1. Why d’Alembert’s Principle? (Intuition First)

Newton’s second law for the $i$-th particle of mass $m_i$ is

\[ \mathbf{F}_i = m_i \ddot{\mathbf{R}}_i. \]

For a system of $N$ particles this leads to many coupled vector equations. In analytical dynamics we prefer to describe the system using generalized coordinates $q_j$, $j = 1,\dots,n$, instead of the individual position vectors $\mathbf{R}_i$.

Difficulties in a direct Newton approach:

• Each position $\mathbf{R}_i(t)$ depends on several generalized coordinates $q_j$.
• Constraint forces appear explicitly and are often messy to compute.
• Many of these constraint forces never do any useful work.

d’Alembert’s idea is to avoid fighting the constraints directly. Instead of considering the actual motion, he considers imaginary instantaneous displacements that are consistent with all constraints, called virtual displacements $\delta \mathbf{R}_i$.

His principle can be paraphrased as:

For all virtual displacements that respect the constraints, the difference between applied forces and inertial forces does no virtual work.

This observation allows us to remove constraint forces entirely from the working equations of motion.

2. Virtual Displacements: What They Are

The position of the $i$-th particle can be written as a function of time and generalized coordinates:

\[ \mathbf{R}_i = \mathbf{R}_i(t, q_1, \dots, q_n). \]

A virtual displacement $\delta \mathbf{R}_i$ is:

• infinitesimal and instantaneous (no time evolution is implied),
• consistent with all constraints at that instant,
• independent of the actual velocities and accelerations.

Mathematically, varying only the coordinates $q_j$ at fixed time gives

\[ \delta \mathbf{R}_i = \sum_{j=1}^{n} \frac{\partial \mathbf{R}_i}{\partial q_j} \,\delta q_j, \]

where the vectors

\[ \mathbf{v}_{ij} \equiv \frac{\partial \mathbf{R}_i}{\partial q_j} \]

are called partial velocities. For each $j$, the set $\{\mathbf{v}_{1j}, \dots, \mathbf{v}_{Nj}\}$ describes how the system moves if only $q_j$ is changed.

3. Virtual Work

Let $\mathbf{f}_i$ be the resultant of all applied (non-constraint) forces on particle $i$. The virtual work of $\mathbf{f}_i$ under the virtual displacement $\delta \mathbf{R}_i$ is

\[ \delta W_i = \mathbf{f}_i \cdot \delta \mathbf{R}_i. \]

The total virtual work of all applied forces is

\[ \delta W = \sum_{i=1}^{N} \mathbf{f}_i \cdot \delta \mathbf{R}_i. \]

Constraint forces $\mathbf{f}_{c,i}$ are normal to the constraint surface, whereas admissible virtual displacements lie in the tangent plane of the constraint surface. Hence

\[ \mathbf{f}_{c,i} \cdot \delta \mathbf{R}_i = 0 \quad \text{for all admissible } \delta \mathbf{R}_i. \]

This orthogonality is the key that allows constraint forces to disappear from the virtual-work equations.

4. d’Alembert’s Principle

Start from Newton’s second law for each particle:

\[ \mathbf{f}_i + \mathbf{f}_{c,i} = m_i \ddot{\mathbf{R}}_i. \]

Rearranging,

\[ \mathbf{f}_i - m_i \ddot{\mathbf{R}}_i + \mathbf{f}_{c,i} = \mathbf{0}. \]

Take the dot product with $\delta \mathbf{R}_i$ and sum over all particles:

\[ \sum_{i=1}^{N} \left( \mathbf{f}_i - m_i \ddot{\mathbf{R}}_i + \mathbf{f}_{c,i} \right) \cdot \delta \mathbf{R}_i = 0. \]

Using $\mathbf{f}_{c,i} \cdot \delta \mathbf{R}_i = 0$ for admissible virtual displacements, the constraint-term drops out and we obtain

\[ \sum_{i=1}^{N} \left( \mathbf{f}_i - m_i \ddot{\mathbf{R}}_i \right) \cdot \delta \mathbf{R}_i = 0. \]

This is the standard form of d’Alembert’s principle:

For any system of particles and any virtual displacement compatible with the constraints, the total virtual work of the applied forces plus the inertial “forces” $-m_i\ddot{\mathbf{R}}_i$ is zero.

5. Generalized Forces and the “Generalized Newton’s Law”

Substitute the expansion

\[ \delta \mathbf{R}_i = \sum_{j=1}^{n} \mathbf{v}_{ij} \,\delta q_j \]

into d’Alembert’s principle:

\[ \sum_{i=1}^{N} \left( \mathbf{f}_i - m_i \ddot{\mathbf{R}}_i \right) \cdot \left( \sum_{j=1}^{n} \mathbf{v}_{ij} \,\delta q_j \right) = 0. \]

Swap the order of summations:

\[ \sum_{j=1}^{n} \left[ \sum_{i=1}^{N} \left( \mathbf{f}_i - m_i \ddot{\mathbf{R}}_i \right) \cdot \mathbf{v}_{ij} \right] \delta q_j = 0. \]

The virtual coordinate variations $\delta q_j$ are independent and arbitrary, so each bracket must vanish separately:

\[ \sum_{i=1}^{N} m_i \ddot{\mathbf{R}}_i \cdot \mathbf{v}_{ij} = \sum_{i=1}^{N} \mathbf{f}_i \cdot \mathbf{v}_{ij}, \qquad j = 1,\dots,n. \]

Define the generalized force associated with $q_j$ as

\[ Q_j \equiv \sum_{i=1}^{N} \mathbf{f}_i \cdot \mathbf{v}_{ij}, \]

and the corresponding generalized inertial term as

\[ \sum_{i=1}^{N} m_i \ddot{\mathbf{R}}_i \cdot \frac{\partial \mathbf{R}_i}{\partial q_j}. \]

Then the equation above can be written compactly as

\[ \sum_{i=1}^{N} m_i \ddot{\mathbf{R}}_i \cdot \frac{\partial \mathbf{R}_i}{\partial q_j} = Q_j, \qquad j = 1,\dots,n. \]

This is the generalized form of Newton’s law. In Domain B.3 it will be re-expressed in terms of the Lagrangian $L = T - V$ to produce the familiar Euler–Lagrange equations.

6. Physical Interpretation: “Dynamics as Statics”

d’Alembert’s principle can be interpreted as a static equilibrium condition in the space of virtual displacements. For each particle,

\[ \underbrace{\mathbf{f}_i}_{\text{applied}} + \underbrace{\mathbf{f}_{c,i}}_{\text{constraint}} + \underbrace{(-m_i \ddot{\mathbf{R}}_i)}_{\text{inertial}} = \mathbf{0}. \]

Projecting this equation onto all admissible virtual displacements and using the fact that $\mathbf{f}_{c,i}$ does no virtual work, we are left with a “balance” between applied forces and inertial forces alone.

In this sense, d’Alembert’s principle turns a dynamics problem into a statics problem in the virtual-work space, which is much easier to handle using generalized coordinates.

7. Summary Box

What d’Alembert’s Principle Gives You

• Automatically eliminates constraint forces from the equations of motion.
• Rewrites Newton’s law directly in generalized coordinates $q_j$.
• Defines generalized forces $Q_j$ that capture the effect of applied forces.
• Provides the bridge from Newton’s law to Lagrange’s equations.

Core Formula (Virtual-Work Form)

\[ \sum_{i=1}^{N} \left( \mathbf{f}_i - m_i \ddot{\mathbf{R}}_i \right) \cdot \delta \mathbf{R}_i = 0 \quad \text{for all admissible } \{\delta \mathbf{R}_i\}. \]