Euler's laws of motion

Euler's laws of motion, formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws about the motion of particles, extends them to rigid body motion.

Overview

Euler's first law

Euler's first law states that the linear momentum of a body, $\mathbf G$ is equal to the product of the mass of the body and the velocity of its center of mass: $\mathbf G \mathrm = m \mathbf v_c$. Internal forces, between the particles that make up a body, do not contribute to changing the total momentum of the body. The law is also stated as $\mathcal F \mathrm = m \mathbf a_G$.

Euler's second law

Euler's second law states that the rate of change of angular momentum about a point, $\mathcal \mathrm{d}(\mathbf H) \over \mathrm{d}t$, is equal to the sum of the external moments about that point: $\mathcal M = {\mathrm{d}(\mathbf H) \over \mathrm{d}t}$. For rigid bodies translating and rotating in only 2D, this can be expressed as $\mathcal M = \mathbf r_{cm} \times \mathbf a m + I \mathbf{\alpha}$, where r_cm is the position vector of the center of mass with respect to the point about which moments are summed.

Explanation and derivation

The density of internal forces at every point in a deformable body are not necessarily equal, i.e. there is a distribution of stresses throughout the body. 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 normally 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. If a body is represented as an assemblage of discrete particles, each governed by Newton’s laws of motion, then Euler’s equations can be derived from Newton’s laws. Euler’s equations can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure.

The total body force applied to a continuous body with mass m and volume V is expressed as

$\mathbf F_B=\int_V\mathbf b\,dm=\int_V \rho\mathbf b\,dV$

where $\mathbf b$ is the body force "density".

Body forces and contact forces acting on the body lead to corresponding moments of force (torques) relative to a given point. Thus, the total applied torque $\mathcal M$ about the origin is given by

$\mathcal M= \mathbf M_B + \mathbf M_C$

where $\mathbf M_B$ and $\mathbf M_C$ indicate moments caused by body and contact forces, respectively.

Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) in the body can be given by

$\mathcal F = \int_V \mathbf a\,dm = \int_S \mathbf T\,dS + \int_V \rho\mathbf b\,dV$

$\mathcal M = \int_S \mathbf r \times \mathbf T\,dS + \int_V \mathbf r \times \rho\mathbf b\,dV$

Let the coordinate system $x_1, x_2, x_3\,\!$ be an inertial frame of reference. Let $\mathbf r$ be the position vector of a particle or point $\mathbf P$ in the continuous body with respect to the origin of the coordinate system, and $\mathbf v$ the velocity vector of point $\mathbf P$.

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 $\mathbf G$ of an arbitrary portion of a continuous body is equal to the total applied force $\mathcal F$ acting on the considered portion, and it is expressed as

$\begin{align} \frac{d\mathbf G}{dt} &= \mathcal F \\ \frac{d}{dt}\int_V \rho\mathbf v\,dV&=\int_S \mathbf T\,dS + \int_V \rho\mathbf b\,dV \\ \end{align}$

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 $\mathbf H$ of an arbitrary portion of a continuous body is equal to the total applied torque $\mathcal M$ acting on the considered portion, and it is expressed as

$\begin{align} \frac{d\mathbf H}{dt} &= \mathcal M \\ \frac{d}{dt}\int_V \mathbf r\times\rho\mathbf v\,dV&=\int_S \mathbf r \times \mathbf T\,dS + \int_V \mathbf r \times \rho\mathbf b\,dV \\\end{align}$

The derivatives of $\mathbf G$ and $\mathbf H$ are material derivatives.

See also

• List of topics named after Leonhard Euler