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Integrating by parts the p(d/dt) bq term we obtain
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M . oH bp(t) = q - op
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bq(t)
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By setting the variation M action I can also be written
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0 we therefore recover Eqs. (1-8). Note that the
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Hence p's and q's playa similar role in the action.
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Let us now compute the action 1 along the stationary trajectory. We assume (q2, t 2) close enough to (qb td so that the latter is unique and use the notation 1(q2, t2; qb t1) or even the shorter form 1(2, 1).
It is by no means infrequent to find several trajectories joining two points in a given time. For an harmonic oscillator, for instance, infinitely many trajectories return to the origin after half a period.
We can easily verify that
M (q2, t 2 ; q1, t1)
where
(P2 bq2 - H2 bt 2) - (P1 bq1 - H1 btl)
H2 = _ 01(2,1)
(1-12)
P2 = 01(2, 1) Oq2
If the Lagrange function depends explicitly on some parameter r:t., we have, furthermore,
-d 1(q2, t2 ; q1, t 1) =
it2 dt oL [a, Q(t, a), Q(t, a)] .
:l va
(1-13)
where the derivative on the right-hand side is taken on the explicit dependence of L on a. As an application of this result, let us add to a given Lagrange function a term qF(t) which will contribute an external driving force F(t) to the equations of motion. The action will become a functional of F(t), and for t1 < t < t2 we obtain from (1-13)
M(2,1)
bF(t)
Q(t)
(1-14)
where Q(t) stands for the trajectory in the presence of F (we can also evaluate both sides for F = 0). This is not a priori the best method to solve the equations of motion, but it illustrates the method of generating functions, a procedure occurring frequently in the following.
Consider, for instance, a particle of mass m constrained to move along a one-dimensional axis under the action of a given force F(t). Its Lagrange function is
L(q, 4) =
m4 2
2 + qF(t)
(1-15)
The solution of the equation of motion can be written
Q(t)
ql(t2 - t) + q2(t - td t2 - tl
f" J"
dt' G(t, t') F(t')
(1-16)
= tl
where the Green function G(t,t'), symmetric in the interchange of t and t' and vanishing at t and t = t 2, is a solution of
-2 G(t, t') = b(t - t')
(1-17)
Here b(t - t') is Dirac's point distribution, the derivative of the step function 8(t - t'). We find
G(t, t') = - - - [(t2 - t)(t' - td8(t - t') t2 - tl
+ (t2
- t')(t - tl)8(t' - t)]
(1-18)
CLASSICAL THEORY
The value of the action along the trajectory reads
f.t. dt f.t. dt' F(t)G(t, t')F(t')
(1-19)
We verify that the functional derivative with respect to F yields the trajectory. If we set F recover
0 we also
8/(2,1) qz - ql ---=m - - - =pz 8qz tz-t l
--= --
8/(2,1) 8tz
m (qZ _ qr)Z -2 tz - tl
-H(pz,qz)
(1-20)
As a second example let us recall how this formalism applies to a free relativistic particle. Let x and v be the position and velocity of the particle written q and 4 up to now. The space-time interval reads (1-21) where r is the proper time along the trajectory dr light. In four-vector notations
x" == (ct, x) and
(1 - vZ/CZ) liZ dt
J1. = 0, 1,2,3
= y-I dt
and c is the velocity of
with the metric tensor
0 ~ x
0 -1 0 0
0 0 -1 0
(1-22)
ct sinh ()(]
used to lower or raise the Lorentz indices. The interval ~sz is invariant under Poincare transformations, combining translations (a) and homogeneous Lorentz transformations (1\):
A Lorentz transformation can be factorized as a product of an ordinary rotation and a boost along a direction n : x'
+ n[(cosh()( -1)x n -
(1-23)
ct cosh ()( - n . x sinh ()(
where v = c tanh ()(n is the velocity of the moving frame. The four-velocity
u"= d;=
(dt dX) dt c dr' dr = dr (c,v)
(1-24)
is a time-like vector of constant length c. Its derivative with respect to proper time dZx"/dr z orthogonal to u", is therefore space-like. For a free particle the four-momentum satisfies
du"/dr,
(1-25)
QUANTIJM FIELD THEORY
Note that p = Ev/e 2 Let us now find the corresponding Lagrange function using as principles (1) the relativistic in variance of the action and (2) the occurrence of only first derivatives of the position in L. The last requirement is, of course, borrowed from nonrelativistic mechanics and means that position and velocity should be sufficient to prescribe the motion. It may be relaxed under more general circumstances to comply with the finite velocity of signals in a relativistic context. In the case of a free particle the irrelevance of the origin of space and time coordinates implies, furthermore, that L should only depend on first derivatives and that in the limit Iv I e we should recover L = ~mv2 up to a total derivative in time. It is clear that L = a ds/dt satisfies (1) and (2) with the constant a given by the nonrelativistic limit. We thus have L
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