# Hamiltons Principle proving newtons laws?

1. Dec 15, 2007

### frenchyc

My teacher asked us too prove that Hamiltons principle proves that newtons equations of motion hold for N particles. I'm not sure that i fully grasp the concept but this is my understanding so far:

Using the lagrangian we can prove newtons law for specific situations, however Hamiltons principle allows us to make the specific situation into generalized coordintes. This is my understanding so far, but i feel that i don't completely understand. Why do we need it to be in generalized coordinates? Why does hamiltons principle allow it to change to generalized coordinates?

2. Dec 15, 2007

### robphy

One of the strengths of the Lagrangian and Hamiltonian formulations is the ability to use generalized coordinates, which can often be chosen to take advantage of symmetries in the system. To recover Newton's equations, you should choose the generalized coordinates to be the rectangular coordinates associated with Euclidean space.

The above formulations can be given a geometrical interpretation in which the "space" associated with the Lagrangian formulation is a certain manifold called the tangent-bundle and the Hamiltonian formulation with the cotangent-bundle (more generally, a symplectic manifold).

Last edited: Dec 15, 2007
3. Dec 15, 2007

### frenchyc

I'm sorry but i was not referring to the Hamiltonian, but to Hamilton's principle. The principle that states that the path that a particle follows is such that the action integral of the Lagrangian from point 1 to point 2 is stationary. Im not sure if you were referring to that and i simply didn't understand it or if there was a miscommunication

4. Dec 15, 2007

### robphy

the configuration of the system is described using a "configuration space" Q (which could be described in any convenient set of coordinates for it)
the tangent-bundle is essentially the "space of configurations and velocities" (TQ)
the cotangent-bundle is essentially the "space of configurations and momenta" (T*Q)

possibly useful:

5. Dec 16, 2007

### frenchyc

thank you. I think my problem stems from me having difficulty wrapping my mind around this concept of changing the way we look at space.