Euler's Laws of Motion

Euler's Laws of Motion

Swiss mathematician Leonhard Euler formulated two laws to describe the motion of a rigid body relative to an inertial reference frame. The first law describes how applied forces change the velocity of the center of mass of a rigid body. The second law describes how the change in angular momentum of a rigid body is influenced by the moment of the applied force and any attachments to other bodies an object may have. These laws were written for bodies of fixed matter. That is the body must be composed of a fixed amount and type of matter that does not change.

The first of Euler's two laws is written as F = d/dt(G) where F is each individual force applied to the body and G is the linear momentum of the body. This form is written relative to an inertial reference frame. The points of the inertial reference frame in this case have no acceleration, neither translational nor rotational. For a particle with mass m its linear momentum G is given by its mass times its velocity, G = mv. For a body composed of many particles its linear momentum is assumed to be the sum of the linear momentum of its particles. G = m_iv_i, where m_i is the individual mass of a particle and v_i is the velocity of that particle. So for a continuous body the summation can be replaced by G = _mvdm The location of the center of mass of a body composed of several particles relative to an inertial frame of reference can be found from r_CM = 1/m _m rdm, where r_CM is the position of the center of mass relative to the inertial frame and r is the position of a particle of the body relative to the inertial frame. When the time derivative of this equation is taken we obtain a form that contains velocity of the center of mass relative to the inertial frame: mv_CM = _mvdm, where v_CM is the velocity of the center of mass with respect to the reference frame and m is the total mass of the body. So the linear momentum of a body is given by: G = mv_CM. This by substitution yields an alternate form of Euler's first law of motion for a rigid body: F = d/dt(mv_CM). Since mass is constant in Newtonian mechanics this becomes F = ma_CM, where a_CM is the acceleration of the center of mass. For a single particle the location of the center of mass and the location of the particle are the same and so Euler's first law of motion is the same as Newton's second law.

Euler's second law describing how the moment of an applied force changes the angular momentum of a rigid body is written as M₀ = d/dt (H₀), where M₀ is the moment of an applied force of the body relative to the frame of reference and H₀ is the angular momentum of the body relative to the reference frame. This inertial frame of reference can be removed if the center of mass is used so that the second law is rewritten as M_CM = d/dt (H_CM). This form of Euler's second law is especially useful if the body is accelerating since the calculation can be performed relative to the body's center of mass as opposed to some inertial reference point.

Euler used these laws to develop an understanding of the motion of the Moon around the Earth. At that time, the motion of the Moon around the Earth was not seen as governed by the same laws that governed the motion of the planets around the Sun. Newton developed ideas concerning the law of universal gravitation and his three laws of motion and used these as the basis for the Principia. These laws were used to explain planetary motion around the Sun but Newton thought the three-bodied problem too complex to be solved. Even if the Earth-Moon system were considered a two-bodied problem the orbits would not be simple ellipses since neither of them is a perfect sphere and so does not behave as a point mass. Euler developed methods of integrating linear differential equations in 1739 and this led him to draw up lunar tables in 1744 demonstrating the gravitational attraction of the Earth, Moon, Sun system. In the 1750s a small perturbation in the precession of the Earth's axis of rotation caused by the gravitational field of the Moon led Euler to develop his mechanics of rigid bodies and formulate his laws of motion. From 1760 onwards Euler was the first to study the general three-bodied problem under mutual gravitation. The Paris Academy Prize of 1772 was won jointly by Lagrange and Euler for their work on the orbit of the Moon.