Question

wCT S the particle's 4-velocity 12.13 Specalize the Darwin Lagrangian (12.82) to the interaction of two charged particles (m, q) and (m, qa). Introduce reduced particle coordinates X1X2, V -2 and also center of mass coordinates. Write out the Lagrangian in the reference frame in which the velocity of the center of mass vanishes and evaluate the canonical momentum components, p, aLlau (a) r = etc. action is known as the Breit interaction (1930). For system of interacting charged particles the complete Darwin can be written down by expanding a Lagrangian, correct to order 1/c2 inclusive, the free-particle Lagrangian (12.7) for each particle and summing up all the in- teraction terms of the form (12.81). The result is 1 mo+ 2 LDarwin mot 8c 2 rii o od n (12.82) 1 + (v + where r= x, - x, , is a unit vector in the direction x, the double summation indicates the omission of the (self-energy) terms, i = Although the Darwin Lagrangian has had its most celebrated application in the quantum-mechanical context of the Breit interaction, it has uses in the purely classical domain. Two examples are cited in the suggested reading at the end of X, and the prime the chapter. See also the problems.

Answer 1

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201j &c. as inst deived by dorusn But this s not Le af the formLL, FL Let the Lagrangian to dhe huncion L Carecion where s a then correspanding addition by H the hamittanian 1s se lated This sugests hat if we wish to exKpress he hami Hanian lerergy dhen ef and ntesms C.) .t) carrect inkerpretation A that mere -Ce) which indicades hat a funtdion the. hamitaniom f e dhe Coesdinales Coes pundine and he Canonical Corre ctian momanta P. So the Smat

be L to he lagrangtan hould expressed io befere subsdracting dems ofp and 4Re rom dAe lowest CYdear derm he hamil- could ecasd then, ene hamiltanian ef and -1anian enegy Tr eems teems ot Ce.1) aesult o destsed which can to ,) . e, Contributins louest esdes hami In the present example, the momenta, ot cancnical tanian Aeems Ho Vi 1,j 2 mi Canection dhe the to Smal desuibe FO lagrungian e Daruin dhe Pi, teams than the Sudtices to arpoxate adher the relatin (3) as oC) have Then ue C 2m;mj

rom uhich oblain the danwtn hamiltaniann we fi 2m; , j 2m,myCe hamiltanian to fist This is he arder charged of two Case particles Cm,a) now, fer dhe ond Lm,) Y =1,-, P, P P we get putir these 4 2t m m3 2m,