D'Alembert's principle

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 ${\displaystyle i}$-th mass is the product of its mass and velocity: $${\displaystyle {\mathbf{p}}_i=m_i{\mathbf{v}}_i}$$ and its time derivative is $${\displaystyle {\dot{\mathbf{p}}}_i={\dot{m}}_i{\mathbf{v}}_i+m_i{\dot{\mathbf{v}}}_i.}$$

In many applications, the masses are constant and this equation reduces to $${\displaystyle {\dot{\mathbf{p}}}_i=m_i{\dot{\mathbf{v}}}_i=m_i{\mathbf{a}}_i.}$$

However, some applications involve changing masses (for example, chains being rolled up or being unrolled) and in those cases both terms ${\displaystyle {\dot{m}}_i{\mathbf{v}}_i}$ and ${\displaystyle m_i{\dot{\mathbf{v}}}_i}$ have to remain present, giving $${\displaystyle \sum _i({\mathbf{F}}_i-m_i{\mathbf{a}}_i-{\dot{m}}_i{\mathbf{v}}_i)\cdot \delta {\mathbf{r}}_i=0.}$$ If the variable mass is ejected with a velocity ${\displaystyle {\mathbf{w}}_i}$ the principle has an additional term: $${\displaystyle \sum _i({\mathbf{F}}_i-m_i{\mathbf{a}}_i-{\dot{m}}_i({\mathbf{v}}_i-{\mathbf{w}}_i))\cdot \delta {\mathbf{r}}_i=0.}$$

Special case with constant mass

Consider Newton's law for a system of particles of constant mass, ${\displaystyle i}$. The total force on each particle is $${\displaystyle {\mathbf{F}}_i^{(T)}=m_i{\mathbf{a}}_i,}$$ where

• ${\displaystyle {\mathbf{F}}_i^{(T)}}$ are the total forces acting on the system's particles,
• ${\displaystyle m_i{\mathbf{a}}_i}$ 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: $${\displaystyle {\mathbf{F}}_i^{(T)}-m_i{\mathbf{a}}_i=\mathbf{0}.}$$

Considering the virtual work, ${\displaystyle \delta W}$, done by the total and inertial forces together through an arbitrary virtual displacement, ${\displaystyle \delta {\mathbf{r}}_i}$, of the system leads to a zero identity, since the forces involved sum to zero for each particle.

$${\displaystyle \delta W=\sum _i{\mathbf{F}}_i^{(T)}\cdot \delta {\mathbf{r}}_i-\sum _i{m}_i{\mathbf{a}}_i\cdot \delta {\mathbf{r}}_i=0}$$

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, ${\displaystyle {\mathbf{F}}_i}$, and constraint forces, ${\displaystyle {\mathbf{C}}_i}$, yields $${\displaystyle \delta W=\sum _i{\mathbf{F}}_i\cdot \delta {\mathbf{r}}_i+\sum _i{\mathbf{C}}_i\cdot \delta {\mathbf{r}}_i-\sum _i{m}_i{\mathbf{a}}_i\cdot \delta {\mathbf{r}}_i=0.}$$

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, $\sum _i{\mathbf{C}}_i\cdot \delta {\mathbf{r}}_i=0$. 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: $${\displaystyle \delta W=\sum _i({\mathbf{F}}_i-m_i{\mathbf{a}}_i)\cdot \delta {\mathbf{r}}_i=0.}$$

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 (not just the center 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). 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.

Some educators caution that attempts to use d'Alembert inertial mechanics lead students to make frequent sign errors. 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 ${\displaystyle \mathbf{a}}$ 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 ${\displaystyle -\mathbf{a}}$ and an apparent inertial, or pseudo or fictitious force ${\displaystyle -m\mathbf{a}}$ will seem to act on it: in this situation the inertial force has a minus sign.

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 ${\displaystyle n}$ rigid bodies with ${\displaystyle m}$ generalized coordinates requires $${\displaystyle \delta W=\left(Q_1+Q_1^{\ast }\right)\delta q_1+\cdots +\left(Q_m+Q_m^{\ast }\right)\delta q_m=0,}$$ for any set of virtual displacements ${\displaystyle \delta q_j}$ with ${\displaystyle Q_j}$ being a generalized applied force and ${\displaystyle Q_j^{\ast }}$ being a generalized inertia force. This condition yields ${\displaystyle m}$ equations, $${\displaystyle Q_j+Q_j^{\ast }=0,j=1,\dots ,m,}$$ which can also be written as $${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }T}{\mathrm{\partial }{\dot{q}}_j}-\frac{\mathrm{\partial }T}{\mathrm{\partial }{q}_j}=Q_j,j=1,\dots ,m.}$$ The result is a set of m equations of motion that define the dynamics of the rigid body system.

Formulation using the Lagrangian

D'Alembert's principle can be rewritten in terms of the Lagrangian ${\displaystyle L=T-V}$ of the system as a generalized version of Hamilton's principle for the case of point particles, as follows, $${\displaystyle \delta {\int }_{t_1}^{t_2}L(\mathbf{r},\dot{\mathbf{r}},t)dt+\sum _i{\int }_{t_1}^{t_2}{\mathbf{F}}_i\cdot \delta {\mathbf{r}}_{i}dt=0,}$$ where:

• ${\displaystyle \mathbf{r}=({\mathbf{r}}_1,...,{\mathbf{r}}_N)}$

• ${\displaystyle {\mathbf{F}}_i}$ are the applied forces

• ${\displaystyle \delta {\mathbf{r}}_i}$ is the virtual displacement of the ${\displaystyle i}$-th particle, consistent with the constraints ${\displaystyle \sum _i{\mathbf{C}}_i\cdot \delta {\mathbf{r}}_i=0}$
• the critical curve satisfies the constraints ${\displaystyle \sum _i{\mathbf{C}}_i\cdot {\dot{\mathbf{r}}}_i=0}$

With the Lagrangian $${\displaystyle L(\mathbf{r},\dot{\mathbf{r}},t)=\sum _i\frac{1}{2}{m}_i{\dot{\mathbf{r}}}_i^2,}$$ the previous statement of d'Alembert principle is recovered.

Generalization for thermodynamics

An extension of d'Alembert's principle can be used in thermodynamics. For instance, for an adiabatically closed thermodynamic system described by a Lagrangian depending on a single entropy S and with constant masses ${\displaystyle m_i}$, such as $${\displaystyle L(\mathbf{r},\dot{\mathbf{r}},S,t)=\sum _i\frac{1}{2}{m}_i{\dot{\mathbf{r}}}_i^2-V(\mathbf{r},S),}$$ it is written as follows $${\displaystyle \delta {\int }_{t_1}^{t_2}L(\mathbf{r},\dot{\mathbf{r}},S,t)dt+\sum _i{\int }_{t_1}^{t_2}{\mathbf{F}}_i\cdot \delta {\mathbf{r}}_{i}dt=0,}$$ where the previous constraints $\sum _i{\mathbf{C}}_i\cdot \delta {\mathbf{r}}_i=0$ and $\sum _i{\mathbf{C}}_i\cdot {\dot{\mathbf{r}}}_i=0$ are generalized to involve the entropy as:

• ${\displaystyle \sum _i{\mathbf{C}}_i\cdot \delta {\mathbf{r}}_i+T\delta S=0}$
• ${\displaystyle \sum _i{\mathbf{C}}_i\cdot {\dot{\mathbf{r}}}_i+T\dot{S}=0.}$

Here ${\displaystyle T=\mathrm{\partial }V/\mathrm{\partial }S}$ is the temperature of the system, ${\displaystyle {\mathbf{F}}_i}$ are the external forces, ${\displaystyle {\mathbf{C}}_i}$ are the internal dissipative forces. It results in the mechanical and thermal balance equations: $${\displaystyle m_i{\mathbf{a}}_i=-\frac{\mathrm{\partial }V}{\mathrm{\partial }{\mathbf{r}}_i}+{\mathbf{C}}_i+{\mathbf{F}}_i,i=1,...,NT\dot{S}=-\sum _i{\mathbf{C}}_i\cdot {\dot{\mathbf{r}}}_i.}$$ Typical applications of the principle include thermo-mechanical systems, membrane transport, and chemical reactions.

For ${\displaystyle \delta S=\dot{S}=0}$ the classical d'Alembert principle and equations are recovered.