| d | (T + V) = 0, or T + V = const., |
| dt |
(20)
in virtue of (13).
As a first application of Lagrange’s formula (11) we may form the equations of motion of a particle in spherical polar co-ordinates. Let r be the distance of a point P from a fixed origin O, θ the angle which OP makes with a fixed direction OZ, ψ the azimuth of the plane ZOP relative to some fixed plane through OZ. The displacements of P due to small variations of these co-ordinates are ∂r along OP, r δθ perpendicular to OP in the plane ZOP, and r sin θ δψ perpendicular to this plane. The component velocities in these directions are therefore ṙ, rθ̇, r sin θψ̇, and if m be the mass of a moving particle at P we have
2T = m (ṙ2 + r2θ;̇2 + r2 sin2 θψ;̇2).
(21)
Hence the formula (11) gives
| m (r̈ − rθ̇2 − r sin2 θψ̇2) | = R, |
| d/dt (mr2θ̇) − mr2 · sin θ cos θψ̇2 | = Θ, |
| d/dt (mr2 sin2 θψ̇) | = Ψ. |
(22)
The quantities R, Θ, Ψ are the coefficients in the expression R δr + Θ δθ + Ψ δψ for the work done in an infinitely small displacement; viz. R is the radial component of force, Θ is the moment about a line through O perpendicular to the plane ZOP, and Ψ is the moment about OZ. In the case of the spherical pendulum we have r = l, Θ = − mgl sin θ, Ψ = 0, if OZ be drawn vertically downwards, and therefore