---
title: "Lagrange's equations from D'Alembert's principle"
subject: "Engineering Mechanics"
chapter: 9
key: "semester_3/engineering_mechanics:lagrange-equations-derivation-dalembert"
source: https://www.bctengineer.com/q/semester_3/engineering_mechanics/lagrange-equations-derivation-dalembert
site: BCT Engineering
---

# Lagrange's equations from D'Alembert's principle

Engineering Mechanics · Chapter 9: Lagrangian Dynamics

- Asked in: year not given, 8 marks
- Video: [lagrange equation from d'almbert principle (hindi)](https://www.youtube.com/watch?v=QGCdoGoZb9A) — Eduphile
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/engineering_mechanics/lagrange-equations-derivation-dalembert
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_mechanics/chapter-9

## Question

Starting from D'Alembert's principle, derive Lagrange's equations of motion for a system of particles with holonomic constraints in terms of generalised coordinates. Show how the equations reduce to the form with the Lagrangian L = T - V for a conservative system.

## Short answer

Starting from D’Alembert’s principle, express virtual work in generalized coordinates; applied forces give generalized forces Q_j; inertia term converts to d/dt(∂T/∂qdot) - ∂T/∂q. Independent virtual displacements yield d/dt(∂T/∂qdot) - ∂T/∂q = Q_j. For conservative systems, substituting Q_j = -∂V/∂q and L = T - V gives d/dt(∂L/∂qdot) - ∂L/∂q = 0.

## Answer

Consider $N$ particles with positions $\mathbf{r}_i(q_1,\dots,q_n,t)$ in terms of $n$ independent generalised coordinates $q_j$ (holonomic constraints). Let $\mathbf{F}_i$ be the applied force on particle $i$ (constraint forces do no virtual work on smooth constraints).

#### Step 1: D'Alembert-Lagrange principle
D'Alembert's principle with virtual displacements: the virtual work of the real forces plus the inertia forces is zero:

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

#### Step 2: Express in generalised coordinates
Since $\delta\mathbf{r}_i = \sum_j \dfrac{\partial\mathbf{r}_i}{\partial q_j}\delta q_j$, the applied-force term becomes $\sum_j Q_j\,\delta q_j$, where the **generalised force** is

$$
Q_j = \sum_i \mathbf{F}_i\cdot\frac{\partial\mathbf{r}_i}{\partial q_j}
$$

The inertia term is $\sum_j\left[\sum_i m_i\ddot{\mathbf{r}}_i\cdot\dfrac{\partial\mathbf{r}_i}{\partial q_j}\right]\delta q_j$.

#### Step 3: Convert the inertia term to kinetic energy
The velocity is $\dot{\mathbf{r}}_i = \sum_j \dfrac{\partial\mathbf{r}_i}{\partial q_j}\dot q_j + \dfrac{\partial\mathbf{r}_i}{\partial t}$. Two identities follow:

$$
\frac{\partial\dot{\mathbf{r}}_i}{\partial\dot q_j} = \frac{\partial\mathbf{r}_i}{\partial q_j}
\qquad
\frac{d}{dt}\left(\frac{\partial\mathbf{r}_i}{\partial q_j}\right) = \frac{\partial\dot{\mathbf{r}}_i}{\partial q_j}
$$

Then

$$
\sum_i m_i\ddot{\mathbf{r}}_i\cdot\frac{\partial\mathbf{r}_i}{\partial q_j}
= \frac{d}{dt}\left(\sum_i m_i\dot{\mathbf{r}}_i\cdot\frac{\partial\dot{\mathbf{r}}_i}{\partial\dot q_j}\right) - \sum_i m_i\dot{\mathbf{r}}_i\cdot\frac{\partial\dot{\mathbf{r}}_i}{\partial q_j}
$$

With kinetic energy $T = \sum_i\tfrac12 m_i\dot{\mathbf{r}}_i^2$, this equals $\dfrac{d}{dt}\left(\dfrac{\partial T}{\partial\dot q_j}\right) - \dfrac{\partial T}{\partial q_j}$.

#### Step 4: Lagrange's equations
Because the $\delta q_j$ are independent, each coefficient must vanish:

$$
\frac{d}{dt}\left(\frac{\partial T}{\partial\dot q_j}\right) - \frac{\partial T}{\partial q_j} = Q_j \qquad (j = 1, \dots, n)
$$

#### Conservative systems
For conservative forces $Q_j = -\dfrac{\partial V}{\partial q_j}$, where $V = V(q)$ does not depend on velocities. Define the **Lagrangian** $L = T - V$. Since $\dfrac{\partial V}{\partial\dot q_j} = 0$:

$$
\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_j}\right) - \frac{\partial L}{\partial q_j} = 0
$$

If some forces are non-conservative, they appear as $Q_j^{nc}$ on the right-hand side. The number of equations equals the number of DOF, and constraint forces never appear.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the starting principle state?
   A. The virtual work of the real forces plus inertia forces is zero: sum (F_i - m_i r_i'') . delta r_i = 0
   B. Total energy is constant
   C. Sum F = 0 for each particle
   D. Momentum is conserved
   Answer: A. It is the principle with virtual displacements, with constraint forces doing no virtual work. (Key point: D'Alembert-Lagrange principle)

2. How is the generalised force Q_j defined?
   A. Q_j = sum F_i . dr_i/dq_j
   B. Q_j = sum m_i r_i''
   C. Q_j = dT/dq_j
   D. Q_j = sum F_i dq_j
   Answer: A. Substituting delta r_i in terms of delta q_j gives Q_j = sum F_i . partial r_i/partial q_j. (Key point: Generalised force)

3. What do the identities lead to for the inertia term?
   A. d/dt(dT/dq_j-dot) - dT/dq_j
   B. dT/dq_j-dot
   C. T - V
   D. d/dt(dT/dq_j)
   Answer: A. The inertia term equals d/dt(partial T / partial q_j-dot) - partial T / partial q_j. (Key point: Kinetic energy form)

4. For conservative forces, why does the equation take the form d/dt(dL/dq-dot) - dL/dq = 0?
   A. Because Q_j = -dV/dq_j and V does not depend on velocity, so dV/dq-dot = 0
   B. Because T is zero
   C. Because V is constant
   D. Because Q_j = 0 for all systems
   Answer: A. Moving -dV/dq to the left and using L = T - V gives the standard form. (Key point: Lagrangian)

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