Euler's laws of motion

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 m, mass density ρ, and volume V, is the volume integral integrated over the volume of the body:

$${\displaystyle {\mathbf{F}}_B={\int }_V\mathbf{b}dm={\int }_V\mathbf{b}\rho dV}$$

where b is the force acting on the body per unit mass (dimensions of acceleration, misleadingly called the "body force"), and dm = ρ dV 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 M about the origin is given by

$${\displaystyle \mathbf{M}={\mathbf{M}}_B+{\mathbf{M}}_C}$$

where M_B and M_C 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:

$${\displaystyle \mathbf{F}={\int }_V\mathbf{a}dm={\int }_V\mathbf{a}\rho dV={\int }_S\mathbf{t}dS+{\int }_V\mathbf{b}\rho dV}$$ $${\displaystyle \mathbf{M}={\mathbf{M}}_B+{\mathbf{M}}_C={\int }_S\mathbf{r}\times \mathbf{t}dS+{\int }_V\mathbf{r}\times \mathbf{b}\rho dV.}$$

where t = t(n) is called the surface traction, integrated over the surface of the body, in turn n denotes a unit vector normal and directed outwards to the surface S.

Let the coordinate system (x₁, x₂, x₃) be an inertial frame of reference, r be the position vector of a point particle in the continuous body with respect to the origin of the coordinate system, and v = ⁠dr/dt⁠ 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 p of an arbitrary portion of a continuous body is equal to the total applied force F acting on that portion, and it is expressed as

$${\displaystyle \begin{array}{cc}\frac{d\mathbf{p}}{dt}& =\mathbf{F}\\ \frac{d}{dt}{\int }_V\rho \mathbf{v}dV& ={\int }_S\mathbf{t}dS+{\int }_V\mathbf{b}\rho dV.\end{array}}$$

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 L of an arbitrary portion of a continuous body is equal to the total applied torque M acting on that portion, and it is expressed as

$${\displaystyle \begin{array}{cc}\frac{d\mathbf{L}}{dt}& =\mathbf{M}\\ \frac{d}{dt}{\int }_V\mathbf{r}\times \rho \mathbf{v}dV& ={\int }_S\mathbf{r}\times \mathbf{t}dS+{\int }_V\mathbf{r}\times \mathbf{b}\rho dV.\end{array}}$$

where ${\displaystyle \mathbf{v}}$ is the velocity, ${\displaystyle V}$ the volume, and the derivatives of p and L are material derivatives.