nLab moment of inertia

Idea

In the mechanics of rigid body dynamics in Cartesian space $\mathbb{R}^n$, the moment of inertia of a rigid body is the analog of mass for rotational dynamics?. In linear dynamics?, we have the formula

$$p = m v$$

which says that the momentum $p$ is proportional to the velocity $v$. Similarly, in rotational dynamics, we have the analogous formula

$$L = I \Omega$$

where $L$ is the angular momentum, $\Omega$ is the angular velocity, and $I$ is the moment of inertia.

However, the rotational equation is somewhat more complicated than the linear one: firstly because $L$ and $\Omega$ are not naturally vectors but bivectors; and secondly because they are not necessarily proportional, so that $I$ cannot be a scalar. In general, the moment of inertia is a linear function

$$I \colon \wedge^2 \mathbb{R}^n \to \wedge^2 \mathbb{R}^n$$

so that the above equation becomes simply

$$L = I(\Omega).$$

This linear function is additionally symmetric with respect to the induced inner product on $\bigwedge^2 \mathbb{R}^n$, so it can be represented in coordinates by a symmetric $\frac{n(n-1)}{2} \times \frac{n(n-1)}{2}$ matrix.

Similarly, differentiating this equation once with respect to time (and assuming that $I$ is constant as it is for a rigid body), we have

$$\tau = I \alpha ,$$

relating the total torque $\tau$ to the angular acceleration? $\alpha$ — this is the rotational analogue of Newton's second law $F = m a$ (where $m$ must be constant).

In low dimensions

In low dimensions, the situation can be (and usually is) simplified.

• In two dimensions, bivectors form a one-dimensional vector space, so that the moment of inertia is simply a scalar.

• In three dimensions, bivectors form a three-dimensional vector space, so that the moment of inertia can be represented by a symmetric $3 \times 3$ matrix. Additionally, in three dimensions, there is an isomorphism between bivectors and vectors (once we choose an orientation to go with our inner product); so that angular velocity and momentum can be (and usually are) identified with vectors, and the moment of inertia with a symmetric rank-2 tensor.

In Hamiltonian dynamics

In terms of the discussion at Hamiltonian dynamics on Lie groups, the rigid body dynamics in $\mathbb{R}^n$ is given by Hamiltonian motion on the special orthogonal group $SO(n)$. It is defined by any left invariant? Riemannian metric

$$\langle -,-\rangle \in Sym^2_{C^\infty(G)} \Gamma(T G)$$

hence a bilinear non-degenarate form on the Lie algebra $\mathfrak{so}(n)$ (not necessarily the Killing form).

This bilinear form is the moment of inertia. (For instance AbrahamMarsden, section 4.6.)

In terms of mass density

If a rigid body has mass density? $\rho$, then its angular momentum is defined in terms of $\Omega$ by the $n$-dimensional integral

$$L = \int \rho \vec{x} \wedge (\vec{x} \cdot \Omega) \,d^n x$$

over all space, where $\vec{x}$ is the vector from the origin to the point of integration, $\cdot$ denotes the interior product? of a vector with a bivector (yielding a vector), and $\wedge$ denotes the exterior product of two vectors (yielding a bivector).

When $\Omega$ is the same everywhere (as for a rigid body), then we may view this as a function from $\Omega$ to $L$; this function is the moment of inertia.

Related pages

• Hamiltonian dynamics on Lie groups

  • rigid body dynamics

    • angular velocity

    • angular momentum

References

A classical textbook discussion is for instance section 4.6 of

• Ralph Abraham, Jerrold Marsden, Foundations of Mechanics

A pedestrian discussion of moment of inertia in terms of bivectors that applies in any dimension of space(spacetime) is around page 74 of

• Chris Doran, Anthony Lasenby, Geometric Algebra for Physicists Cambridge University Press

or around page 56 of

• Chris Doran, Anthony Lasenby, Physical applications of geometric algebra (pdf)

and around slide 6 in

• Anthony Lasenby, Chris Doran and Robert Lasenby, Rigid Body Dynamics and Conformal Geometric Algebra (pdf)

These authors amplify the canonical embedding of bivectors into the Clifford algebra, which they call “Geometric Algebra”.