To within second order 
6x(T) = Ax, - £ AT (1.34) 
Within this order of approximation, (1.33) reduces to the following: 
ae [ Sian! q coll Ope AP bs qf du do 
+ [q' ' 
fey CE) sh Cae WE Aas, 
AES) ' 
qe [l= @" Cit) ze sp CG. ar iu My + F(T, Xp uy) | AT 
If gq is now determined so that the coefficient 
of 6x vanishes, 
This is the same differential Equation (1.23) that p satisfied; our 
Lagrange multiplier can then be identified with p 
a> (1.35) 
Moreover, since the relationship must hold independent of du, 
= ' oo 
FY pif, 0 (1. 36) 
Since there are no longer restrictions on Ax and AT, 
(is Ss) 
Introducing the Hamiltonian (1.25) yields 
18 
