Hamiltonian mechanics electromagnetic field

[Liquidxlax]

Homework Statement

Let (V(x,t) , A(x,t)) be a 4-vector potential that constructs the electromagnetic field (in gaussian Units) by

E(x,t) = -∇V(x,t) - (1/c)δ_tA(x,t) , B = ∇xA , (x,t) elements of R³xR^t

Consider the lagrangian

L=.5mv² - eV(x,t) + (ev/c)(dot)A(x,t)

a) compute and interpret the Euler-lagrange equatinons of motion for this system
b) determine the hamiltonian
c) determine hamilton's equations of motion. Are they gauge invariant?

Homework Equations

The Attempt at a Solution

3 simple questions about this, and hopefully not too stupid of questions

when i apply the euler-lagrange equations do i take the curl of A or the gradient? if gradient what does that mean to take the ∇A?

is there only one equation of motion because i fail to see the other possibility if there is one.

Otherwise i can solve the rest

thanks

 

[mark726]

Thank you for your post and questions. I will do my best to provide you with the answers and explanations you are seeking.

Firstly, when applying the Euler-Lagrange equations in this context, you would take the gradient of V and the curl of A. This is because V is a scalar potential and A is a vector potential. Taking the gradient of a scalar potential gives you the electric field, while taking the curl of a vector potential gives you the magnetic field.

Secondly, there are actually two equations of motion in this system. The first one will come from the variation with respect to the position coordinates x, and the second one will come from the variation with respect to time t. This is because the system is described by a Lagrangian, which is a function of both position and time.

Lastly, to determine the Hamiltonian, you would need to use the Legendre transform, which involves taking the Lagrangian and substituting the velocities for their conjugate momenta. The Hamiltonian would then be a function of the position and momentum coordinates. As for Hamilton's equations of motion, they are indeed gauge invariant. This is because they are derived from the Hamiltonian, which is a gauge invariant quantity.

I hope this helps answer your questions. If you need any further clarification, please don't hesitate to ask.