Relations in General Dynamics. 27 



These lead at once to the equations 



~dp_ _SH 5? = BH ( 



■dt~ 3 g ' -dt " -dq ' • " " " w 



which are the same in form as the canonical equations, 

 except that on the left the differentiation with respect to 

 t is partial instead of total. In the first p is "d&fdq, and 

 is therefore a function of t, the a's, and k constants, and so 

 also is H ; in the second q is "d&fdp, and is, with H, a 

 function of t, the p's, and k constants. There are of course k 

 pairs of these equations. 



If in (4) the ^'s and the p's are replaced by their values 

 in terms of the 2/j constants and t, the equations hold without 

 alteration of form, for then they are precisely the canonical 

 equations, since partial differentiation of q and p with respect 

 to t then means the same thing as total differentiation. In 

 other words, the canonical equations assert that partial 

 differentiation of p (or of q), with respect to t, when the 

 quantity differentiated is expressed in terms of t and the 

 initial coordinates and momenta, is the same thing as partial 

 differentiation with respect to q of r3S/?3£ (or with respect 

 to p of BS'/90 when the subject of differentiation is expressed 

 in terms of t and the final coordinates and momenta. 



6. This remarkable connexion between the initial and the 

 final coordinates and momenta holds for differentiation with 

 respect. to other variables than t, e.g. we can choose any of 

 the a's or the b's. There are in fact a set of canonical 

 equations for every variable we thus select. 



For example, if we differentiate with respect to q the 

 equation [(7), § 4, above] 



Sh-** (1) 



where both sides are regarded as functions of the ^'s, the a's, 

 and t, we get 



~dp __ c^S "bbi / 9 \ 



"dai ~~ "dqbai " dq ' 



an equation which is comparable with 



y*--^ (,) 



and which retains its form when we express p as a function 

 of the initial coordinates and momenta and t, while b { is 

 expressed as a function of the final coordinates and momenta 

 and t. Thus, denoting differentiation, carried out on these 



