506 Prof. P. Ehrenfest on Adiabatic Invariants 



0. For systems of: one degree of freedom we have 

 according to (7) : 



?-f 



dq dp = adiabatic invariant. . . (8) 



i. e. for systems of one degree of freedom the area enclosed 

 by the phase-curve in the q-p diagram is an invariant (in 

 this case there exists no other invariant which is independent 

 of the former). 



D. A theorem by P. Hertz (1910) *. Imagine a system 

 of n degrees of freedom and consider any motion belonging 

 to a set of given values a l0 , a 2o . . . of the parameters. The 

 corresponding phase-curve in the 2n-dimensional q-p space 

 lies wholly on a certain hypersurface of constant energy, 

 e(q, p, a Q ) = e , which encloses a certain 2n-dimensional 

 volume i 



V =f... {dq...cCq n . .... (9) 



An adiabatic reversible affection a — ■*a l : firstly, changes 

 the value of the energy (by the amount of the work performed 

 on the system) ; secondly, alters the form and position of 

 the hypersurfaces e(q,p, a) —const. Let the volume enclosed 

 by that surface of constant energy on which lies the phase- 

 curve of the system after the affection be V. Then the 

 theorem of P. Hertz asserts that 



V=V (10) 



For systems of one degree of freedom (]0) and (8) coincide, 

 for more degrees of freedom this is not the case. 



§ 6. Connexion with the formulce of the Theory of Quanta^ 

 as proposed by Planck, Debye and others for systems of one 

 degree of freedom. 



Planck's hypothesis of energy elements (1901) asserts 

 that an harmonically vibrating resonator of frequency v 

 can contain only the following amounts of energyf : 



e=0,hv Q ,2hv (11) 



Hence the adiabatic invariant of the resonator may take 

 only the values : 



- = — =((dqdp = 0,h,2h.. . (12) 



Let us consider a resonator with a non-linear equation of 



motion : . /».«.*•> /io\ 



q = -(v *q + ai q 2 -\-a 2 q s ...).. . (13) 



* P. Hertz, Ann. d. Fhys. vol. xxxiii. (1910) pp. 225, 537, § 11. 

 t Comp. note *, § 2, p. 503. 



