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23
2.1. Relativistic classical mechanics
related to the proper time τ by a Lorentz transformation, where
we assume the particle coordinate is zero in the particle rest frame.
Substituting (2.5) into (2.4), it follows that
L(x, v; t) = −m c
2
1 − v 2 /c 2 + q v · A(x, t) − q φ(x, t) (2.9)
in the lab frame. We have made use of the vector notation v · A
to express the inner product of the two three-vectors v and A. In
Cartesian coordinates this is v · A = v x A x + v y A y + v z A z .
The Lagrangian L is a scalar function of the position x, and the
velocity v. The time t is regarded as a parameter which uniquely
specifies a point along the particle trajectory. The position and
velocity depend implicitly on the time. Indeed, the central problem is to solve for this dependence. In the case where the electric
and magnetic fields vary with time, the electromagnetic potentials
have explicit time dependence. For static fields, these potentials
have no explicit time dependence. The Lagrangian therefore has
no explicit time dependence in this case.
The integral in (2.8) can be abbreviated as
t b
S ab =
L(x, v; t) dt.
(2.10)
ta
It is a scalar quantity with units of energy times time, or action.
The integral S ab is therefore known as the action integral. The expression (2.8) says that the action integral has an extremum for
the physically allowable trajectory. This trajectory exists among
many hypothetical trajectories, each displaced infinitesimally from
the physical trajectory. The expression (2.8) is known as Hamilton’s principle of least action.
Forming a Taylor expansion of the variation (2.7) in the lab frame,
and retaining only terms to first order in δx i and δv i , we find
3
t b
t b 4 ∂L
∂L
δ
L dt =
δx i +
δv i dt.
(2.11)
ta
ta i=1 ∂x i
∂v i
�
23
2.1. Relativistic classical mechanics
related to the proper time τ by a Lorentz transformation, where
we assume the particle coordinate is zero in the particle rest frame.
Substituting (2.5) into (2.4), it follows that
L(x, v; t) = −m c
2
1 − v 2 /c 2 + q v · A(x, t) − q φ(x, t) (2.9)
in the lab frame. We have made use of the vector notation v · A
to express the inner product of the two three-vectors v and A. In
Cartesian coordinates this is v · A = v x A x + v y A y + v z A z .
The Lagrangian L is a scalar function of the position x, and the
velocity v. The time t is regarded as a parameter which uniquely
specifies a point along the particle trajectory. The position and
velocity depend implicitly on the time. Indeed, the central problem is to solve for this dependence. In the case where the electric
and magnetic fields vary with time, the electromagnetic potentials
have explicit time dependence. For static fields, these potentials
have no explicit time dependence. The Lagrangian therefore has
no explicit time dependence in this case.
The integral in (2.8) can be abbreviated as
t b
S ab =
L(x, v; t) dt.
(2.10)
ta
It is a scalar quantity with units of energy times time, or action.
The integral S ab is therefore known as the action integral. The expression (2.8) says that the action integral has an extremum for
the physically allowable trajectory. This trajectory exists among
many hypothetical trajectories, each displaced infinitesimally from
the physical trajectory. The expression (2.8) is known as Hamilton’s principle of least action.
Forming a Taylor expansion of the variation (2.7) in the lab frame,
and retaining only terms to first order in δx i and δv i , we find
3
t b
t b 4 ∂L
∂L
δ
L dt =
δx i +
δv i dt.
(2.11)
ta
ta i=1 ∂x i
∂v i