From: bds@ipp-garching.mpg.de (Bruce Scott TOK )
Newsgroups: sci.physics.plasma
Subject: Re: lagrangian
Organization: Rechenzentrum der Max-Planck-Gesellschaft in Garching


Richard Logan (rjl@honeywell.com) wrote:

: Can someone please post the lagranian for a charged particle in a
: nonuniform magnetic field? Thanks in advance.

The general action integral for a charged particle in an electromagnetic
field is given by

S = \int (e/c) dx^a A_a

where \int is the integral along the particle's world line, x is the
particle's four-position, and A is the four-potential of the
electromagnetic field. The Lagrangian is given by

S = \int d\tau L, L = (e/c) u^a A_a,

where \tau is the proper time and u is the four particle's velocity.

If you want to do this nonrelativistically, set \tau --> t and write

S = \int dt L, L = (e/c) v^a A_a - e \phi,

where now the space x^a is three-space, v^a is the particle's velocity,
and A and \phi are the magnetic vector potential and electrostatic
potential.

If you apply the variation principle to S, you will get the Lorentz
force for the particle's motion.

If you want to specify the magnetic field, you have to convert A_a to
B^a by taking the curl,

B^a = \eps^{abc} \grad_b A_c,

where \eps is the totally antisymmetric pseudotensor of rank three,
normalised such that its elements in a space with a Cartesian metric are
1, 0, or -1, and \grad is the covariant derivative operator. Since no
gradient operator appears in the Lagrangian, you are pretty much stuck
with using A, so the best way to proceed is to specify B and then
calculate A. This is what every text I've ever seen does. For example,
if (x,y,z) are Cartesian coordinates, and if you have a magnetic field
in the z-direction whose strength varies in x, then you have A in the
y-direction with dependence on x. You can proceed to more general
fields from there.

If you would like to see a write up on how to do this, take a look at

http://www.rzg.mpg.de/~bds/lectures/field-particle.html

It is pretty much Landau-Lifshitz material, but collected into a smaller
space (and with special attention on how to set the signs and units).

--
Mach's gut!
Bruce Scott The deadliest bullshit is
Max-Planck-Institut fuer Plasmaphysik odorless and transparent
bds@ipp-garching.mpg.de -- W Gibson