How to use Lagrange approach to solve elastic collision?

[kof9595995]

I tried to use Lagrangian and Hamiltonian to solve 1-D elastic collision, but I got nothing but constant velocity motion. Is it because I miss some constraint? Such as the motion is colinear or something?But how to write a constraint like colinear?
Or it's not actually solvable with Hamiltonian or Lagrangian?(I think this is a possibility because when I learned Newtonian mechanics we actually didn't solve this with Newton's 3 laws directly, but with conservation laws.)

 

[Dale]

There are two solutions to the elastic collision problem. One is constant velocity motion, and the other is the one you are interested in. Look back at your steps and find out where you accidentally threw away the wrong solution.

 

[kof9595995]

Well,thanks for the reply, but honestly I can't see how it would help. I didn't find anything about Hamilton's or Lagrange's mechanics in the link.

 

[Dale]

The error has nothing to do with Hamiltonian or Lagrangian mechanics, it is just math. You have an equation which has more than one solution (e.g. x² = 4 has the solutions x = 2 and x = -2). You simply threw away the wrong solution. Go back, find where you threw away a solution and keep the one you threw away.

 

[kof9595995]

Ok, let me clarify my problem:
I used to think to solve a classical system like this, all you need is Lagrangian and some constraints, then you can solve for all the details of the motion.
But in this problem how should I write a constraint?

 

[Dale]

I would do it by making the potential energy increase very sharply for r<R where r is the separation between the two objects and R is the sum of their radii. This will have the effect of modelling the elastic potential energy in the collision. There may be a more advanced way that has some advantage, but that would be how I would approach it.

 

[kof9595995]

Ok, I'll try, thanks