Hence
| 2 | dT | = ṗ1q̇1 + ṗ2q̇2 + ... + ṗnq̇n + ṗ1q̈1 + ṗ2q̈2 + ... + ṗnq̈n |
| dt |
| = ( | ∂T | + Q1 ) q̇1 + ( | ∂T | + Q2 ) q̇2 + ... + ( | ∂T | + Qn ) q̇n + | ∂T | q̈1 + | ∂T | q̈2 + ... + | ∂T | q̈n |
| ∂q̇1 | ∂q̇2 | ∂q̇n | ∂q̇1 | ∂q̇2 | ∂q̇n |
| = | dT | + Q1q̇1 + Q2q̇2 + ... + Qnq̇n, |
| dt |
(18)
or
| dT | = Q1q̇1 + Q2q̇2 + ... + Qnq̇n. |
| dt |
(19)
This equation expresses that the kinetic energy is increasing at a rate equal to that at which work is being done by the forces. In the case of a conservative system free from extraneous force it becomes the equation of energy