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.
Euler's first law
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 is 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:
For rigid bodies translating and rotating in only 2d, this can be expressed as:
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.
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
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:
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
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
The derivatives of p and L are material derivatives.
- List of topics named after Leonhard Euler
- Euler's laws of rigid body rotations
- Newton-Euler equations of motion with 6 components, combining Euler's two laws into one equation.
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