Euler's first law states that the linear momentum of a body, is equal to the product of the mass of the body and the velocity of its center of mass : Internal forces between the particles that make up a body do not contribute to changing the total momentum of the body as there is an equal and opposite force resulting in no net effect. The law is also stated as: where is the acceleration of the center of mass and is the total applied force on the body. This is just the time derivative of the previous equation.
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 is the position vector of the center of mass with respect to the point about which moments are summed, is the angular acceleration of the body about its center of mass, and is the moment of inertia of the body about its center of mass. See also Euler's equations.
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. 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 distribution. 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.
