4
4
L =
(m U µ U µ + q A µ U µ ).
(2.4)
µ=1
Here U µ and A µ are the four-vector velocity and electromagnetic
potential, respectively, given by
U µ = (γ v, iγc)
A µ = (A, iφ/c),
(2.5)
where v is the three-vector particle velocity, A is the magnetic
three-vector potential, and φ is the electrostatic scalar potential.
We have defined a quantity γ as
1
γ =
.
(2.6)
1 − v 2 /c 2
21
2.1. Relativistic classical mechanics
position. The reader is referred to the book by Goldstein [35] for
a thorough and detailed discussion of classical mechanics.
2.1.1 Hamilton’s principle of least action
We seek a general condition governing the motion of a particle
with charge q and rest mass m in external electric and magnetic
fields. We require that this condition be covariant with respect
to the Lorentz transformation of special relativity. This ensures
that the equations of motion have the same form in all frames of
reference in uniform motion with respect to one another. To this
end, following Goldstein, et. al. [35], we define a function L, called
the invariant Lagrangian, as
We notice from the form of (2.4) that the invariant Lagrangian L is
a sum of inner products of two four-vectors. It is straightforward to
show that the inner product of two four-vectors is invariant under
a Lorentz transformation. It follows that L is Lorentz invariant,
and has the same value in every uniformly moving reference frame.
The proof of this is left to the reader in the problems.
4
L =
(m U µ U µ + q A µ U µ ).
(2.4)
µ=1
Here U µ and A µ are the four-vector velocity and electromagnetic
potential, respectively, given by
U µ = (γ v, iγc)
A µ = (A, iφ/c),
(2.5)
where v is the three-vector particle velocity, A is the magnetic
three-vector potential, and φ is the electrostatic scalar potential.
We have defined a quantity γ as
1
γ =
.
(2.6)
1 − v 2 /c 2
21
2.1. Relativistic classical mechanics
position. The reader is referred to the book by Goldstein [35] for
a thorough and detailed discussion of classical mechanics.
2.1.1 Hamilton’s principle of least action
We seek a general condition governing the motion of a particle
with charge q and rest mass m in external electric and magnetic
fields. We require that this condition be covariant with respect
to the Lorentz transformation of special relativity. This ensures
that the equations of motion have the same form in all frames of
reference in uniform motion with respect to one another. To this
end, following Goldstein, et. al. [35], we define a function L, called
the invariant Lagrangian, as
We notice from the form of (2.4) that the invariant Lagrangian L is
a sum of inner products of two four-vectors. It is straightforward to
show that the inner product of two four-vectors is invariant under
a Lorentz transformation. It follows that L is Lorentz invariant,
and has the same value in every uniformly moving reference frame.
The proof of this is left to the reader in the problems.