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:
“I am trembling:
I am suddenly charged with their language, these six strings,
Suddenly made to see they can declare
Nothing but harmony, and may not move
Without a happy stirring of the air
That builds within this room a second room....”
—Philip Larkin (1922–1986)
“Experience is never limited, and it is never complete; it is an immense sensibility, a kind of huge spider-web of the finest silken threads suspended in the chamber of consciousness, and catching every air-borne particle in its tissue.”
—Henry James (1843–1916)
“And through the field the road runs by
To many-towered Camelot;”
—Alfred Tennyson (1809–1892)