How Do You Calculate a(t+dt) Using the Velocity Verlet Algorithm?

[shad0w2000]

Homework Statement

I am going to use the velocity verlet algorithm to simulate the position, velocity and acceleration at time t of some particle.

Homework Equations

We got the Langevin equation: a(t) = -v(t)/tau - U'(x) / m + F_f(t) / m

Where tau is the mass divided by the friction coefficient, U'(x(t)) the external force and F_f(t) the fluctuating force. Let's say the external force is harmonic, ½kx(t)^2, so U'(x(t)) / m = kx(t)/m.

I have so far typed in x(t+dt), v(t+dt/2) and I need to find what a(t+dt) is. It's easier if you go see http://en.wikipedia.org/wiki/Verlet_integration" for the equations i refer to :)

As for a(t+dt) I am in doubt how I should find this, what I tried so far is:

a(t+dt) = -1/m * dV(x(t+dt))/dx

= -v(t) / tau - k*x(t+dt) / m + F_f(t)

(Where I take the value of the fluctuating force randomly from a normaldistribution)

My doubt is about the use of v(t), can I use v(t) in the above equation? I don't see how many other options I have since we need a(t+dt) to be able to calculate v(t+dt), or have I missed something?

I hope someone can help me a bit, if you need more info or anything is unclear, please tell me here.

 

[shad0w2000]

No one able to help? :)

 

[genneth]

The verlet integrator is semi-implicit, which means exactly what you've noticed: you need $$v(t+dt)$$ to calculate $$a(t+dt)$$. In general this means that you need to solve an equation at each step (numerically). Do you have to use Verlet? The usual Runge-Kutta family is pretty good, or even just plain ol' midpoint rule?