Applying velocity Verlet algorithm

Question

I want to implement a simple particules system using the velocity form of the Verlet algorithm as integrator.

Initial conditions at $t=0$ for a given particule $p$:

• mass: $ m $
• position: $\overrightarrow x(t=0) = \overrightarrow x_0$
• velocity: $\overrightarrow v(t=0) = \overrightarrow v_0$
• forces applied to it: $ \overrightarrow F(t=0) = \overrightarrow F_0 $

Algorithm's recipe says:

1. Calculate: $\overrightarrow v(t+\frac{1}{2}\Delta t) = \overrightarrow v(t) + \frac{1}{2}\overrightarrow a(t)\Delta t$
2. Calculate: $\overrightarrow x(t+\Delta t) = \overrightarrow x(t) + \overrightarrow v(t+\frac{1}{2}\Delta t) \Delta t$
3. Derive: $\overrightarrow a(t+\Delta t)$ from the interaction potential using $\overrightarrow x(t+\Delta t)$
4. Calculate: $\overrightarrow v(t+\Delta t) = \overrightarrow v(t + \frac{1}{2}\Delta t) + \frac{1}{2}\overrightarrow a(t+\Delta t)\Delta t$

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Let's apply it in order to find $\overrightarrow x(t=1)$ and $\overrightarrow v(t=1)$, so with $\Delta t = 1$:

1. $\overrightarrow v(0+\frac{1}{2}1) = \overrightarrow v(0) + \frac{1}{2}\overrightarrow a(0)1 = \overrightarrow v_0 + \frac{1}{2}\frac{\overrightarrow F_0}{m}$ [OK using Newton's second law]
2. $\overrightarrow x(0+1) = \overrightarrow x(0) + \overrightarrow v(0+\frac{1}{2}1) 1 = \overrightarrow x_0 + (1.)$ [OK using previous result (1.)]
3. ???
4. $\overrightarrow v(0+1) = \overrightarrow v(0 + \frac{1}{2}1) + \frac{1}{2}\overrightarrow a(0+1)1 = (1.) + \frac{1}{2}(3.)$ [OK using (1.) and (3.)]

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So I'm stuck with (3.)

I have to calculate $\overrightarrow a(0+1)$ but I don't know how... I can't apply Newton's second law here since I don't know $\overrightarrow F(t=1)$. Algorithm says "from the interaction potential using (1.)" but I don't understand what it means...

Can you help?

Thank you

Answer 1

I think you all your steps are correct, I would suggest adding units though, otherwise adding $\vec{x}$ and $\vec{v}$ can be misleading.

For step 3. You can just calculate $a = F/m$, where $F$ can either be gravitation $F=m g$, so it does not depend on $x$, or for example for a spring dependent on $x$, so $F = -k x$, here you need the next position from step 2.

Answer 2

To integrate your equation, you need to know the force law: F(x)=-dU(x)/dx F(x)= (U(x)-U(x+Dx))/Dx

What kind of potential to you deal with ?