D'Alembert's principle

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

$${\displaystyle \sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i}-{\dot {m}}_{i}\mathbf {v} _{i})\cdot \delta \mathbf {r} _{i}=0.}$$

Consider Newton's law for a system of particles of constant mass, ${\displaystyle i}$. The total force on each particle is[7]

$${\displaystyle \mathbf {F} _{i}^{(T)}=m_{i}\mathbf {a} _{i},}$$

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:[7]

$${\displaystyle \mathbf {F} _{i}^{(T)}-m_{i}\mathbf {a} _{i}=\mathbf {0} .}$$

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.[7]

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

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[7]

$${\displaystyle \delta W=\sum _{i}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}+\sum _{i}\mathbf {C} _{i}\cdot \delta \mathbf {r} _{i}-\sum _{i}m_{i}\mathbf {a} _{i}\cdot \delta \mathbf {r} _{i}=0.}$$

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, ${\textstyle \sum _{i}\mathbf {C} _{i}\cdot \delta \mathbf {r} _{i}=0}$. Such displacements are said to be consistent with the constraints.[8] 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:[7]

$${\displaystyle \delta W=\sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i})\cdot \delta \mathbf {r} _{i}=0.}$$

There is also a corresponding principle for static systems called the principle of virtual work for applied forces.

An extension of d'Alembert's principle can be used in thermodynamics.[4] 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 ${\textstyle \sum _{i}\mathbf {C} _{i}\cdot \delta \mathbf {r} _{i}=0}$ and ${\textstyle \sum _{i}\mathbf {C} _{i}\cdot {\dot {\mathbf {r} }}_{i}=0}$ 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 ${\displaystyle T=\partial V/\partial S}$ 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:[4]

$${\displaystyle m_{i}\mathbf {a} _{i}=-{\frac {\partial V}{\partial \mathbf {r} _{i}}}+\mathbf {C} _{i}+\mathbf {F} _{i},\;\;i=1,...,N\qquad \qquad T{\dot {S}}=-\sum _{i}\mathbf {C} _{i}\cdot {\dot {\mathbf {r} }}_{i}.}$$

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.