbf{q}} \cdot \nabla_{\mathbf{p}} H - \nabla_{\mathbf{p}} \cdot \nabla_{\mathbf{q}} H = \frac{\partial^2 H}{\partial q^i \partial p_i} - \frac{\partial^2 H}{\partial p^i \partial q_i} = 0 \ ,\]

under the assumption of sufficient regularity required by the Schwartz theorem about mixed partial derivatives.

Algorithms - examples:

• Verlet: simplectic, time reversible

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Applications - examples:

• Integration of spatial dynamics: motion of planets,…

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