62 The Law of Distribution [CH. v 



We shall suppose that in addition to these intermolecular forces, the 

 whole gas is under the influence of a permanent external field of force. No 

 assumption will be made about this field of force, except that the potential 

 energy of a molecule in the field is a single-valued function of the coordinates 

 of the molecule, this assumption being required to ensure conservation of 

 energy. 



Liouvilles Theorem. 



72. Let a conservative dynamical system of the most general kind be 

 specified by n coordinates q lt q z ... q n . Let the corresponding momenta be 

 denoted by pi, p 2 ...p n , these being given by 



dE 

 PS = 3 , etc ............................... (126), 



VQs 



where E is the energy of the system expressed as a function of the q's and 

 qs. We can represent this dynamical system in a space of 2n dimensions 

 corresponding to 2n variable coordinates 



Pi,P---Pn, qi, qa-'-qn ............... ............ (127). 



The equations of motion of the system expressed in the Hamiltonian 

 form are of the type 



dp g _ ?>E 



-~ 



These, therefore, are the equations of the trajectories in this generalised 

 space. Let us suppose the space filled with fluid, initially of density p. Let 

 Dp/Dt be the rate of increase of p as we follow any element of fluid in its 

 motion. In three dimensions, the hydrodynamical equation of continuity 

 can, with the usual notation, be put in the form 



In our present 2?i-dimensional case the last bracket must be replaced by 



where f is any one of the 2ra coordinates (127), and the summation extends to 

 all. This again may be replaced by 



'/ I J." \ 



3p a dq 8 J ' 

 and the general equation of continuity for our present space is therefore 



:0 (130). 



