thumb|Traité de dynamique by [[Jean Le Rond d'Alembert, 1743. In it, the French scholar enunciated the principle of the quantity of movement, also known as "D'Alembert's principle".]]
thumb|right|[[Jean d'Alembert (1717–1783)]]
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.
D'Alembert's principle can be applied in cases of kinematic constraints that depend 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,
<math display="block">\sum_i ( \mathbf F_i - m_i \dot\mathbf{v}_i - \dot{m}_i\mathbf{v}_i)\cdot \delta \mathbf r_i = 0,</math>
where:
- <math>i</math> is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,
- <math>\mathbf {F}_i</math> is the total applied force (excluding constraint forces) on the <math>i</math>-th particle,
- <math> m_i </math> is the mass of the <math>i</math>-th particle,
- <math>\mathbf v_i</math> is the velocity of the <math>i</math>-th particle,
- <math>\delta \mathbf r_i</math> is the virtual displacement of the <math>i</math>-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 <math>\mathbf Q_j</math> 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 <math>i</math>-th mass is the product of its mass and velocity:
<math display="block">\mathbf p_i = m_i \mathbf v_i</math>
and its time derivative is
<math display="block">\dot{\mathbf{p_i = \dot{m}_i \mathbf{v}_i + m_i \dot{\mathbf{v_i.</math>
In many applications, the masses are constant and this equation reduces to
<math display="block">\dot{\mathbf{p_i = m_i \dot{\mathbf{v_i = m_i \mathbf{a}_i.</math>
However, some applications involve changing masses (for example, chains being rolled up or being unrolled) and in those cases both terms <math>\dot{m}_i \mathbf{v}_i</math> and <math>m_i \dot{\mathbf{v_i</math> have to remain present, giving
<math display="block">\sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i - \dot{m}_i \mathbf{v}_i)\cdot \delta \mathbf r_i = 0.</math>
If the variable mass is ejected with a velocity <math>\mathbf{w}_i</math> the principle has an additional term:
<math display="block">\sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i - \dot{m}_i (\mathbf{v}_i - \mathbf{w}_i))\cdot \delta \mathbf r_i = 0.</math>
Special case with constant mass
Consider Newton's law for a system of particles of constant mass, <math>i</math>. The total force on each particle is
<math display="block">\mathbf {F}_{i}^{(T)} = m_i \mathbf {a}_i,</math>
where
- <math>\mathbf {F}_{i}^{(T)}</math> are the total forces acting on the system's particles,
- <math>m_i \mathbf {a}_i</math> 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: 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: A potential cause for these errors is the sign of the inertial forces. Inertial forces can be used to describe an apparent force in a non-inertial reference frame that has an acceleration <math>\mathbf{a}</math> with respect to an inertial reference frame. In such a non-inertial reference frame, a mass that is at rest and has zero acceleration in an inertial reference system, because no forces are acting on it, will still have an acceleration <math>-\mathbf{a}</math> and an apparent inertial, or pseudo or fictitious force <math>-m\mathbf{a}</math> will seem to act on it: in this situation the inertial force has a minus sign.