# TSBASICSYMPLECTIC#

ODE solver using basic symplectic integration schemes https://en.wikipedia.org/wiki/Symplectic_integrator These methods are intended for separable Hamiltonian systems

\begin{align*} qdot = dH(q,p,t)/dp \\ pdot = -dH(q,p,t)/dq \end{align*}

where the Hamiltonian can be split into the sum of kinetic energy and potential energy

$H(q,p,t) = T(p,t) + V(q,t).$

As a result, the system can be generally represented by

\begin{align*} qdot = f(p,t) = dT(p,t)/dp \\ pdot = g(q,t) = -dV(q,t)/dq \end{align*}

and solved iteratively with

\begin{align*} q_new = q_old + d_i*h*f(p_old,t_old) \\ t_new = t_old + d_i*h \\ p_new = p_old + c_i*h*g(p_new,t_new) \\ i = 0,1,...,n. \end{align*}

The solution vector should contain both q and p, which correspond to (generalized) position and momentum respectively. Note that the momentum component could simply be velocity in some representations. The symplectic solver always expects a two-way splitting with the split names being “position” and “momentum”. Each split is associated with an IS object and a sub-TS that is intended to store the user-provided RHS function.

TS: Scalable ODE and DAE Solvers, TSCreate(), TSSetType(), TSRHSSplitSetIS(), TSRHSSplitSetRHSFunction(), TSType