27 
1920-21.] The Equations of Motion of a Single Particle. 
system will have the same form independently of the system of coordinates 
used, and all forces will appear as geometrical constraints. 
§ 2. For the present we shall consider only the motion of a single particle 
whose position is defined by rectangular coordinates x v x 2 , x 3 , the time 
variable being denoted by x 4 and the potential function by V. The 
Newtonian equations of motion are then 
( 4 ) 
= dx% {h= 1, 2, 3), d%= 0, 
OXi, 
and our problem is to determine a symmetric matrix \js = (a rs ) such that 
the equations 
(5) d <2 x k +^ j | \ 
or the equivalent set 
(5') . . 
dx i dx i — 0, 
(*= 1, 2, 3, 4) 
* 3 
k 
dxidxj = 0, (k= 1, 2, 3, 4) 
are approximately satisfied when (4) is satisfied, and vice versa. To 
determine the conditions for this, we substitute the values of d 2 x k from (4) 
in the first three equations of (5'), thus obtaining 
(6) 
0V 
ddij, 'da at. da. 
dx n dXn dxj 
so that the term under the 2 on the right is zero except when i and j are 
both 4. It follows immediately that the part of the matrix whose sub- 
scripts do not exceed three, i.e. the part which refers to the subspace 
x v x 2 , x 3> is independent of x v x 2 , x 3 ; and therefore, since \/s is symmetric, 
there is a real orthogonal transformation, with coefficients independent of 
x v x 2 , x 3 , which reduces this part of \fs to the main diagonal. We may 
therefore assume without loss of generality that a ij = 0 , (i,j = 1 , 2, 3; i-^j) 
—% 4" 
and da 4i /dXj = 0 (i } j = l, 2, 3). Further, since ^ 
= 0 (i, 4), we have 
and similarly 
t da i1c da H = Q 
dxj dx i dx k 5 
ca>j 4 da^ k da k4 
dx k dx 4 dx^ ’ 
so that da ile /dx i = 0 (showing that the coefficients of the orthogonal trans- 
formation used above are also independent of x 4 ) and 
so that we may set 
(7) . . . 
da i4 
dx k 
dajg 
dx, 
0A 
(i, k=f= 4) 
a n = ( z== fj 2, 3). 
