| H = − | ∂S | . |
| ∂τ |
(6)
S is called by Hamilton the principal function; if its general form for any system can be found, the preceding equations suffice to determine the motion resulting from any given conditions. If we substitute the values of p1, p2, ... and H from (5) and (6) in the expression for the kinetic energy in the form T′ (see § 1), the equation
T¹ + V = H
(7)
becomes a partial differential equation to be satisfied by S. It has been shown by Jacobi that the dynamical problem resolves itself into obtaining a “complete” solution of this equation, involving n + 1 arbitrary constants. This aspect of the subject, as a problem in partial differential equations, has received great attention at the hands of mathematicians, but must be passed over here.
There is a similar theory Characteristic function. for the function
A = 2 ∫ Tdt = S + Hτ
(8)
It follows from (4) that