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This is on my electrodynamics homework and I haven't been able to get anywhere with it. Here it is:

The Hamiltonian of a particle of mass m, charge q, position

**r**, momentum

**p**, in an external field defined by a vector potential

**A**(

**r**,t) and scalar potential [tex]\phi[/tex](

**r**,t) is given below:

H(

**r**,

**p**) = (1/2m)[

**p**- q

**A**(

**r**,t)]

^{2}+ q[tex]\phi[/tex](

**r**,t) = (1/2m)(p

_{j}p

_{j}- 2qp

_{j}A

_{j}+q

^{2}A

_{j}A

_{j}) + q[tex]\phi[/tex]

Calculate Hamilton's equations of motion. You can use the relations

[tex]\frac{\partial p_{j}}{\partial p_{i}} = \delta _{ij}[/tex]

and

[tex]\frac{\partial p_{j}}{\partial r_{i}} = 0.[/tex]

So, attempted solution...

according to my notes, these equations of motion are

[tex]\frac{\partial H}{\partial p_{i}} = \frac{dr_{i}}{dt}[/tex]

and

[tex]\frac{\partial H}{\partial r_{i}} = -\frac{dp_{i}}{dt}[/tex]

What I tried to do is stick the Hamiltonian into the two equations, but I am a bit confused about what the j and i subscripts are all about. I assumed they meant j = x,y,z, so I put H into those equations to get expressions for j = x, y and z.

I got

[tex]\frac{dH}{dp_{x}} = 1/m(p_{x} - qA_{x})[/tex] and equivalent for y and z,

and

[tex]\frac{dH}{dr_{x}} = 1/2m(-2qp_{x}(d/dr_{x})A_{x} + q^{2}(d/dr_{x})A^{2}_{x}[/tex] and equivalent for y and z.

This is pretty much as far as I have got with this (ie. nowhere). I don't really understand what I'm supposed to be doing (ie. what the point of putting H into these equations is), and I dont see what use the 2 relations I was given in the question are. I have given all the information the question has. Can anyone shed any light on what the hell this question is on about?

Is there some way of rewriting the vector and scalar potentials in terms of momentum or something?

I am totally lost here and my notes barely even touch on this stuff.

Thanks.