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Appendices
A.1 Euler’s equations
A.1.1 Angular momentum and the equations of angular motion
We define the relative angular momentum, h, of a system of particles comprising a
body by
h = Σr × (mvr) (A.1.1)
where r is the position vector of a particle of mass m and vrel is the velocity of the
particle relative to the origin, which may be that of a moving frame. The summation
is taken over all the particles of the system. Then
dh/dt = Σ(dr/dt) × (mvrel) + Σr × (mdvrel/dt) (A.1.2)
Now, if v is the absolute velocity of the particle and v0 the velocity of the origin
of the moving frame,
vrel = dr/dt = v – v0
and dvrel/dt = dv/dt – dv0/dt
Since
(dr/dt) × vrel = (dr/dt) × (dr/dt)
the first term of eqn A.1.2 is zero; therefore,
dh/dt = Σ r × (mdv/dt) – Σr × (mdv0/dt)
But mdv/dt = F, where F is the resultant external force acting on the particle.
Hence
Σr × (mdv/dt) = Σr × F = T
where T is the moment of the external forces about the origin.
Also, since dv0/dt is constant over the system of particles,
Appendices 361
Σr × (mdv0/dt) = Mrg × a0
since Σmr is the mass moment of the system relative to the origin, M being the total
system mass, rg the position vector of the centre of gravity of the system, and a0 the
acceleration of the origin of the moving frame. Thus, finally,
dh/dt = T – Mrg × a0 (A.1.3)
We can obtain an alternative formula by defining the absolute angular momentum
by
H = Σr × (mv) (A.1.4)
where, as above, v is the absolute velocity of the particle. By an argument similar to
that above we obtain the equation
dH/dt = T – v0 × Mvg (A.1.5)
where vg is the velocity of the centre of gravity of the system of particles.
We now expand the general vector eqns A.1.3 and A.1.5 in terms of components
measured in the chosen frame of reference.
From the defining equations A.1.1 and A.1.2 we find that
H = h + Σr × mv0 = h + Mrg × v0
For subsequent applications we shall choose axes which are fixed in the body, as
it is clearly convenient that the inertial properties of the body should remain constant
with time. We can then write for the particle velocity vr
　
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