D'Alembert

D'Alembert's-Lagrange principle is a statement of the fundamental classical laws of motion. It is equivalent to Newton's second law. It is named after its discoverer, the French physicist Jean le Rond d'Alembert. stolen from french wikipedia File links The following pages link to this file: Jean le Rond dAlembert Talk:Gravity Categories: Images with unknown copyright status ... stolen from french wikipedia File links The following pages link to this file: Jean le Rond dAlembert Talk:Gravity Categories: Images with unknown copyright status ... Jean le Rond dAlembert, pastel by Maurice Quentin de la Tour Jean Le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... Classical physics is physics based on principles developed before the rise of quantum theory, including the special theory of relativity. ... Newtons first and second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... A physicist is a scientist trained in physics. ... Jean le Rond dAlembert, pastel by Maurice Quentin de la Tour Jean le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ...

The principle states that the sum of the differences between the generalized forces acting on a system and the time derivative of the generalized momenta of the system itself along an infinitesimal displacement compatible with the constraints of the system (a virtual displacement), is zero. That is: In physics, a force is an external cause responsible for any change of a physical system. ... In mathematics, the derivative is one of the two central concepts of calculus. ... In physics, momentum is the product of the mass and velocity of an object. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... In Newtonian mechanics, displacement is one of two subtly different quantities measuring distance. ... A constraint is a limitation of possibilities. ... The concept of a virtual displacement is meaningful only when discussing a physical system subject to contraints on its motion. ...

The principle is also known as the principle of virtual work. Virtual work describes a variational approach to solving physics problems. ...

This above equation is often called d'Alembert's principle but it was first written in this variational form by Joseph Louis Lagrange. D'Alembert should be credited with demonstrating that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces need not consider constraint forces. Joseph Louis Lagrange Joseph Louis Lagrange (January 25, 1736 – April 10, 1813; born Giuseppe Luigi Lagrangia in Turin, Lagrange moved to Paris (1787) and became a French citizen, adopting the French translation of his name, Joseph Louis Lagrange) was an Italian mathematician and astronomer who made important contributions to classical...

D'Alembert's Principle of inertial forces

D'Alembert showed that one can transform an accelerating system into an equivalent static system by adding the so-called "inertial force" and "inertial torque" or moment. The inertial force must act through the centre of mass and the inertial torque can act anywhere. The system can then be analysed exactly as a static system subjected to this "inertial force and moment" and the external forces. The advantage is that, in the equivalent static system' one can take moments about any point (not just the centre of mass). This often leads to simpler calculations because any force (in turn) can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation (sum of moments = zero). In textbooks of engineering dynamics this is sometimes referred to as D'Alembert's principle.

Example for plane 2D motion of a rigid body

For a planar rigid body, moving in the plane of the body (the x-y plane), and subjected to forces and torques causing rotation only in this plane, the inertial force is

where is the position vector of the centre of mass of the body, and m is the mass of the body. The inertial torque (or moment) is

where I is the moment of inertia of the body. If, in addition to the external forces and torques acting on the body, the inertia force acting through the center of mass is added and the inertial torque is added (acting around the centre of mass is as good as anywhere) the system is equivalent to one in static equilibrium. Thus the equations of static equilibrium

hold good. The important thing is that is the sum of torques (or moments, including the inertial moment and the moment of the inertial force) taken about any point. The direct application of Newton's laws requires that the angular acceleration equation be applied only about the center of mass.