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Let us consider a rigid body which is rotating about a stationary point P fixed in the body. The angular momentum of the body about point P is

L_P = Σ r_i × m_i v_i

where, r_i and v_i denote the position and velocity vectors of ith particle with respect to point P. (We use symbols r_i and v_i instead of for simplicity of notation).

Since the magnitude of r_i is constant, velocity v_i of the particle arises solely from the rotation of the rigid body about P, that is, we have

v_i= × r_i

Hence, we get

L_P = Σ m_i r_i × ( × r_i)

= Σ m_i [ (r_i . r_i) – (r_i . ) r_i]

Or, L_P = Σ m_i r_i² – Σ m_i r_i (r_i . )

Let us now put up a Cartesian co-ordinate system (XYZ) rigidly attached to the body, with origin at P; that is, this co-ordinate system rotates with the body (and translates if P translates). Taking the components of r_iand vectors in this system, we have

r_i = x_i i + y_i j + z_i k

and, = ω_x i + ω_y j + ω_z k

therefore, we get,

(L_P)_x = ω_x Σ m_i r_i² – Σ m_i x_i (x_i ω_x + y_i ω_y + z_i ω_z)

= ω_x Σ m_i (r_i² – x_i²) – ω_y Σ m_i x_i y_i – ω_z Σ m_ix_i z_i

= ω_x I_xx + ω_y I_xy + ω_z I_xz

Similarly,

(LP)_y = ω_x I_yx + ω_y I_yy + ω_z I_yz

and, (L_P)_z = ω_zx I_zx+ ω_zy I_zy + ω_zz I_zz

Note the following points in the above analysis:

(i) It shows that the angular momentum vector L_P is not parallel to angular velocity vector , in general.

(ii) The XYZ axes are rigidly fixed with the body. These are therefore called body-axes, in contrast to space-axes of the external, inertial, co-ordinate system at O.

(iii) Since the axes (XYZ) are fixed with the body and rotates along with it, the components x_i, y_i, z_i do not change with time. Therefore, the coefficients of inertia are constants and are defined with respect to body axes.

(iv) On the other hand, the vector , in general, may change both in magnitude and direction with time. That is, in general, the direction of instantaneous axis of rotation changes with time and the body may be rotating around different directions from moment to moment. Hence, values of ω_x, ω_yand ω_z are not necessarily constant.

Thus, the general dynamics of a rigid body is quite complicated with nine coefficients of inertia, and changing its direction with time.

We shall (therefore) study only some special simple systems performing simple motions (and leave the rest for higher studies).