Chapter 14
Dynamics
14.1
Relativistic Action
As stated in Section 4.4, all of dynamics is derived from the principle of
least action. Thus it is our chore to find a suitable action to produce the
dynamics of objects moving rapidly relative to us. It will be advantageous
if that action possess the maximum amount of symmetry. This will produce
the largest number of conserved quantities which in turn will simplify the
analysis. In other words, in addition to having the usual symmetries of space
and time translation, it would be nice to have the action be symmetric
under Lorentz transformations.
Remember that the classical actions are
not symmetric under Galilean transformations but are invariant instead, see
Section 5.4.4. This will expand the set of conserved quantities available for
the solution of dynamical problems.
In the following sections, we will be
more careful in our handling of the notation and remember that there are
three spatial directions, i. e.
the position is
x
. Where it is unimportant for
the interpretation, we will suppress the vector designation.
14.1.1
The Action for a Free Particle
In order to discover the action for rapidly moving particles, we should look at
simple situations. For the free particle, we know what the natural trajectory
in space time is – a straight line. We want this to be the least action. In
addition, if we want the set of Lorentz transformations to be a symmetry for
for this action, we should construct it from form invariants of the Lorentz
Transformations, see Section
??
, and Section 12.5. For timelike trajectories,
the form invariant that characterizes the trajectory is the proper time. The
action for the free particle should be dependent on the proper time.
The
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CHAPTER 14.
DYNAMICS
simplest possibility is linearity. Since action has the dimensions of an energy
times a time, we have to multiply the proper time by something with the
dimensions of an energy. Fortunately, the relevant dimensionful parameters
are available. One is the mass of the particle. In fact, when you think about
it this is the definition of mass. Well, actually only that mass that is called
the inertial mass.
We will expand on this idea in Chapter 15 and below
in Section 14.3.
Also, since in the case of the free particle, the straight
trajectory is the longest worldline between two events, see Section 12.7, the
action should be proportional to the negative of the proper time.
In this
way, the greatest proper time will correspond to the least action. The unique
combination that we have been led to is
S
(
x
0
, t
0
, x
f
, t
f
;
trajectory
)
=

mc
2
τ
(
x
0
,t
0
,x
f
,t
f
;
trajectory
)
(14.1)
=

mc
2
x
f
,t
f
trajectory,x
0
,t
0
Δ
τ
(14.2)
Place Figure Here
This form is inappropriate for the interpretation of an action since it is
not time sliced. To transform from segment slicing which is what we have
in Equation 14.2 to time slicing, we use the fact that we can relate proper
time intervals to coordinate time intervals as
Δ
τ
=
(
Δ
t
)
2

(
Δ
x
)
2
c
2
. If we
now factor out the (
Δ
t
)
2
and realize that the velocity in space time is the
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 Spring '07
 Anderson
 Energy, Mass, Special Relativity, Lorentz Transformations, tra jectory

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