qdot = dH(q,p,t)/dp
pdot = -dH(q,p,t)/dq
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 genearlly represented by
qdot = f(p,t) = dT(p,t)/dp
pdot = g(q,t) = -dV(q,t)/dq
and solved iteratively with
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)
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.
Reference: wikipedia (https://en.wikipedia.org/wiki/Symplectic_integrator)