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Momentum and energy follow from the general expressions
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and agree, of course, with (1-25). The free equation of motion is dp/dt generalized force J" = dp"/dr vanishes. The action along the trajectory 1(2,1) =
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Sl) = -
me 2[e2(t 2 - tl)2 -
(X2 -
XI)2] 1/2
is a Lorentz scalar. The standard relations
-~~-=E2=E
01(2,1) Ot2
01(2,1)
~~-=P2=P
mean that the relativistic momentum -oI/ox. = p. is a four-vector. An alternative form of the action may be used: 1= - f2 dr
~ [dx(r)]2
2 dr
(1-27)
where r is a priori an arbitrary parameter along the trajectory. The equation of motion obtained by requiring I to be stationary, m d 2x"/dr2 = 0, implies that r is proportional to proper time and can be chosen such that u 2 = (dx/dr)2 = e 2 In the following we shall set e = 1, unless otherwise stated.
One of the virtues of the Lagrange-Hamilton formalism is to suggest a wide class of transformations leaving the structure of the equations of motion invariant. These go beyond simple reparametrizations of the configuration space since they naturally mix positions and momenta. A transformation (p, q) +-->(p', q') (possibly time dependent) is canonical if there exists a function H' of p', q' (and perhaps t) such that in terms of the new variables the equations of motion also read
4' = op'
[3'= - - oq'
where indices have again been suppressed. A sufficient condition for this property is suggested by the least-action principle. We require that the differential forms p' dq' - H' dt and p dq - H dt differ at most by a total differential. In turn this
CLASSICAL THEORY
means the equality of the external derivatives of these forms.:
L dPi 1\ dqi i
L dpi
dqi - dR'
(1-28)
This condition implies that functions on phase space have equal Poisson brackets expressed in terms of old (q, p) or new (q', p') variables:
{f,g} =
L (Of
og _ of Og) = 0Pi Oqi Oqi 0Pi
L (Of
og _ of Og) opi oqi oqi opi
(1-29)
From (1-28) we can also derive equations for R' which turn out to form an integrable system by virtue of (1-29):
oR' L [OPk (OR Oqk) opi = k opi OPk - at oR' oqi
+ opi Oqk + at
oqi Oqk ot
Oqk (OR
OPk)]
(1-30)
L [OPk (OR _
k oqi Oqk
Oqk) ot
+ Oqk (OR + OPk)]
A glance at (1-29) and (1-30) enables us to recognize that a typical example of canonical transformation is given by the solution of the equations of motion where (q', p') correspond to initial data at time to and (q, p) to the phase space position at time t. It is clear that R' vanishes in this case and that Poisson brackets are left invariant. Assume that R is time independent. The 2N functions q', P' expressed in terms of q, p, and t are constants of motion. Eliminating t one obtains 2N - 1 (local) constants of motion. Only a small number of them can be extended as well-defined functions over all phase space. A theorem due to Poincare states that the latter are associated with symmetries of the motion. Canonical transformations are, of course, not limited to solutions of the equations of motion. Their set generates an infinite group. Finally we remark that restricting (1-28) at fixed time and taking the Nth external power of both sides shows that canonical transformations preserve the measure ili dPi 1\ dqi on phase space; this is Liouville's theorem.
1-1-2 Electromagnetic Field as an Infinite Dynamical System
Systems with infinitely many degrees of freedom are familiar in fluid mechanics, electromagnetism, solid-state physics, etc. We shall discuss here how the lagrangian formalism extends to the electromagnetic field. To obtain the Lagrange function we start from the field equations in the presence of fixed external sources. We use Heaviside's units (with the Coulomb force given by QQ' /4nr2) and take c = 1. In terms of the electric field E, magnetic induction B, charge and current density p and j, Maxwell's equations read
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