(15)
This follows independently from (14), assuming the special variations δx = ẋdt, &c., and therefore δq1 = q˙1dt, δq2 = q˙2dt, ...
Again, if the values of the velocities and the momenta Reciprocal theorems. in any other motion of the system through the same configuration be distinguished by accents, we have the identity
p1q˙′1 + p2q˙′2 + ... = p′1q˙1 + p′2q˙2 + ...,
(16)
each side being equal to the symmetrical expression
A11q˙1q˙′1 + A22q˙2q˙′2 + ... + A12(q˙1q˙′2 + q˙′1q˙2) + ...
(17)
The theorem (16) leads to some important reciprocal relations. Thus, let us suppose that the momenta p1, p2, ... all vanish with the exception of p1, and similarly that the momenta p′1, p′2, ... all vanish except p′2. We have then p1q˙′1 = p′2q˙2, or
q˙2 : p1 = q˙′1 : p′2