To derive the Euler equations of motion for a rigid body we must first set up a schematic representing the most general case of rigid body motion, as shown in the figure below.

In the schematic, two coordinate systems are defined:

The first coordinate system used in the Euler equations derivation is the global XYZ reference frame. This reference frame is fixed to the ground. It is an inertial reference frame. It is useful to define a global (fixed) reference frame since quantities will only make sense if they are measured from a fixed frame. For example, we can only observe (and make sense of) the change in direction of an acceleration vector over time, if it is measured from a fixed reference frame.

The second coordinate system used in the Euler equations derivation is the local

For both reference frames a sign convention is assigned, as shown in the figure below. The derivation of the Euler equations will be based on this sign convention.

For both reference frames, the positive direction of the three individual axes is defined as shown. The choice of positive direction for these axes is important, especially for the local

For the global XYZ reference frame there are instances where the choice of positive direction for the axes is not important. But instances where the choice does matter are where (for example) cross-product multiplication is used. For example, let’s say we were to calculate the acceleration of the center of mass

This equation involves vector cross-product multiplication, which (as mentioned before) is based on the choice of sign convention for XYZ.

So, even though there are instances where the choice of sign convention for XYZ does not matter, it is not practical to list them all. So to be on the safe side it is better to always use the sign convention shown here for every dynamics problem you solve, using equations of motion (such as the Euler equations). Then you can’t go wrong.

The following variables are used in the derivation of the Euler equations (these are shown in the figure above):

To begin the derivation of the Euler equations, apply Newton’s Second law to the small mass element:

Where:

Σ

We can rewrite the vector sum of the forces as

where Σ

Therefore,

The next step in the derivation of the Euler equations is to take the cross product of both sides of this equation with the vector

The left side of this equation is defined as the sum of the moments acting on the small mass element

where Σ

The acceleration term

Since the small mass element

Therefore,

Substitute equation (2) into equation (1). This gives us

Set:

Where:

Set:

Where:

Set:

Where:

Set:

Where:

Rewrite equation (3):

The final step in the derivation of the Euler equations is to sum both sides of equation (4) over all the small mass elements in the rigid body. This sum is as follows:

where

On the left side of equation (5) internal forces cancel in pairs (by Newton’s Third Law), and the result is only external moments remaining (on the left side). The external moments are due to the external forces

It now remains to carry out the messy cross-product multiplications on the right side of equation (5) and then sum the resulting terms using integration (over the rigid body). This will not be shown here, since it is merely a mathematical exercise.

The following (scalar) equations of motion result:

Where:

Σ

Σ

Σ

Note that

Where:

Σ

For more information on moments see moment of a force.

Now, we can express Σ

where Σ

Also, note that

In other words, the angular acceleration vector is the derivative of the angular velocity vector. See vector derivative for more information.

The three general equations (6)-(8) describe the motion of a rigid body at an instant. The Euler equations will follow from these, as will be shown. If any of the variables (such as the sum-of-moments, angular velocity, or angular acceleration) in these equations change, the equations must be re-solved to find the new unknowns (corresponding to the new variables). In general, when solving real-world problems these equations must be evaluated using numerical techniques, such as for the physics of bowling.

The following six terms in the equations are called “inertia terms”. They are a function of how the mass is distributed in the rigid body. These terms are calculated relative to the local

Since the local

Now, if

As a result, equations (6)-(8) simplify to:

These three equations are called the

The equations (6)-(8) and the Euler equations apply for moments summed about the center of mass

For this situation the inertia terms would be calculated relative to point O.

When using the equations of motion (6)-(8) or the Euler equations to solve dynamics problems, the following applies:

If you know the direction of spin of the angular velocity, you must use the right-hand rule to assign the correct direction for its corresponding vector, and thus determine the correct values for its three components (

The right-hand rule also applies to the sum-of-moments vector (e.g. Σ

For more information on moments see moment of a force.

It is very interesting that one can derive the (somewhat complicated) Euler equations of motion simply from a clever application of Newton’s Second Law (

The key to deriving equations (6)-(8) and the Euler equations is to take the cross-product of both sides of Newton's Second Law equation with the position vector

Taking the cross-product is simply a way of accounting for rotation. And this is true even if the forces acting on a body cancel out. For example, let's say we have two equal and opposite external forces that are offset a certain distance apart, and are acting on a body. These forces cancel out, which means that the acceleration of the center of mass of the body is zero (By Newton's Second Law). However, the body will still rotate. So there needs to be some way to account for the effect of these two forces on the rotation (since Newton's Second Law, by itself, provides no information on that). Application of the cross-product is a mathematical way of accounting for rotation caused by these two forces. By taking the cross-product of each of these two forces with a position vector (say, from the center of mass of the body to where the forces act), and then summing them together, we find that they do not cancel out. So this alone gives a clue that the cross-product can account for the rotation of a body due to the forces acting on it. And as we have seen, equations (6)-(8) and the Euler equations are the grand result of applying the cross-product to Newton's Second Law equation.

As mentioned before, equations (6)-(8) and the Euler equations are based on the sign convention used here (i.e. that is how they are derived).

An example of a sign convention that does not match the one used here is shown below.

If you use this sign convention to solve problems using equations (6)-(8) or the Euler equations, you might get the wrong answer!

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