D’Alembert’s Principle

At a Glance

• Frequency: 1 sub-part across 1 of 13 years (2021)
• Priority tier: T4
• Marks (count): 10 (1)
• Average solve time: ~15 min
• Difficulty mix: medium 1
• Section: A | Dominant type: proof

Why This Chapter Matters

D’Alembert’s principle is the conceptual bridge between Newton’s laws and Lagrangian mechanics: it restates dynamics as a statics problem by introducing inertia forces, and from it Lagrange’s equations follow directly. UPSC 2021 tested the statement of the principle and its application to a standard constrained system — a compact Section A question that rewards concise, rigorous presentation.

Minimum Theory

Newton’s Second Law and the Inertia Force

For a system of $N$ particles with masses $m_i$, positions $\mathbf{r}_i$, applied forces $\mathbf{F}_i$, and constraint forces $\mathbf{R}_i$:

$m_i \ddot{\mathbf{r}}_i = \mathbf{F}_i + \mathbf{R}_i, \quad i = 1, \ldots, N.$

Rearranging: $\mathbf{F}_i + \mathbf{R}_i - m_i\ddot{\mathbf{r}}_i = \mathbf{0}$.

The term $-m_i\ddot{\mathbf{r}}_i$ is called the inertia force (or reversed effective force) on particle $i$.

D’Alembert’s Principle (Statement)

D’Alembert’s Principle. For a system of particles subject to constraints, the total virtual work done by the applied forces and the inertia forces in any virtual displacement consistent with the constraints is zero:

$\sum_{i=1}^{N} \left(\mathbf{F}_i - m_i\ddot{\mathbf{r}}_i\right) \cdot \delta\mathbf{r}_i = 0.$

Here $\delta\mathbf{r}_i$ is a virtual displacement — an infinitesimal displacement compatible with the constraints at the current instant, with time held fixed.

Why constraint forces vanish. For ideal (workless) constraints — smooth surfaces, inextensible strings, rigid rods — the constraint forces do no virtual work: $\sum_i \mathbf{R}_i \cdot \delta\mathbf{r}_i = 0$. Hence the constraint forces drop out of D’Alembert’s equation, leaving only the applied forces $\mathbf{F}_i$.

Derivation of the Equation of Motion for a Constrained System

Express the virtual displacements in terms of generalised coordinates $q_j$ ($j = 1,\ldots,n$):

$\delta\mathbf{r}_i = \sum_{j=1}^{n} \frac{\partial \mathbf{r}_i}{\partial q_j}\,\delta q_j.$

Substituting into D’Alembert’s principle and collecting coefficients of each independent $\delta q_j$:

$\sum_{j=1}^{n} \left[\sum_{i=1}^{N} \left(\mathbf{F}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q_j} - m_i\ddot{\mathbf{r}}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q_j}\right)\right]\delta q_j = 0.$

Since the $\delta q_j$ are independent, each bracket is zero, leading (after algebraic manipulation) to Lagrange’s equations of motion:

$\frac{d}{dt}\frac{\partial T}{\partial \dot{q}_j} - \frac{\partial T}{\partial q_j} = Q_j,$

where $T$ is the kinetic energy and $Q_j = \sum_i \mathbf{F}_i \cdot \partial\mathbf{r}_i/\partial q_j$ is the generalised force.

Application: Atwood Machine

Two masses $m_1$ and $m_2$ connected by a light inextensible string over a frictionless pulley. Let $x$ be the displacement of $m_1$ downward; $m_2$ moves upward by $x$.

Generalised coordinate: $q = x$. Virtual displacement: $\delta\mathbf{r}_1 = \delta x\,\hat{j}$ (down), $\delta\mathbf{r}_2 = -\delta x\,\hat{j}$ (up).

D’Alembert’s principle:

$\left[(m_1 g - m_1\ddot{x})\cdot\delta x\right] + \left[(-m_2 g + m_2\ddot{x})\cdot(-\delta x)\right]^* = 0$

$\bigl[(m_1 g - m_1\ddot{x}) - (m_2 g - m_2\ddot{x})\bigr]\,\delta x = 0.$

Since $\delta x$ is arbitrary:

$(m_1 - m_2)g = (m_1 + m_2)\ddot{x} \implies \ddot{x} = \frac{(m_1 - m_2)g}{m_1 + m_2}.$

Question Archetypes

state-and-apply (1 question; 2021)

Recognition Cues

• Problem names D’Alembert’s principle explicitly.
• A constrained mechanical system is given (Atwood machine, compound pendulum, bead on wire, inclined plane).
• Asked to state the principle and derive the equation of motion.

Solution Template

1. State D’Alembert’s principle: $\sum_i (\mathbf{F}_i - m_i\ddot{\mathbf{r}}_i)\cdot\delta\mathbf{r}_i = 0$; state that ideal constraint forces do no virtual work.
2. Define the system: label masses, coordinates, degrees of freedom.
3. Express virtual displacements in terms of the single generalised coordinate (for a one-degree-of-freedom system).
4. Substitute into D’Alembert’s equation.
5. Factor out the independent virtual displacement and set its coefficient to zero.
6. Obtain the equation of motion.

Worked Example

2021 Paper 2, 2021-P2-QMF (10 marks)

State D’Alembert’s principle. Using it, derive the equation of motion of a compound pendulum (rigid body of mass $M$, moment of inertia $I$ about the pivot, centre of mass at distance $\ell$ from pivot).

Statement of D’Alembert’s Principle.

For a system of particles, the virtual work of all applied forces and inertia forces through any virtual displacement consistent with the constraints is zero:

$\sum_i (\mathbf{F}_i - m_i\ddot{\mathbf{r}}_i)\cdot\delta\mathbf{r}_i = 0.$

Ideal constraint forces are excluded because they do no virtual work.

Setup.

Let $\theta$ be the angle the pendulum makes with the vertical. The generalised coordinate is $q = \theta$. For a rigid body, D’Alembert’s principle in angular form reads:

$\left(\tau_{\text{applied}} - I\ddot{\theta}\right)\delta\theta = 0,$

where $\tau_{\text{applied}}$ is the moment of applied forces about the pivot and $I\ddot{\theta}$ is the “inertia torque”.

Applied torque.

Gravity $Mg$ acts downward at the centre of mass, which is at distance $\ell$ from the pivot. The restoring torque (opposing positive $\theta$) is:

$\tau_{\text{applied}} = -Mg\ell\sin\theta.$

Apply D’Alembert’s principle.

Since $\delta\theta$ is an arbitrary virtual displacement:

$-Mg\ell\sin\theta - I\ddot{\theta} = 0.$

Equation of motion:

$\boxed{I\ddot{\theta} + Mg\ell\sin\theta = 0.}$

For small oscillations ($\sin\theta \approx \theta$): $\ddot{\theta} + (Mg\ell/I)\,\theta = 0$, giving period $T = 2\pi\sqrt{I/(Mg\ell)}$. $\blacksquare$

Common Traps

• Forgetting to state that constraint forces drop out because they do no virtual work — this is the key step that must be stated explicitly.
• Applying the principle as $\sum \mathbf{F}_i = m_i\ddot{\mathbf{r}}_i$ (Newton’s law) without the virtual displacement structure — this is not D’Alembert’s principle.
• Sign errors: the inertia force is $-m_i\ddot{\mathbf{r}}_i$ (negative acceleration), not $+m_i\ddot{\mathbf{r}}_i$.
• For compound pendulum: using $M\ell^2$ instead of the correct moment of inertia $I$ about the pivot.

Marks-Aware Writing

This is a 10-mark Section A question. Be concise but complete:

1. State the principle in equation form — 2–3 marks.
2. Identify the system’s degree(s) of freedom and generalised coordinate — 1–2 marks.
3. Write out the virtual work equation and cancel the virtual displacement — 3–4 marks.
4. State the equation of motion cleanly in a box — 1 mark.

Do not derive Lagrange’s equations unless asked; it wastes time and reads as off-topic.

Practice Set

Only one historical question on this atom (shown above).