Applying forces

In many robotics, graphics, and physics applications some object must be 'moved' by applying a force. Doing this the naive way will often result in very unstable behaviour. There are a number of very simple steps that can be taken to improve the situation. First of all, don't just directly update the position from some overall calculated force (or rather, impulse in this case), instead accumulate a velocity using standard integration. That is:

velocity += acceleration*dt
position += velocity*dt

This is a good first step, and many people stop there and discover that they are still left with large oscillation problems. This is due to the poor first-order integrator (Euler) and the lack of damping. Adding damping and velocity limiting will help:

velocity += acceleration*dt
velocity = clamp(velocity, min, max)
velocity *= dampen //eg: 0.98
position += velocity*dt

Hopefully your system is acting closer to how you expected. At this point if you still have some minor oscillation and odd edge cases you can probably solve them through more advanced techniques. If your still not getting a behaviour close to what you want then try a completely different approach.

The final quick and easy step to improve performance is to improve the integrator. A simple leap-frog (velocity-verlet) integrator will provide decent performance (no need to go to RK45!):

velocity += acceleration*dt
velocity = clamp(velocity, min, max)
velocity *= dampen //eg: 0.98
position += velocity*dt + 0.5*acceleration*dt*dt

Done!

Now all your robot path-planning, navigation, graphics particle systems, and cloth-simulations will be up and running. Plenty of research papers to read if you want to keep improving it!

I'll leave you with this real life video illusion of marbles rolling up a slope: