Symplectic integrators of the pendulum equation?

[exmachina]

In particular, a symplectic integrator to solve:

[itex]\ddot{\theta} + \dfrac{g}{l} \sin(\theta) = 0[/itex]

I'm currently using velocity verlet - by realizing that

[itex]\ddot{\theta} = -\nabla (-cos(\theta)) = A(\theta(t))[/itex]

ie. letting x = theta
v = dtheta/dt
a = d^2 theta /dt^2

is it safe to apply verlet integration to generalized coordinates? In particular, does this hold true for a generalized coordinate theta:

[itex]\theta_{t+dt} \approx \theta_t + \dot{\theta}_t dt + \frac{1}{2} \ddot{\theta}_t (dt)^2[/itex]

 

[LightHero]

Yes, it is safe to apply velocity Verlet integration to generalized coordinates. This equation is the same as the one for theta. The only difference is that the acceleration term now depends on theta instead of a constant.