Lorentz Force and Generalized Potential
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Vectorial
Approach
The
Lorentz force on a charged particle moving in an electromagnetic field is given
by
Substituting
the values of the electric and magnetic fields in terms of the Coulomb and
vector potential, respectively, i.e.
the
Lorentz force becomes
Recall
the vector identity
Letting a
= v and b = A we obtain
The
terms including spatial derivatives of the velocity are zero which leaves
Solving
for v´(Ñ´A)
gives
Substitute
this expression into Eq. (3)
The
first term on the right hand side is an example of what is known as a generalized
potential and usually labeled U, i.e.
The
second term is the convective derivative of A whose value is the
total derivative of A, i.e.
Substituting
the expressions in Eqs. (9) and (10) into Eq. (8) we obtain
Eq. (12)
can be rearranged to give
The canonical
or generalized 3-momentum, P, for a charged particle in an EM
field has the value
Therefore
in terms of P and U the Lorentz force in Eq. (1) can be written as
The time
rate of change of the canonical momentum is called the canonical 3-force
and will here be labeled F. Thus Eq. (1) now takes on the simple form
Analytical Approach
The
result obtained above in Eq. (15) can more readily be obtained using Lagrangian
mechanics. Start with the Lagrangian for a charged particle in an
electromagnetic field
If we
define the following quantities
Then the
Lagrangian can be expressed in a more familiar form
The canonical
momentum of a particle, p,
is defined in terms of the Lagrangian as
Inserting
these into Lagrange’s equations
we get
F
is the canonical force defined as the time rate of change of the
canonical momentum. It can readily be shown that p has the value
The
result in Eq. (21) is identical to the result obtained above in Eq. (15).
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