4. THE DYNAMICS OF THE EPIGENETIC SYSTEM 37 



simple, being a sum of// integrals all having the same functional form. This 

 is analogous to the case of the ideal gas, where there are n molecules, each of 

 which has an energy expression — i.e. an integral. As a pattern of strong 

 interactions between components is introduced, G will get more complex and 

 a decomposition into // dependent functions will no longer be possible. 



In order to obtain equations (17) from the general integral G, we can use 

 the following equations which are easily verified : 



1 



(20) 



These relations are formally identical with Hamilton's equations in dynamics, 

 where .v,- corresponds to momentum, /?,•, and j^,- corresponds to position, 

 qi. Our set of 2// first-order equations are therefore already in "Hamiltonian" 

 form, and there is no need to use the transformation theory required in dyna- 

 mics to define a set of generalized momenta which are conjugate to the corre- 

 sponding variables of position. It would, of course, be possible to complete 

 the analogy between the set of equations (10) for / = 1, 2, , . ., // and Newton's 

 equations of motion. To do this, observe that 



d'^yi dxi 



so that we can write 



dt^ 



'<^.-') 



This second-order equation is now analogous to the equations derived by 

 Newton to describe the motion of a particle, expressed in terms of acceleration 

 and force. We could then proceed from this point with the Hamiltonian 

 argument, ending up with a function identical with G except for the coefficients 

 of the variables. Such a procedure is artificial and unnecessary, since the 

 equations (17) give immediately a Hamiltonian function, G, with which we can 

 construct a statistical mechanics. 



However, it is of some interest to show that the equations (20) are compre- 

 hended under a variational principle similar to Hamilton's principle in 

 mechanics. Consider the function 



^ = i[s A',J>/-S i/J,]-(7(x,>;) (x, = ^, y, = ^-^ 



The vanishing of the variation in the time-integral of Zl, with fixed end-points, 

 viz. 



h 



S j Adt = 



