Discovering the Origins of Conservation Laws in Particle Simulations

[dyb]

Let's assume I simulate a number of particles using a computer program. I teach the particles to move according to F=ma. The F acting on each particle will be the sum of all forces to other particles according to F=m1*m2/distance^2.

I give the particles a set of initial positions and velocities and I automatically get conservation of linear momentum, conservation of angular momentum, and conservation of energy, just like that. I haven't actually programmed those properties into code.

Where does the conservation come from?

Basically I've only used F=m*a, v=a*t, p=v*t, action=reaction, the pythagorean theorem and some sums. Does it come from F=m*a, from action=reaction, is it a property of isometric space, or all of them?

 

[DrClaude]

Look up Noether's[/PLAIN] theorem.

 

[dyb]

To understand the theorem, I'd need to understand what a differentiable symmetry is, what Lagrangian is, and so forth. If I knew all that, I wouldn't be asking.

Can it be pinpointed down to something more simple? Can you give a simplified answer?

 

[DrClaude]

The simple answer is that each conservation law is related to a symmetry. Conservation of energy comes from the symmetry of translation in time, conservation of momentum translation symmetry in space, and angular momentum rotation symmetry. If your equations (and importantly their numerical implementation) have these symmetries, then the quantities will be conserved.

Note that depending on the implementation (for instance, because of time discretization), some quantities might be conserved only on average.

 

[dyb]

Thanks!