CLASSICAL THEORY in .NET framework

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CHAPTER
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CLASSICAL THEORY
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In this chapter we survey the lagrangian formalism in classical mechanics and its extension to systems with infinitely many degrees of freedom, emphasizing symmetries and conservation laws in their local form. Using electromagnetism as an example, we introduce Green functions and propagators. Elementary radiation problems are presented, and we close the chapter by studying the inconsistencies of self-interaction. As classical field configurations playa growing role in modern developments we will return to this subject in later chapters.
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1-1 PRINCIPLE OF LEAST ACTION 1-1-1 Classical Motion
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In classical mechanics the equations of motion follow from a principle of least action. If q stands for a finite collection of configuration variables, q == {q1,Q2, .. :,qN}, with q their velocities at time t, the action is defined as
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dt L[q(t), q(t)]
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(1-1)
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The Lagrange function L depends on the positions, velocities, and sometimes also explicitly on time for open systems, i.e., systems subjected to given external forces. The principle of least action states that among all trajectories q(t) which join q1 at time t1 to q2 at time t 2, the physical one yields a stationary value for the
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QUANTUM FIELD THEORY
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action. This stationary value is a unique minimum if q1, t1 and q2, t2 are close enough. The action is therefore to be considered as a functional of all regular functions q(t) satisfying the boundary conditions q(t 1) = q1, q(t2) = q2' If Q(t) is the actual trajectory, a nearby one is written q(t) = Q(t) + bq(t). Expanding the action in powers of bq as
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l(q)
I(Q)
+ Jtl
r dt bq(t) (Q)bq(t) + ... M
(1-2)
we express the principle by setting
(1-3)
To compare this expression with the Euler-Lagrange equations we note that since
bq(t) = :t bq(t)
(1-4)
the variation of (1-1) reads
r M r ['OL oL d ] Jtl dt bq(t) bq(t) = Jtl dt oq(t) bq(t) + oq(t) dt bq(t)
t2 t2
An integration by parts of the last term, taking into account boundary conditions, yields the familiar form
M _ oL d oL _ 0 bq(t) = oq(t) - dt oq(t) (1-5)
which is to be interpreted as a vector equation in the case of several degrees of freedom. In the simplest cases L is the difference between a kinetic part quadratic in the velocities and a potential part. The equations are unchanged if we add to L a total derivative with respect to time, and the action is only modified by contributions depending on the boundary conditions. This observation shows that the Lagrange function is not an intrinsic object to determine the motion and leads to a more abstract formalism, developed in particular by E. Cartan. The hamiltonian formulation is obtained through the definition of conjugate momenta
.) Pi= oL (q,q ~ uqi
(1-6)
Assume that we are able to invert this equation, i.e., to express the velocities in terms of the momenta and the positions. We shall see later what happens if the jacobian of this transformation vanishes. Hamilton's function is then obtained by a Legendre transformation
H(p, q) = Piqi(P, q) - L[q, q(p, q)]
(1-7)
CLASSICAL THEORY
where summation over dummy indices is understood. By differentiation
. [ qi
OL)] 04j -----:+ 04j ( Pj - -----:- dPi + [OL - - (OL - Pj)] dqi - 0Pi oqj Oqi Oqi oqj
We see, using the expression of conjugate momenta, that the Euler-Lagrange equations take the form . oH . oH p.- - q.-(1-8) ,- Oqi ,- Pi More generally the variation of a function defined on phase space (the space of the 2N variables p, q) is
df = of dt ot
+ oH of _ oH of
0Pi Oqi Oqi 0Pi
= of
+ {H f}
(1-9)
We have introduced the Poisson bracket notation
{j, g} = of og _ of og
, 0Pi Oqi Oqi 0Pi
(1-10)
It follows from (1-9) that a function without an explicit dependence on time, and such that its Poisson bracket with H vanishes, is a constant of the motion. It is remarkable that Hamilton's Eqs. (1-8) also follow from a principle of stationarity. Indeed, let us insert
L(q, 4) dt
P dq - H(p, q) dt
in (1-1) so that the action reads
It2 P dq t,
H dt
(1-11)
and can be thought of as a functional of 2N independent functions q(t) and p(t). Assume again that q(td = ql, q(t2) = q2 without any restriction on p(td or P(t2)' We then have
[bP(4 -
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