Why is Verlet integration better than Euler integration?

Question

Can someone explain to me why Verlet integration is better than Euler integration? And why RK4 is better than Verlet? I don't understand why it is a better method.

Answer 1

The Verlet method is is good at simulating systems with energy conservation, and the reason is that it is symplectic. In order to understand this statement you have to describe a time step in your simulation as a function, f, that maps the state space into itself. In other words each timestep can be written on the following form.

(x(t+dt), v(t+dt)) = f(x(t),v(t))

The time step function, f, of the Verlet method has the special property that it conserves state-space volume. We can write this in mathematical terms. If you have a set A of states in the state space, then you can define f(A) by

f(A) = {f(x)| for x in A}

Now let us assume that the sets A and f(A) are smooth and nice so we can define their volume. Then a symplectic map, f, will always fulfill that the volume of f(A) is the same as the volume of A. (and this will be fulfilled for all nice and smooth choices of A). This is fulfilled by the time step function of the Verlet method, and therefore the Verlet method is a symplectic method.

Now the final question is. Why is a symplectic method good for simulating systems with energy conservation, but I am afraid that you will have to read a book to understand this.