164 
Proceedings of the Royal Society of Edinburgh. [Sess. 
XIV.— On General Dynamics: — I. Hamilton’s Partial Differential 
Equations and the Determination of their Complete 
Integrals. By Professor A. Gray, F.R.S. 
(MS. received February 19, 1912. Read February 19, 1912.) 
The present paper contains the first part of a series of notes on general 
dynamics which, if it is found worth while, may be continued. In § 1 
I have shown how the first Hamiltonian differential equation is led up to 
in a natural and elementary manner from the canonical equations of 
motion for the most general case, that in which the time t appears 
explicitly in the function usually denoted by H. The condition of 
constancy of energy is therefore not assumed. In § 2 it is proved that 
the partial derivatives of the complete integral of Hamilton’s equation 
with respect to the constants which enter into the specification of that 
integral do not vary with the time, so that these derivatives equated to 
constants are the integral equations of motion of the system.* 
In § 3 the alternative form of differential equation, also given by 
Hamilton, and the equations which flow from it, are dealt with in a 
similar manner ; while § 4 contains some remarks on the dual system 
thus obtained and its applications. 
1. Derivation of the first 'partial differential equation from the 
canonical equations. — Let T denote the kinetic energy of a system of 
connected particles, V the potential energy, and H as usual the value of 
the expression 'Z(pq) — T + V. Here q denotes any one of the k 
generalised co-ordinates which with the equations of connection define 
the configuration of the particles at time t, and p denotes the corresponding 
element of generalised momentum. V is the potential energy, or more 
properly the function of q v q 2 , . . . ., q k from which the generalised forces 
at time t are given by the partial derivatives — 0V jdq v — 0V /dq 2 , . . . ., 
— dV/dq k . In the general case here considered V may be also an explicit 
function of the time t. 
The complete solution of any dynamical problem regarding such a 
system gives 
9\—f\{ a \) • • 
• -5 ^2’ ' * 
. k ty 
q 2 =f 2 (a v a 2 , • . 
. ., a k , b 2 , . . 
■ ; K t) 
- . 
• • (i) 
9k = B"p ®2> * • 
• •) ® k ■> ^ ’ 
■ ; K t) j 
• 
* This discussion was suggested by a passage regarding a particular case in Darboux, 
Theorie des Surfaces, t. ii., § 563. 
