| d | ( | ∂T | ) − | ∂T | = Qr. |
| dt | ∂q̇r | ∂qr |
(11)
If we put r = 1, 2, ... n in succession, we get the n independent equations of motion of the system. These equations are due to Lagrange, with whom indeed the first conception, as well as the establishment, of a general dynamical method applicable to all systems whatever appears to have originated. The above proof was given by Sir W. R. Hamilton (1835). Lagrange’s own proof will be found under [Dynamics], § Analytical. In a conservative system free from extraneous force we have
Σ (X δx + Y δy + Z δz) = −δV,
(12)
where V is the potential energy. Hence
| Qr = − | ∂V | , |
| ∂qr |
(13)
and
| d | ( | ∂T | ) − | ∂T | = − | ∂V | . |
| dt | ∂q̇r | ∂qr | ∂qr |