(1)
Now let x + δx, y + δy, z + δz be the co-ordinates of m in any arbitrary motion of the system differing infinitely little from the actual motion, and let us form the equation
Σm (ẍδx + ÿδy + z¨δz) = Σ (Xδx + Yδy + Zδz)
(2)
Lagrange’s investigation consists in the transformation of (2) into an equation involving the independent variations δq1, δq2, ... δqn.
It is important to notice that the symbols δ and d/dt are commutative, since
| δẋ = | d | (x + δx) − | dx | = | d | δx, &c. |
| dt | dt | dt |
(3)
Hence
| Σm(ẍδx + ÿδy + z¨δz) = | d | Σm (ẋδx + ẏδy + z˙δz) − Σm (ẋδẋ + ẏδẏ + z˙δz˙) |
| dt |