From Newton to Lagrange: d'Alembert's Principle
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You should be comfortable with Newton’s second law for systems of particles, the concept of virtual displacements, and the distinction between holonomic and non-holonomic constraints.
Why this matters
d'Alembert's principle provides the critical bridge between force-based dynamics and the elegant, coordinate-free language of analytical mechanics.
While Newton's laws are powerful, they rely on vector forces that can be difficult to identify in complex constrained systems. By reformulating dynamics as a statement about virtual work, we eliminate the need to calculate constraint forces explicitly. This shift allows us to choose any convenient set of generalized coordinates, reducing the problem to scalar algebra and setting the stage for the Euler-Lagrange equations.
The Core Idea: Dynamics as Statics
Imagine a particle moving under the influence of a force. Newton tells us that the net force equals mass times acceleration. d'Alembert’s insight was to move the acceleration term to the left-hand side, treating it as an "inertial force" that opposes the motion. By doing this, the equation of motion looks exactly like a static equilibrium condition. The system is in "dynamic equilibrium" when the sum of the applied forces and the reversed inertial forces performs zero net work for any virtual displacement consistent with the constraints. This conceptual shift is profound: it transforms a problem of changing motion into a problem of balance, allowing us to leverage the powerful tools of statics and variational calculus.
Virtual Work and Ideal Constraints
To apply this principle, we must distinguish between real displacements, which occur over time, and virtual displacements, which are hypothetical infinitesimal changes in position at a fixed instant. For a system of \( N \) particles, the virtual displacement \( \delta\mathbf{r}_i \) of the \( i \)-th particle must be compatible with the constraints. If the constraints are ideal, meaning they do no work (such as a rigid rod or a smooth surface), the virtual work done by the constraint forces is zero. This allows us to focus solely on the applied forces, such as gravity or springs, which are often conservative.
This is the principle of virtual work for a system in equilibrium: for any virtual displacement allowed by the constraints, the total virtual work of the applied forces is zero. d’Alembert’s step, below, extends it to moving systems by adding the inertial term \(-m_i\ddot{\mathbf{r}}_i\) to each force. This is a scalar equation, which is far easier to handle than the vector equations of Newton's laws, especially when the number of constraints is large.
Generalized Coordinates and the Principle
Instead of using Cartesian coordinates \( x, y, z \), we describe the system using \( n \) independent generalized coordinates \( q_1, \dots, q_n \). The position of each particle is a function of these coordinates: \( \mathbf{r}_i = \mathbf{r}_i(q_1, \dots, q_n, t) \). The virtual displacement is then related to the changes in the generalized coordinates: \( \delta\mathbf{r}_i = \sum_{j=1}^{n} \frac{\partial \mathbf{r}_i}{\partial q_j} \delta q_j \). Substituting this into the virtual work equation and using the fact that the \( \delta q_j \) are independent, we arrive at d'Alembert's principle in its generalized form.
This set of \( n \) equations is equivalent to Newton's laws but is expressed in terms of the generalized coordinates. The term \( \mathbf{F}_i - m_i \ddot{\mathbf{r}}_i \) represents the effective force, combining the applied force and the inertial reaction. The beauty of this formulation is that it automatically accounts for the constraints, as they are built into the definition of the generalized coordinates.
From d'Alembert to Lagrange
To connect this to the Lagrangian formalism, we assume the applied forces are conservative, meaning they can be derived from a potential energy \( V(q, t) \). The virtual work of the applied forces is then \( -\sum_{j=1}^{n} \frac{\partial V}{\partial q_j} \delta q_j \). The kinetic energy \( T \) is a function of the generalized coordinates and their time derivatives. By manipulating the inertial term using the chain rule and the definition of the Lagrangian \( L = T - V \), we can show that d'Alembert's principle is exactly equivalent to the Euler-Lagrange equations.
This result is the culmination of the transition from Newtonian to Lagrangian mechanics. It shows that the dynamics of the system are encoded in the Lagrangian, a single scalar function. The Euler-Lagrange equations are derived from the principle of stationary action, which is a more general and powerful framework than d'Alembert's principle, but the latter provides the essential physical intuition for why the Lagrangian works.
The derivation
Euler-Lagrange Equations from d'Alembert
For a system of N particles subject to ideal (workless) holonomic constraints and described by n independent generalized coordinates \( q_1,\dots,q_n \), d'Alembert's principle \( \sum_i \left(\mathbf{F}_i - m_i \ddot{\mathbf{r}}_i\right)\cdot \delta\mathbf{r}_i = 0 \) is exactly equivalent to the n Euler–Lagrange equations \[ \frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}_j}\right) - \frac{\partial L}{\partial q_j} = 0, \qquad L = T - V, \] provided the applied forces derive from a potential \( V=V(q,t) \) that is independent of the generalized velocities.
A common pitfall is assuming that d'Alembert's principle applies to non-ideal constraints, such as friction. If the constraint forces do work, they must be included in the virtual work sum, which complicates the derivation and may prevent the system from being described by a simple Lagrangian. Always verify that your constraints are workless before applying the standard Lagrangian formalism. C
Worked example
Problem. A bead of mass \( m = 0.5\ \mathrm{kg} \) slides without friction on a vertical hoop of radius \( R = 0.2\ \mathrm{m} \). The hoop is fixed in a uniform gravitational field \( g = 9.8\ \mathrm{m/s^2} \). Let \( \theta \) be the angle from the vertical downward direction. Find the equation of motion for \( \theta(t) \).
- Identify the generalized coordinate: \( q = \theta \). The position of the bead is \( \mathbf{r} = (R \sin\theta, -R \cos\theta) \).
- Calculate the kinetic energy: \( T = \frac{1}{2} m \dot{\mathbf{r}} \cdot \dot{\mathbf{r}} = \frac{1}{2} m R^2 \dot{\theta}^2 \).
- Calculate the potential energy: \( V = m g y = -m g R \cos\theta \). (Taking the center of the hoop as the reference level).
- Form the Lagrangian: \( L = T - V = \frac{1}{2} m R^2 \dot{\theta}^2 + m g R \cos\theta \).
- Apply the Euler-Lagrange equation: \( \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}}\right) - \frac{\partial L}{\partial \theta} = 0 \).
- Compute derivatives: \( \frac{\partial L}{\partial \dot{\theta}} = m R^2 \dot{\theta} \) and \( \frac{\partial L}{\partial \theta} = -m g R \sin\theta \).
- Substitute and simplify: \( \frac{d}{dt}(m R^2 \dot{\theta}) - (-m g R \sin\theta) = 0 \implies m R^2 \ddot{\theta} + m g R \sin\theta = 0 \).
- Divide by \( m R \): \( R \ddot{\theta} + g \sin\theta = 0 \).
Units check. \( \ddot{\theta} \) has units \( \mathrm{rad/s^2} \) (dimensionless angle, so \( \mathrm{s^{-2}} \)). \( \frac{g}{R} \) has units \( \frac{\mathrm{m/s^2}}{\mathrm{m}} = \mathrm{s^{-2}} \). The units match.
Three quick tests
- What is the key difference between a real displacement and a virtual displacement?
Answer
A real displacement occurs over a finite time interval and satisfies the equations of motion, while a virtual displacement is an infinitesimal change in position at a fixed instant that is consistent with the constraints but does not necessarily satisfy the equations of motion. - Why are ideal constraints important for the Lagrangian formulation?
Answer
Ideal constraints do no work, so they do not appear in the virtual work equation. This allows us to ignore them when deriving the equations of motion, simplifying the problem significantly. - If the potential energy \( V \) depends on the generalized velocities \( \dot{q} \), can we still use the standard Euler-Lagrange equations?
Answer
Yes, provided the forces can be obtained from a generalized, velocity-dependent potential \(U(q,\dot{q})\) in the right way — the electromagnetic force on a charged particle is the standard example, with \(L = T - U\). Forces that cannot be written this way (such as friction) are instead added as generalized forces \(Q_j\) on the right-hand side of the equations.
Problem set for PU-101 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.