D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classicallaws of motion. It is named after its discoverer, the French physicist and mathematicianJean le Rond d'Alembert. It is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamilton's principle, avoiding restriction to holonomic systems. A holonomic constraint depends only on the coordinates and time. It does not depend on the velocities. If the negative terms in accelerations are recognized as inertial forces, the statement of d'Alembert's principle becomes The total virtual work of the impressed forces plus the inertial forces vanishes for reversible displacements. The principle does not apply for irreversible displacements, such as sliding friction, and more general specification of the irreversibility is required.
Statement of the principle
The principle states that the sum of the differences between the forces acting on a system of massive particles and the time derivatives of the momenta of the system itself projected onto any virtual displacement consistent with the constraints of the system is zero. Thus, in mathematical notation, d'Alembert's principle is written as follows, where : Newton's dot notation is used to represent the derivativewith respect to time. 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's contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces need not include constraint forces. It is equivalent to the somewhat more cumbersome Gauss's principle of least constraint.
Derivations
General case with variable mass
The general statement of d'Alembert's principle mentions "the time derivatives of the momenta of the system." By Newton's second law, the first time derivative of momentum is the force. The momentum of the -th mass is the product of its mass and velocity: and its time derivative is In many applications, the masses are constant and this equation reduces to which appears in the formula given above. However, some applications involve changing masses and in those cases both terms and have to remain present, giving
Special case with constant mass
Consider Newton's law for a system of particles of constant mass,. The total force on each particle is where Moving the inertial forces to the left gives an expression that can be considered to represent quasi-static equilibrium, but which is really just a small algebraic manipulation of Newton's law: Considering the virtual work,, done by the total and inertial forces together through an arbitrary virtual displacement,, of the system leads to a zero identity, since the forces involved sum to zero for each particle. The original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements. Separating the total forces into applied forces,, and constraint forces,, yields If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, the constraint forces don't do any work,. Such displacements are said to be consistent with the constraints. This leads to the formulation of d'Alembert's principle, which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work:. There is also a corresponding principle for static systems called the principle of virtual work for applied forces.
D'Alembert's principle of inertial forces
D'Alembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called "inertial force" and "inertial torque" or moment. The inertial force must act through the center of mass and the inertial torque can act anywhere. The system can then be analyzed 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. This often leads to simpler calculations because any force can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation. Even in the course of Fundamentals of Dynamics and Kinematics of machines, this principle helps in analyzing the forces that act on a link of a mechanism when it is in motion. In textbooks of engineering dynamics this is sometimes referred to as d'Alembert's principle.
Dynamic equilibrium
D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that is to be for any set of virtual displacements δqj. This condition yields m equations, which can also be written as The result is a set of m equations of motion that define the dynamics of the rigid body system.
