When the origin of the moving axes is also in motion with a velocity whose components are u, v, w, the dynamical equations are
| dξ | − rη + qζ = X, | dη | − pζ + rχ = Y, | dζ | − qχ + pη = Z, |
| dt | dt | dt |
(11)
| dλ | − rμ + qν − wη + vζ = L, | dμ | − pν + rλ- uζ + wξ = M, |
| dt | dt |
| dν | − qλ + pμ − vξ + uη = N. |
| dt |
(12)
To prove these, we may take fixed axes O′x′, O′y′, O′z′ coincident with the moving axes at time t, and compare the linear and angular momenta ξ + δξ, η + δη, ζ + δζ, λ + δλ, μ + δμ, ν + δν relative to the new position of the axes, Ox, Oy, Oz at time t + δt with the original momenta ξ, η, ζ, λ, μ, ν relative to O′x′, O′y′, O′z′ at time t. As in the case of (2), the equations are applicable to any dynamical system whatever. If the moving origin coincide always with the mass-centre, we have ξ, η, ζ = M0u, M0v, M0w, where M0 is the total mass, and the equations simplify.
When, in any problem, the values of u, v, w, p, q, r have been determined as functions of t, it still remains to connect the moving axes with some fixed frame of reference. It will be sufficient to take the case of motion about a fixed point O; the angular co-ordinates θ, φ, ψ of Euler may then be used for the purpose. Referring to fig. 36 we see that the angular velocities p, q, r of the moving lines, OA, OB, OC about their instantaneous positions are
p = θ̇ sin φ − sin θ cos φψ̇, q = θ̇ cos φ + sin θ sin φψ̇,
r = φ̇ + cos θψ̇,
(13)