General Multiply Periodic Systems. Uniqueness of the Action Variables

Hitherto wc have applied the quantum theory only to mechanical systems whose motion may be calculated by separation of the variables. We proceed now to deal in a general manner with the question of when it is possible to introduce the angle and action variables wk and Jfc so admirably suited to the application of the quantum theory. For this purpose it is necessary, in the first place, to fix the J s by suitable postulates so that only integral linear transformations with the determinant 1 are possible for it is only in such cases that the quantum conditions (1) Jk=nkh 

Generalising our former considerations, we fix our attention on mechanical systems1 whose Hamiltonian functions H(gq, pt. . . ) do not involve the time explicitly. We assume further that it is possible to find new variables wk, Jfc derived from the qk, pk by means 

1 The following conditions according to J. M Burgers, Hcl Aloommodcl van Rutherford-Bohr (Dies. Leyden), Haarlem, 1918, 10. 

It follows from this that the wk s arc linear functions of the time, and that the 3ka are constant. The functions qk(w1.. . w,) possess a periodicity Lattice in the w-spacc, the cells of the lattice being cubes with sides 1. 

Now it may be easily shown that the quantities J (apart from being indeterminate to the extent of a linear integral transformation with the determinant J l) fire not yet uniquely determined by the two conditions (A) and (B). 

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General Multiply Periodic Systems. Uniqueness of the Action Variables... 

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