D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert, and Italian-French mathematician Joseph Louis Lagrange. D'Alembert's principle generalizes the principle of virtual work from static to dynamical systems by introducing forces of inertia which, when added to the applied forces in a system, result in dynamic equilibrium.
The principle does not apply for irreversible displacements, such as sliding friction, and more general specification of the irreversibility is required. D'Alembert's principle is more general than Hamilton's principle as it is not restricted to holonomic constraints that depend only on coordinates and time but not on velocities.
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: is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system, is the total applied force (excluding constraint forces) on the -th particle, is the mass of the -th particle, is the velocity of the -th particle, is the virtual displacement of the -th particle, consistent with the constraints.
Newton's dot notation is used to represent the derivative with respect to time. The 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
However, some applications involve changing masses (for example, chains being rolled up or being unrolled) 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
are the total forces acting on the system's particles, are the inertial forces that result from the total forces.
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 (which is not usually the case, so this derivation works only for special cases), 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.