# Rigid body dynamics

Dynamics
$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
Newton's Second Law

Classical mechanics
$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
Newton's Second Law
History of ...

In physics, rigid body dynamics is the study of the motion of rigid bodies. Unlike particles, which move only in three degrees of freedom (translation in three directions), rigid bodies occupy space and have geometrical properties, such as a center of mass, moments of inertia, etc., that characterize motion in six degrees of freedom (translation in three directions plus rotation in three directions). Rigid bodies are also characterized as being non-deformable, as opposed to deformable bodies. As such, rigid body dynamics is used heavily in analyses and computer simulations of physical systems and machinery where rotational motion is important, but material deformation does not have a major effect on the motion of the system.

## Rigid body linear momentum

Newton's Second Law states that the rate of change of the linear momentum of a particle with constant mass is equal to the sum of all external forces acting on the particle:

$\frac{\mathrm{d}(m \mathbf{v})}{\mathrm{d}t}=\sum_{i=1}^N \mathbf{f}_i$

where m is the particle's mass, v is the particle's velocity, their product mv is the linear momentum, and fi is one of the N number of forces acting on the particle.

Because the mass is constant, this is equivalent to

$m \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=\sum_{i=1}^N \mathbf{f}_i.$

To generalize, assume a body of finite mass and size is composed of such particles, each with infinitesimal mass dm. Each particle a position vector r. There exist internal forces, acting between any two particles, and external forces, acting only on the outside of the mass. Since velocity v is the derivative of position r, the derivative of velocity dv/dt is the second derivative of position d2r/dt2, and the linear momentum equation of any given particle is

$\mathrm{d}m \frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2}= \sum_{i=1}^M \mathbf{f}_{i,\text{internal}} + \sum_{j=1}^N \mathbf{f}_{j,\mathrm{external}}.$

When the linear momentum equations for all particles are added together, the internal forces sum to zero according to Newton's third law, which states that any such force has opposite magnitudes on the two particles. By accounting for all particles, the left side becomes an integral over the entire body, and the second derivative operator can be moved out of the integral, so

$\frac{\mathrm{d}^2}{\mathrm{d}t^2} \int \mathbf{r}\, \mathrm{d}m = \sum_{j=1}^N \mathbf{f}_{j,\mathrm{external}}$.

Let M be the total mass, which is constant, so the left side can be multiplied and divided by M, so

$M \frac{\mathrm{d}^2}{\mathrm{d}t^2}\!\left(\frac{\int \mathbf{r}\, \mathrm{d}m}{M}\right) = \sum_{j=1}^N \mathbf{f}_{j,\mathrm{external}}$.

The expression $\frac{\int \mathbf{r}\, \mathrm{d}m}{M}$ is the formula for the position of the center of mass. Denoting this by rcm, the equation reduces to

$M \frac{\mathrm{d}^2 \mathbf{r}_\mathrm{cm}}{\mathrm{d}t^2} = \sum_{j=1}^N \mathbf{f}_{j,\mathrm{external}}.$

Thus, linear momentum equations can be extended to rigid bodies by denoting that they describe the motion of the center of mass of the body. This is known as Euler's first law.

## Rigid body angular momentum

The most general equation for rotation of a rigid body in three dimensions about an arbitrary origin O with axes x, y, z is

$M b_{G/O} \times \frac{\mathrm{d}^2 R_O}{\mathrm{d}t^2} + \frac{\mathrm{d}(\mathbf{I}\boldsymbol{\omega})}{\mathrm{d}t} = \sum_{j=1}^N \tau_{O,j}$

where the moment of inertia tensor, $\mathbf{I}$, is given by

$\mathbf{I} = \begin{pmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{pmatrix}$
$\mathbf{I} = \begin{pmatrix} \int (y^2+z^2)\, \mathrm{d}m & -\int xy\, \mathrm{d}m & -\int xz\, \mathrm{d}m\\ -\int xy\, \mathrm{d}m & \int (x^2+z^2)\, \mathrm{d}m & -\int yz\, \mathrm{d}m \\ -\int xz\, \mathrm{d}m & -\int yz\, \mathrm{d}m & \int (x^2+y^2)\, \mathrm{d}m \end{pmatrix}$

and the angular velocity, $\boldsymbol{\omega}$, is given by

$\quad \boldsymbol{\omega} = \omega_x \mathbf{\hat{i}} + \omega_y \mathbf{\hat{j}} + \omega_z \mathbf{\hat{k}}$

where $\scriptstyle{(\mathbf{\hat{i}},\ \mathbf{\hat{j}},\ \mathbf{\hat{k}})}$ is a set of mutually perpendicular unit vectors fixed in a reference frame.

Moving any rigid body is equivalent to moving a Poinsot's ellipsoid.

## Angular momentum and torque

Similarly, the angular momentum $\mathbf{L}$ for a system of particles with linear momenta pi and distances ri from the rotation axis is defined

$\mathbf{L} = \sum_{i=1}^{N} \mathbf{r}_{i} \times \mathbf{p}_{i} = \sum_{i=1}^{N} m_{i} \mathbf{r}_{i} \times \mathbf{v}_{i}$

For a rigid body rotating with angular velocity ω about the rotation axis $\mathbf{\hat{n}}$ (a unit vector), the velocity vector $\mathbf{v}_{i}$ may be written as a vector cross product

$\mathbf{v}_{i} = \omega \mathbf{\hat{n}} \times \mathbf{r}_{i} \ \stackrel{\mathrm{def}}{=}\ \boldsymbol\omega \times \mathbf{r}_{i}$

where

angular velocity vector $\boldsymbol\omega \ \stackrel{\mathrm{def}}{=}\ \omega \mathbf{\hat{n}}$
$\mathbf{r}_{i}$ is the shortest vector from the rotation axis to the point mass.

Substituting the formula for $\mathbf{v}_{i}$ into the definition of $\mathbf{L}$ yields

$\mathbf{L} = \sum_{i=1}^{N} m_{i} \mathbf{r}_{i} \times (\boldsymbol\omega \times \mathbf{r}_{i}) = \boldsymbol\omega \sum_{i=1}^{N} m_{i} r_{i}^{2} = I \omega \mathbf{\hat{n}}$

where we have introduced the special case that the position vectors of all particles are perpendicular to the rotation axis (e.g., a flywheel): $\boldsymbol\omega \cdot \mathbf{r}_{i} = 0$.

The torque $\mathbf{N}$ is defined as the rate of change of the angular momentum $\mathbf{L}$

$\mathbf{N} \ \stackrel{\mathrm{def}}{=}\ \frac{d\mathbf{L}}{dt}$

If I is constant (because the inertia tensor is the identity, because we work in the intrinsecal frame, or because the torque is driving the rotation around the same axis $\mathbf{\hat{n}}$ so that I is not changing) then we may write

$\mathbf{N} \ \stackrel{\mathrm{def}}{=}\ I \frac{d\omega}{dt}\mathbf{\hat{n}} = I \alpha \mathbf{\hat{n}}$

where

α is called the angular acceleration (or rotational acceleration) about the rotation axis $\mathbf{\hat{n}}$.

Notice that if I is not constant in the external reference frame (ie. the three main axes of the body are different) then we cannot take the I outside the derivate. In this cases we can have torque-free precession.

## Applications

Computer physics engines use rigid body dynamics to increase interactivity and realism in video games.