D'Alembert's principle

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Image:Jean le Rond d'Alembert 250px.jpg 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.

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:

<math>

\sum_{i}\left({ {\mathbf F}_{i} - \dot {\mathbf p}_{i} }\right) \cdot \delta{\mathbf r}_{i} = 0. </math>

The principle is also known as the principle of virtual work.

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 <math>{\mathbf F}_{i}</math> need not consider constraint forces.

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

<math> \mathbf{F}_i = - m\ddot{\mathbf{r}_c}</math>

where <math>\mathbf{r}_c</math> is the position vector of the centre of mass of the body, and <math>m</math> is the mass of the body. The inertial torque (or moment) is

<math>T_i = -I\ddot{\theta}</math>

where <math>I</math> 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

<math>\sum F_x = 0</math>

<math>\sum F_y = 0</math>

<math>\sum T = 0</math>

hold. The important thing is that <math>\sum T</math> 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.

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