Homework Help: Hamiltonian Mechanics

1. Sep 16, 2010

iori1234599

1. A particle of mass m is in the environment of a force field with components: Fz=-Kz, Fy=Fx=0 for some constant K. Write down the Hamiltonian of the particle in Cartesian coordinates .What are the constant of motion?

2. H=kinetic energy +potential energy

3. Is the Hamiltonian H is just E= int(-K)dz = -Kz??
Also, I would like to ask what is the meaning of the constant of motion?? I really don't know where to start .

That is the first time that I deal with the Hamiltonian Mechanics. I don't sure how to do the problem.

2. Sep 16, 2010

vela

Staff Emeritus
No, that's not the Hamiltonian. The Hamiltonian, as you noted, can be written as the sum of the kinetic and potential energies. The kinetic energy is just 1/2 mv2, which you will want to express in terms of the components of the particle's momentum. For the potential energy, you want to find the function V(x,y,z) such that F = -∇V.

A constant of motion is simply a conserved quantity. If you write down Hamilton's equations for your Hamiltonian, you'll find the time derivative of two variables is equal to 0. Those two variables are therefore constant.