Hamiltonian Theory and Action Variables

Here we shall l ri( iiy cnnsidor the mcichanics of multiply periodic motions a.nd the corres]lending ( ua,ntum conditions. According to Hamilton, the motion of a, syst( . m is (h serihed completely by stating the energy as a function of t m co-ordinates and the momenta the so-called Hamiltonian function H q, . , . , p2 0- (1  

S rom these the theorem of the conservation of energy follows immediately for if we form the total difierential coefficient of H with respect to the time, and make nse of the equations of motion, we have 

Wk— Vk = is also constant as time goes on, owing to tlix coiLstancy dJ k 

A system is said to be singly or multiply periodic if the variables just defined can be found such that each rectangular co-ordinate is periodic m the quantities Wk i.e. can be represented as a Fourier series 

Ehrenfest has proved that the action variables are adiabatic invariants, i.e. that the quantities J can be quantised. We now postulate the quantum conditions 

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