\(^\)Show that if a line passing through the centre of mass of a rigid body is a principal axis at one point of the line, then it is a principal axis at every point of the line.

Understanding the inertia tensor is crucial for grasping the dynamics of rigid bodies. It's a mathematical representation that encapsulates how a body's mass is distributed with respect to an axis of rotation.

To calculate the inertia tensor, we consider a matrix that contains the moments of inertia about the principal axes and the products of inertia. For a point P in the body, the inertia tensor, denoted as \(I_P\), can be found using the parallel axis theorem:
\[I_P = I_C + M [ d^2 - (P \cdot P) I₃ ]\]
where \(I_C\) is the inertia tensor at the center of mass \(C\), \(M\) represents the body's total mass, \(d\) is the distance from \(C\) to \(P\), and \(I₃\) is the identity matrix. This equation helps account for the body's rotation not just at the center of mass but at any point \(P\).

In essence, the inertia tensor is a fundamental concept in mechanics, describing how a rigid body resists rotational acceleration about various axes. Each element of this tensor gives insight into how different axes influence the body's rotational characteristics.

When it comes to understanding rotational motion, eigenvectors and eigenvalues play a prominent role. In the context of a rigid body, eigenvectors of the inertia tensor represent the principal axes—lines about which the body can rotate freely without experiencing any twisting due to off-axis torques.

An eigenvalue, on the other hand, is associated with its respective eigenvector and represents the moment of inertia about the principal axis. The critical relationship between the inertia tensor \(I_P\), an eigenvector \(u\), and an eigenvalue \(\lambda\) is given by:
\[I_P \cdot u = \lambda u\]
This equation signifies that rotating the body about a principal axis (eigenvector) requires a torque proportional to the body's rotational inertia along that axis, quantified by the eigenvalue.

By determining the eigenvalues and eigenvectors of a body's inertia tensor, we can find the body's principal axes and moments of inertia, providing a clear understanding of the body's rotational behavior.

The moment of inertia is a pivotal concept in rotational dynamics, reflecting how a rigid body's mass is distributed relative to an axis of rotation. It governs the angular acceleration that the body will experience under applied torque. The greater the moment of inertia, the more resistant the body is to changes in rotational velocity.

For a point mass, the moment of inertia is simply the product of the mass \(m\) and the square of its perpendicular distance \(r\) from the axis of rotation:
\[I = m r^2\]
However, for a rigid body, we must integrate this quantity over the entire volume of the body to consider the mass distribution. The tasks of finding eigenvalues and identifying the principal axis of a rigid body are intimately connected to its moments of inertia. Each principal axis has a corresponding eigenvalue, which is the body's moment of inertia about that axis.

When we dive into the inertia tensor calculation for any point on the principal axis, as shown in the exercise, we solidify our understanding that the moment of inertia, as an eigenvalue, remains unchanged. This concept reassures us of the stability of rotational motion around the principal axes and underscores the integral role the moment of inertia plays in dynamics.