# Euler's laws of motion

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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 linear momentum of a body, p (also denoted G) is equal to the product of the mass of the body m and the velocity of its center of mass vcm: 

${\mathbf {p} }=m{\mathbf {v} }_{\rm {cm}}$ .

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:

$\mathbf {F} =m\mathbf {a} _{\rm {cm}}$ .

where acm = dvcm/dt is the acceleration of the centre of mass and F = dp/dt is the total applied force on the body. This is just the time derivative of the previous equation (m is a constant).

### Euler's second law

Euler's second law states that the rate of change of angular momentum L (also denoted H) about a point that is fixed in an inertial reference frame or the mass center of the body, is equal to the sum of the external moments of force (torques) M (also denoted τ or Γ) about that point:

$\mathbf {M} ={d\mathbf {L} \over dt}$ .

Note that the above formula holds only if both M,L 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 2d, this can be expressed as:

${\mathbf {M} }={\mathbf {r} }_{\rm {cm}}\times {\mathbf {a} }_{\rm {cm}}m+I{\boldsymbol {\alpha }}$ ,

where rcm is the position vector of the center of mass with respect to the point about which moments are summed, α is the angular acceleration of the body, and I is the moment of inertia. See also Euler's equations (rigid body dynamics).

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

$\mathbf {F} _{B}=\int _{V}\mathbf {b} \,dm=\int _{V}\mathbf {b} \rho \,dV$ where b is the force acting on the body per unit mass (dimensions of acceleration, misleadingly called the "body force"), and dm = ρdV is an infinitesimal mass element of the body.

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 M about the origin is given by

${\mathbf {M} }={\mathbf {M} }_{B}+{\mathbf {M} }_{C}$ where MB and MC 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) in the body can be given as the sum of a volume and surface integral:

${\mathbf {F} }=\int _{V}{\mathbf {a} }\,dm=\int _{V}{\mathbf {a} }\rho \,dV=\int _{S}{\mathbf {t} }dS+\int _{V}{\mathbf {b} }\rho \,dV$ ${\mathbf {M} }=\int _{S}{\mathbf {r} }\times {\mathbf {t} }dS+\int _{V}{\mathbf {r} }\times {\mathbf {b} }\rho \,dV.$ 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 the considered portion, and it is expressed as

{\begin{aligned}{\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{aligned}} 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 the considered portion, and it is expressed as

{\begin{aligned}{\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{aligned}} The derivatives of p and L are material derivatives.