Finding the Hamiltonian and constants of motion for a particle in a 1D force field

[iori1234599]

1. A particle of mass m is in the environment of a force field with components: F_z=-K_z, F_y=F_x=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.

 

[vela]

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.