If we assume that the wave is damped at large in .NET framework

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If we assume that the wave is damped at large
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u"( (0) = u"(O)
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2 2 [ e e u(O) e 1 (0)] - 1(0) ~~ - ~ -~-~n" m n u(O) 2m 2 n . u(O)
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(1-82)
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We note the appearance of nonlinear effects (terms in 1 2 ). For a monochromatic plane wave such as I(~) = a sin straightforward way:
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x"(~) =
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for instance, we can compute x" in a
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2e sin ~/2 u"(O)r - - a - - - e"
n' u(O)
+ -e [
n" [ 2 ~ 2 2ae' u(O) sin m n' u(O)] 2
ea + -(~ 4m
sin 2~)
(1-83)
The action is expressed as
I(~) =
dr'(m
+ eA'u)
= -mr -
n' u(O)
-mu(oo)'x-
f' d~' I(~'){e. f'
u(O) +
~ [J(~') m
I(O)]}
2 [ eA'u(oo)-~e A2] on' u 2m
(1-84)
In the last form we have used u(oo) defined by (1-82) under the assumption that I(~) vanishes at infinity (recall that n' u is conserved). From (1-84) we can obtain the conjugate momentum as
n" = -
aI(~) =
mu"(oo)
+ ~ [eA' u(oo) - ~ A2]
(1-85)
e- I<I g@
For certain potentials it is meaningful to define averages such that A = 0 [for instance if I@ where g is periodic and 1] --> 0]. We then derive from (1-85) that
(1-86) This last formula is to be interpreted with care because of the anisotropies arising from the plane wave. It shows, however, that a particle in a strong periodic field will respond to external perturbations with a larger inertia. Linear accelerators provide a typical instance of such a motion of electrons in a traveling wave.
1-2 SYMMETRIES AND CONSERVATION LAWS 1-2-1 Fundamental Invariants
We return to systems with finitely many degrees of freedom. The main task is, of course, to solve the equations of motion, with appropriate boundary conditions. General properties of the motion such as symmetries are helpful since they simplify
QUANTUM FIELD THEORY
the calculations. They can also be used to restrict the class of dynamical models. We have already seen examples of this type with Lorentz invariance. Symmetries may playa double role. They enable one to generate families of solutions from a given one if some transformations leave the dynamical equations invariant. Or they lead to the conservation of quantities such as charge, energy, momentum, etc. The deep connection between these two aspects is our present subject. We start from a very simple example. A nonrelativistic point particle moves in a force field deriving from a time-independent potential. The position and velocity at time t given initial conditions at time zero are, of course, the same as those at time t + r if the same initial conditions were given at time r. The problem is invariant under time translation. We also know that in such a case energy (i.e. the value of Hamilton's function) is conserved. Let us see how the two properties are related. Any function on phase space varies along the motion according to
Time-translation invariance is equivalent to the statement oHlot also has {H, H} = 0 it indeed follows that
dH dt
O. Since one
(1-87)
and energy is conserved. This simple remark is sufficient, for instance, to find the motion explicitly if the particle is restricted to move in one dimension. Alternatively, consider the action computed along the stationary trajectory leading from (qb t1) to (q2, t2)' Invariance under time translation means
J(q2' t2
+ r; q1, t1 + r) =
oJ _ oJ ot2 ot1
J(q2' t2; qb t1)
or in differential form (1-88)
Taking into account Eq. (1-12) this is indeed
(1-89)
The conservation law clearly follows from the existence of a continuous invariance group. Similarly, for space translations
J({qn
+ ah, t2; {qn + ah, t1) =
J({qnh, t2; {qnh t1)
Recalling again Eq. (1-12) and differentiating with respect to a, we get the conservation law of total momentum
(2: Pnh = (2: Pnh
(1-90)
The previous examples can be thought of as special cases of the formula giving the variation of the stationary action when an external parameter a is
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