# What the difference is between Størmer Verlet and regular Verlet method?

I was wondering what the difference is between the Størmer Verlet method and the regular Verlet method, if there is any.

My impression is that the two methods are the same, in both cases the position is updated using $$x_{n+1}=2x_n-x_{n-1}+a_n\Delta t^2$$ with $$a_n$$ the acceleration.

Usually, however, I hear "Verlet integration" to be about the velocity Verlet method, where the velocity at the half-step is used:

1. Compute $$x(t+\Delta t)=v(t)+\frac12a(t)\Delta t^2$$
2. Compute $$a(t+\Delta t)$$ using the updated position
3. Compute $$v(t+\Delta t)=v(t)+\frac12\left(a(t)+a(t+\Delta t)\right)\Delta t$$

This, of course, assumes the acceleration does not depends on velocity, which would require a slight modification to the algorithm (I discuss this here). The link Qmechanic's added to your post can provide more details to the base method.

• The method was rediscovered several times, including by Störmer, and by Verlet. Sometimes it is called Newton-Störmer-Verlet, because it appeared in Newton's Principia Mathematica. Some of the historical background can be found here also doi:10.1017/S0962492902000144. – user197851 Dec 13 '18 at 13:13
• @LonelyProf yes, that is my understanding. In fact, the Wikipedia entry on Stormer says that he used the Verlet method for a problem and that's why it's also called the Stormer method. – Kyle Kanos Dec 13 '18 at 13:15
• And my apologies for mis-spelling Størmer, too late to correct now.... – user197851 Dec 13 '18 at 13:19
• @LonelyProf I can't even get it on my cellphone keyboard, so I'm totally okay ;)... curiously, the guy's name uses a "slashed o" whereas the method uses an "umlaut o". – Kyle Kanos Dec 13 '18 at 13:22