Euler's laws of motion

In classical mechanics, Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to rigid body motion. They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws.

Overview

Euler's first law

Euler's first law states that the rate of change of linear momentum Template:Math of a rigid body is equal to the resultant of all the external forces Template:Math acting on the body:

$F_{\text{ext}}=\frac{d𝐩}{dt}.$

Internal forces between the particles that make up a body do not contribute to changing the momentum of the body as there is an equal and opposite force resulting in no net effect.

The linear momentum of a rigid body is the product of the mass of the body and the velocity of its center of mass Template:Math.

Euler's second law

Template:Main Euler's second law states that the rate of change of angular momentum Template:Math about a point that is fixed in an inertial reference frame (often the center of mass of the body), is equal to the sum of the external moments of force (torques) acting on that body Template:Math about that point:

$𝐌=\frac{d𝐋}{dt}.$

Note that the above formula holds only if both Template:Math and Template:Math are computed with respect to a fixed inertial frame or a frame parallel to the inertial frame but fixed on the center of mass. For rigid bodies translating and rotating in only two dimensions, this can be expressed as:

$𝐌={𝐫}_{\mathrm{c}\mathrm{m}}\times {𝐚}_{\mathrm{c}\mathrm{m}}m+I\mathit{\alpha },$

where:

• Template:Math is the position vector of the center of mass of the body with respect to the point about which moments are summed,
• Template:Math is the linear acceleration of the center of mass of the body,
• Template:Math is the mass of the body,
• Template:Math is the angular acceleration of the body, and
• Template:Math is the moment of inertia of the body about its center of mass.

See also Euler's equations (rigid body dynamics).

Explanation and derivation

The distribution of internal forces in a deformable body are not necessarily equal throughout, i.e. the stresses vary from one point to the next. This variation of internal forces throughout the body is governed by Newton's second law of motion of conservation of linear momentum and angular momentum, which for their simplest use are applied to a mass particle but are extended in continuum mechanics to a body of continuously distributed mass. For continuous bodies these laws are called Euler's laws of motion.

The total body force applied to a continuous body with mass Template:Math, mass density Template:Math, and volume Template:Math, is the volume integral integrated over the volume of the body:

${𝐅}_B=\underset{V}{\int }𝐛dm=\underset{V}{\int }𝐛\rho dV$

where Template:Math is the force acting on the body per unit mass (dimensions of acceleration, misleadingly called the "body force"), and Template:Math is an infinitesimal mass element of the body.

Body forces and contact forces acting on the body lead to corresponding moments (torques) of those forces relative to a given point. Thus, the total applied torque Template:Math about the origin is given by

$𝐌={𝐌}_B+{𝐌}_C$

where Template:Math and Template:Math respectively indicate the moments caused by the body and contact forces.

Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) acting on the body can be given as the sum of a volume and surface integral:

$𝐅=\underset{V}{\int }𝐚dm=\underset{V}{\int }𝐚\rho dV=\underset{S}{\int }𝐭dS+\underset{V}{\int }𝐛\rho dV$

$𝐌={𝐌}_B+{𝐌}_C=\underset{S}{\int }𝐫\times 𝐭dS+\underset{V}{\int }𝐫\times 𝐛\rho dV.$

where Template:Math is called the surface traction, integrated over the surface of the body, in turn Template:Math denotes a unit vector normal and directed outwards to the surface Template:Math.

Let the coordinate system Template:Math be an inertial frame of reference, Template:Math be the position vector of a point particle in the continuous body with respect to the origin of the coordinate system, and Template:Math be the velocity vector of that point.

Euler's first axiom or law (law of balance of linear momentum or balance of forces) states that in an inertial frame the time rate of change of linear momentum Template:Math of an arbitrary portion of a continuous body is equal to the total applied force Template:Math acting on that portion, and it is expressed as

$\begin{array}{cc}\frac{d𝐩}{dt}& =𝐅\\ \frac{d}{dt}\underset{V}{\int }\rho 𝐯dV& =\underset{S}{\int }𝐭dS+\underset{V}{\int }𝐛\rho dV.\end{array}$

Euler's second axiom or law (law of balance of angular momentum or balance of torques) states that in an inertial frame the time rate of change of angular momentum Template:Math of an arbitrary portion of a continuous body is equal to the total applied torque Template:Math acting on that portion, and it is expressed as

$\begin{array}{cc}\frac{d𝐋}{dt}& =𝐌\\ \frac{d}{dt}\underset{V}{\int }𝐫\times \rho 𝐯dV& =\underset{S}{\int }𝐫\times 𝐭dS+\underset{V}{\int }𝐫\times 𝐛\rho dV.\end{array}$

where $𝐯$ is the velocity, $V$ the volume, and the derivatives of Template:Math and Template:Math are material derivatives.

See also

• List of topics named after Leonhard Euler
• Euler's laws of rigid body rotations
• Newton–Euler equations of motion with 6 components, combining Euler's two laws into one equation.