Rigid body motion

A rigid body is a system of particles in which the distances between the particles do not vary.  To describe the motion of a rigid body we use two systems of coordinates, a space-fixed system X, Y, Z, and a moving system x, y, z, which is rigidly fixed in the body and participates in its motion.

Let the origin of the body-fixed system be the body's center of mass (CM).  The orientation of the axes of that system relative to the axes of the space-fixed system is given by three independent angles.  The vector R points from the origin of the spaced-fixed system to the CM of the body.  Thus a rigid body is a mechanical system with six degrees of freedom.

Let r denote the position of an arbitrary point P in the body-fixed system.  In the space fixed system its position is given by r + R, and its velocity is
v = d(R + r)/dt = dR/dt + dr/dt = V + Ω × r.
Here V is the velocity of the CM and Ω is the angular velocity of the body.  The direction of Ω is along the axis of rotation and Ω = dφ/dt.

The kinetic energy of the body is
T = ½Σm_iv_i² = ½Σm_i(V + Ω × r_i)² = ½MV² + ½Σm_i(Ω×r_i)²) + Σm_iV·(Ω × r_i)
Σm_iV·(Ω × r_i) = V·(Ω × Σm_ir_i) = 0, since Σm_ir_i =  0.

We rewrite
T = ½MV² + ½Σm_i(Ω×r_i)²) and using (A×B)² = A²B² - (A·B)²,
T= ½MV² + ½Σm_i(Ω²r_i² - (Ω·r_i)²),  M = Σm_i.
We find T = T_CM + T_rot, i.e. the kinetic energy is the sum of the kinetic energy of the motion of the CM and the kinetic energy of the rotation about the CM.

In component form we write
T_rot = ½Σ_ijI_ijΩ_iΩ_j.
where
I_ij = Σ_km_k[Σ_l(x_k)_l²δ_ij - (x_k)_i(x_k)_j] is the inertia tensor. 
The Ω_i are the components of Ω along the axis of the body fixed system.  For a continuous system Σ_km_k --> ∫dm = ∫ρdV.

By appropriate choice of the orientation of the body-fixed coordinate system the inertia tensor can be reduced to diagonal form.  The directions of the axes x_i are then called the principal axes of inertia and the diagonal components of the tensor are then called the principal moments of inertia.
Then T_rot = ½[I₁Ω₁² + I₂Ω₂² + I₃Ω₃²].

Definitions:

The total angular momentum of a rigid body about a point is

L_tot = Σm_ir_i×(V + Ω × r_i) = MR×V + Σm_ir_i×(Ω × r_i).
L_tot = angular momentum of the CM about the point plus angular momentum about the CM.
Let L denote the angular momentum about the CM of the body.
L = Σ_km_kr_k×(V + Ω × r_k) = Σ_km_k[r_k²Ω - r_k(r_k·Ω)],
which in component form yields
L_i = Σ_jΣ_km_k[Σ_l(x_k)_l²δ_ij - (x_k)_i(x_k)_j]Ω_j = Σ_jI_ijΩ_j.

If x₁, x₂, and x₃ are the principal axes of inertia, then
L₁ = I₁Ω₁,  L₂ = I₂Ω₂,  L₃ = I₃Ω₃.

How do we find the principal moments of inertia and the principal axes of inertia?
We diagonalize the inertia tensor.

L_i  = Σ_jI_ijΩ_j.  If Ω points along a principal axis then L and Ω are aligned and L_i = IΩ_i, where I is one of the principal moments of inertia.
Then Σ_j(I_ij - I δ_ij)Ω_j = 0,   |I_ij - I δ_ij|= 0.

When we consider the motion of a rigid body, we distinguish two situations.
(a)  No point of the body is fixed by constraints.  This problem is best treated by separating the motion of the body into motion of the CM and motion about the CM.
Then  T = T_CM + T_rot, and L(about a point) =  angular momentum of the CM about the point plus angular momentum about the CM.

(b)  One point of the body is fixed by constraints.  The problem is often best treated by treating the motion as a pure rotation about an axis through this point.
Then T = T_rot = ½Σ_ijI_ijΩ_iΩ_j,  L_i = Σ_jI_ijΩ_j,  with I_ij = Σ_km_k[Σ_l(x_k)_l²δ_ij - (x_k)_i(x_k)_j].  The origin of the coordinate system then is the fixed point.

The parallel axes theorem (Steiner's theorem)

Let I be the moment of inertia tensor of a body of mass M, calculated in a body fixed system with origin S at the CM.  Let I' be the moment of inertia tensor calculated in a body fixed system with parallel axes, but with origin S' = S + a.  Then I and I' are related by
I'_ij = I_ij + M (a²δ_ij - a_ia_j).
For the diagonal elements we have I'_ii = I_ii + M(a² - a_i²) = I_ii + Ma_⊥².

Non-inertial frames:

In non-inertial frames fictitious forces appear. 
Consider a particle moving with velocity v in a reference frame K which moves with velocity V(t) relative to the inertial frame K₀ and rotates with angular velocity Ω(t).
The equations of motion are

mdv/dt = -∂U/∂r - mdV/dt + mr × dΩ/dt - 2mΩ × v - mΩ × (Ω × r).

Here

• -∂U/∂r = force as observed in an inertial frame
  (If the force in an inertial frame is is not derivable from a potential, replace -∂U/∂r by F_inertial.)
• -mdV/dt = fictitious force due to acceleration of frame
• mr × dΩ/dt = fictitious force due to non-uniform rotation of frame
• -2mΩ × v = Coriolis force
• -mΩ × (Ω × r) = centrifugal force

For a uniformly rotating frame dΩ/dt = 0, dV/dt, and the equations of motion are
mdv/dt = F_inertial - 2mΩ × v - mΩ × (Ω × r).