Is the Hamiltonian Equation of Motion Gauge Dependent?

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Discussion Overview

The discussion revolves around the gauge dependence of the Hamiltonian equation of motion in the context of classical mechanics and electromagnetism. Participants explore the implications of gauge transformations on the Hamiltonian and the resulting equations of motion derived from a given Lagrangian.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a Lagrangian and derives the Lorentz force, noting its invariance under gauge transformations.
  • Another participant asserts that the Hamiltonian is gauge-dependent, referencing the arbitrary constant that can be added to the scalar potential.
  • A concern is raised about the velocity derived from the Hamiltonian equations of motion appearing gauge dependent, which is questioned by others.
  • Some participants argue that the solutions to the equations of motion are gauge invariant, despite the presence of the vector potential A.
  • There is a discussion about the role of momentum, with one participant suggesting that the canonical momentum is not a physical quantity, while the kinetic momentum (p + eA) is considered physical.

Areas of Agreement / Disagreement

Participants express differing views on the gauge dependence of the equations of motion and the physical interpretation of momentum. There is no consensus on whether the solutions should be considered gauge independent.

Contextual Notes

Participants reference the gauge dependence of potentials and momenta without resolving the implications for measurable quantities like velocity. The discussion highlights the complexity of gauge transformations in classical mechanics.

Who May Find This Useful

This discussion may be of interest to those studying classical mechanics, electromagnetism, and the implications of gauge theories in physics.

evilcman
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If we consider the following Lagrangian:
L = m * (dot x)^2 / 2 - e A dot x - e phi
with A the vectorpotential and phithe scalar potential, then
the Euler-Lagrange equations, reduce to the known formula
of Lorentz-force:
m ddot x = e dot x X B + e E
know, this equation is invariant under gauge transformations:
A->A + nabla F
phi -> phi - parcd phi / parcd t
Know this is great, but if we construct the Hamiltonian with the
usual Legendre transform we will get
H = (p+eA)^2 / 2m + e phi
My problem with this is, if a write the (Hamiltonian) equation of
motion now, it seems gauge dependent:
dot x = (p + eA)/m

Isn't this a problem? What's the resolution?
 
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A Hamiltonian is always gauge-dependent. Remember the arbitrary constant C that can be added to the scalar potential? It is of the same nature.
 
The gauge dependence of the Hamiltonian is not my problem. My problem is that, from these equations of motion, the velocity, which is something that should be mesurable, also seems to be gauge dependent...
 
No, the equation solutions are gauge invariant. Moreover, the equations themselves, expressed via EM filed tensions, are gauge invariant.
 
I don't see why the solutions should be gauge independent when A(and not its curl or something) is explicitly in the equations of motion...
 
If you speak of velocity equation, it contains also p which is gauge-dependent. Write the equation for acceleration d2x/dt2. You will see that this equation will contain the electric and magnetic fields rather than potentials.
 
You are right. And if I understand it correctly p is no longer a physical quantity, only p+eA is.
 
Yes, there are two momenta: a canonically conjugated p (a formal mathematical construction) and a kinetic P = p + eA (physical quantity).
 
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