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 of a rigid body is equal to the resultant of all the external forces acting on the body:
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.

Euler's second law

Euler's second law states that the rate of change of angular momentum about a point that is fixed in an inertial reference frame, is equal to the sum of the external moments of force acting on that body about that point:
Note that the above formula holds only if both and 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:
where:
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, mass density, and volume, is the volume integral integrated over the volume of the body:
where is the force acting on the body per unit mass, and is an infinitesimal mass element of the body.
Body forces and contact forces acting on the body lead to corresponding moments of those forces relative to a given point. Thus, the total applied torque about the origin is given by
where and respectively indicate the moments caused by the body and contact forces.
Thus, the sum of all applied forces and torques acting on the body can be given as the sum of a volume and surface integral:
where is called the surface traction, integrated over the surface of the body, in turn denotes a unit vector normal and directed outwards to the surface.
Let the coordinate system be an inertial frame of reference, be the position vector of a point particle in the continuous body with respect to the origin of the coordinate system, and be the velocity vector of that point.
Euler's first axiom or law states that in an inertial frame the time rate of change of linear momentum of an arbitrary portion of a continuous body is equal to the total applied force acting on that portion, and it is expressed as
Euler's second axiom or law states that in an inertial frame the time rate of change of angular momentum of an arbitrary portion of a continuous body is equal to the total applied torque acting on that portion, and it is expressed as
where is the velocity, the volume, and the derivatives of and are material derivatives.