(( UNIVERSITY J 



ELEMENTARY PRINCIPLES IN 

 STATISTICAL MECHANICS 



CHAPTER I. 



GENERAL NOTIONS. THE PRINCIPLE OF 

 OF EXTENSION-IN-PHASE. 



WE shall use Hamilton's form of the equations of motion for 

 a system of n degrees of freedom, writing q l , . . ,q n for the 

 (generalized) coordinates, qi , . . . q n for the (generalized) ve- 

 locities, and 



for the moment of the forces. We shall call the quantities 

 F l9 ...F n the (generalized) forces, and the quantities p 1 . . . p n , 

 defined by the equations 



Pl = ^- t p 2 = ^, etc., (2) 



dqi dq 2 



where e p denotes the kinetic energy of the system, the (gen- 

 eralized) momenta. The kinetic energy is here regarded as 

 a function of the velocities and coordinates. We shall usually 

 regard it as a function of the momenta and coordinates,* 

 and on this account we denote it by e p . This will not pre- 

 vent us from occasionally using formulae like (2), where it is 

 sufficiently evident the kinetic energy is regarded as function 

 of the g's and ^'s. But in expressions like de p /dq 1 , where the 

 denominator does not determine the question, the kinetic 



* The use of the momenta instead of the velocities as independent variables 

 is the characteristic of Hamilton's method which gives his equations of motion 

 their remarkable degree of simplicity. We shall find that the fundamental 

 notions of statistical mechanics are most easily defined, and are expressed in 

 the most simple form, when the momenta with the coordinates are used to 

 describe the state of a system. 



