By Ronald A. Mann
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Additional info for The Classical Dynamics of Particles. Galilean and Lorentz Relativity
35) by a "Lagrange Multiplier" Αα(τ), add the equations together and replace the original Lagrangian L by a modified Lagrangian L: L = L + λ0(τ)/6(<7α, # α , τ), A = 1, 2, . . 36) L has the same magnitude as L. The modified Lagrangian is then substituted into the action and the variational procedure is performed as before except that the varied functions are now the qa(r) and the Λ6(τ), all of which are considered independent. 39) determine the motion of the system. What the multiplier technique actually does is to construct a Lagrangian that derives the constraint equations and thereby guarantees a motion in conformity with the constraints.
Quantities as measured in the lab frame are related to quantities as measured in the center-of-momentum frame and vice versa by the Lorentz transformation connecting the two frames. It is often simpler, however, to make use of the invariance of the scalar products of four-vectors to determine the relation between quantities. 126) where Ρμ' is the four-momentum as measured by the center-of-momentum frame and Ρμ is the four-momentum as measured by the lab frame. 127) K. Distribution Functions and Scattering Cross Sections Many experiments are designed to determine the distribution functions of physical variables.
D [_d_ IdG _d_ dx [ dqa \dqb dG\ . 34) B. 26) it was assumed that the qa were all independent. 26a). 25). , if the constraint conditions can be expressed as equations connecting the coordinates and the time, the equations of constraint may be used to eliminate the dependent variables and the variational procedure proceeds as before with the new set of independent functions. It often happens that the elimination procedure is either difficult to perform because of the intractability of the constraint equations or undesirable because it removes certain functions that one may for one reason or another prefer to keep.