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2 ' ) Linearity of V in the # 's gives, on the other hand, a transformation of the momenta between themselves fc ; Pa Ar leads to J>*=/*(pi, Pa - (11) Tt appears selves, the if ) -- the variables of the one kind transform new among them- variables of the second kind will be linear functions of the old variables of the second kind, the coefficients of be determined functions of the variables of the free first which will kind, and the terms arbitrary functions of the variables of the first kind.
Ft=ftIt therefore interchanges co-ordinates and momenta. A transformation frequently employed is given by THE MECHANICS OF THE ATOM 34 it leads to (15) y=V2p cos q and transforms the expression q*+p 2 into 2p. The somewhat more general function m V = ~o>j a cotg (16) 2i gives * (16') p=V'2nia)p cos q and transforms mco 2 2 2 into co/J. We shall illustrate now, by means of an example, how the canonical substitutions can be used to integrate the equations of motion. Linear Harmonic Oscdlalor. In this case where q denotes the displacement, Introducing tho m the mass, and # the elastic constant.
The volume in the phase space is consequently invariant with respect to a canonical transformation. 8. The Hamilton-Jacob! The idea underlying the method Differential Equation which is so parof atomic mechanics to suited the (just as it is to ticularly problems those of celestial mechanics) will be clear from the example of the of integration THE THEORY OF HAMILTON AND JACOBI oscillator given in 7. appears very awkward in this it Although 37 on the other hand, it is powerful enough to lead to the end even for some quite complicated (particularly periodic) required case, yet, motions.