# What are the Principles and Equations of Euler's Theory in Physics?

• Greg Bernhardt
In summary, Euler's three equations relate the net torque on a rotating rigid body to the moment of inertia tensor and the angular momentum. They are valid only in a frame of reference fixed in the body along three principal axes, and are valid only if the origin of the axes is at the centre of mass or is stationary. These equations are components of a single tensor equation and can also be expressed in a frame of reference fixed in space, but with an added "cross" term. The reason for using Euler's equations instead of the fixed-frame equations is due to the changing moment of inertia tensor being more difficult to work with.
Definition/Summary

Euler's three equations relate the net torque on a rotating rigid body to the moment of inertia tensor and the angular momentum.

They are valid only in a frame of reference fixed in the body along three (perpendicular) principal axes, and therefore rotating with it.

They are valid only if the origin of the axes is at the centre of mass, or is stationary.

They are the three components along those axes of a single tensor equation.

Equations

$$\tau_1\ =\ I_1\,\frac{d\omega_1}{dt} + (I_3\ -\ I_2)\omega_2\omega_3$$

$$\tau_2\ =\ I_2\,\frac{d\omega_2}{dt} + (I_1\ -\ I_3)\omega_3\omega_1$$

$$\tau_3\ =\ I_3\,\frac{d\omega_3}{dt} + (I_2\ -\ I_1)\omega_1\omega_2$$

Extended explanation

Frame of reference fixed in space:

The rate of change of the angular momentum vector of a rigid body about any point equals the net torque vector about that point (the moment of all the external forces acting on that body):

$$\mathbf{\tau}_{net}\ =\ \frac{d\mathbf{L}}{dt}$$

This is the rotational version of Newton's second law.

The angular momentum vector of a rigid body about a point equals the moment of inertia tensor about that point "times" the angular momentum vector ($\mathbf{L}\ =\ \tilde{I}\,\mathbf{\omega}$), and so the net torque vector about that point is:

$$\mathbf{\tau}_{net}\ =\ \frac{d\mathbf{L}}{dt}\ =\ \frac{d}{dt}\left(\tilde{I}\,\mathbf{\omega}\right)\ =\ \tilde{I}\,\frac{d}{dt}\left(\mathbf{\omega}\right)\ +\ \frac{d\tilde{I}}{dt}\left(\mathbf{\omega}\right)$$

This only applies if that point is stationary or is the centre of mass or has a velocity v parallel to the velocity of the centre of mass: in any other case, there is an additional term, v x mvc.o.m..

Although the moment of inertia tensor, $\tilde{I}$, is fixed in the body, it is not fixed in space, and so $d\tilde{I}/dt$ is not zero (unless ω lies along a principal axis).

Frame of reference fixed in the body:

However, if we change to a frame of reference fixed in the body, then $d\tilde{I}/dt$ is zero, although there is an added "cross" term:

$$\mathbf{\tau}_{net}\ =\ \frac{d\mathbf{L}}{dt}\ +\ \mathbf{\omega}\times \mathbf{L}\ =\ \frac{d}{dt}\left(\tilde{I}\,\mathbf{\omega}\right)\ \ +\ \ \mathbf{\omega}\times \left(\tilde{I}\,\mathbf{\omega}\right)\ =\ \tilde{I}\,\frac{d}{dt}\left(\mathbf{\omega}\right)\ \ +\ \ \mathbf{\omega}\times \left(\tilde{I}\,\mathbf{\omega}\right)$$

which, expressed relative to three perpendicular axes fixed in the body along principal axes with moments of inertia $I_1\ I_2\ \text{and}\ I_3$, gives the three Euler's equations:

$$\tau_1\ =\ I_1\,\frac{d\omega_1}{dt} + (I_3\ -\ I_2)\omega_2\omega_3$$

$$\tau_2\ =\ I_2\,\frac{d\omega_2}{dt} + (I_1\ -\ I_3)\omega_3\omega_1$$

$$\tau_3\ =\ I_3\,\frac{d\omega_3}{dt} + (I_2\ -\ I_1)\omega_1\omega_2$$

Comparison:

Euler's equations give the same net torque as the fixed-frame equations.

The only reason why Euler's equations may be preferred to the fixed-frame equations is that the moment of inertia tensor is changing in the latter, but only the angular momentum vector is changing in the former …

and a non-zero derivative of a tensor is a lot nastier to work with than a non-zero derivative of a vector!

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

Thanks for the overview of Euler's equations!

## 1. What are Euler's equations?

Euler's equations are a set of mathematical equations that describe the motion of a rigid body in three-dimensional space. They are named after the Swiss mathematician Leonhard Euler who first derived them in the 18th century.

## 2. What is the significance of Euler's equations?

Euler's equations are important in the field of mechanics and engineering as they help to accurately predict the motion of a rigid body. They are also used in areas such as robotics, aerospace engineering, and computer graphics.

## 3. How do Euler's equations differ from Newton's laws of motion?

Euler's equations are a more general form of Newton's laws of motion, specifically for the motion of a rigid body. While Newton's laws only apply to particles, Euler's equations can be used to describe the complex motion of rigid bodies in three-dimensional space.

## 4. What are the variables in Euler's equations?

The variables in Euler's equations include the angular velocity, the moment of inertia, and the torque acting on the rigid body. These variables are used to calculate the rotational motion of the body.

## 5. How are Euler's equations used in real-world applications?

Euler's equations are used in various real-world applications such as designing spacecrafts, analyzing the motion of objects in flight, and developing control systems for robots. They are also used in computer simulations and video game physics engines to create realistic movement of objects.

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