Relativistic Charged Particle in An Electromagnetic Field
The Lagrangian for a relativistic charged particle is given by:
Thus the particle's canonical (total) momentum is
that is, the sum of the kinetic momentum and the potential momentum.
Solving for the velocity, we get
So the Hamiltonian is
From this we get the force equation (equivalent to the Euler–Lagrange equation)
from which one can derive
An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, is
This has the advantage that can be measured experimentally whereas cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), plus the potential energy,
Read more about this topic: Hamiltonian Mechanics
Famous quotes containing the words charged, particle and/or field:
“When a man hath taken a new wife, he shall not go out to war, neither shall he be charged with any business: but he shall be free at home one year, and shall cheer up his wife which he hath taken.”
—Bible: Hebrew Deuteronomy 24:5.
“You dont hold any mystery for me, darling, do you mind? There isnt a particle of you that I dont know, remember, and want.”
—Noël Coward (18991973)
“They talk about a womans sphere,
As though it had a limit.
Theres not a place in earth or heaven.
Theres not a task to mankind given ...
Without a woman in it.”
—Kate Field (18381896)