02 May 2026

D'Alembert Principle and Lagrange Equations

D'Alembert's principle, generalized coordinates, Lagrange equations, and simple applications.

msc semester-i classical-mechanics lagrange-equation dalembert-principle

Newton’s equation is written for each particle in terms of vector coordinates and forces. This is straightforward for a free particle, but it becomes inconvenient when the motion is constrained, as in a pendulum, bead on a wire, or rigid body. Constraint forces may be unknown, and solving for them is often not the main aim.

D’Alembert’s principle gives a way to remove ideal constraint forces from the calculation. Lagrange’s equations then describe the motion directly in terms of independent generalized coordinates.

D’Alembert’s principle

For the $i$-th particle,

\[\mathbf F_i=m_i\ddot{\mathbf r}_i.\]

D’Alembert rewrites this as a virtual-work statement:

\[\boxed{ \sum_i\left(\mathbf F_i-m_i\ddot{\mathbf r}_i \right)\cdot\delta\mathbf r_i=0. }\]

For ideal constraints, constraint forces do no virtual work. Hence only applied forces remain in the variational equation.

The displacement $\delta\mathbf r_i$ is virtual: it is an imagined infinitesimal change compatible with the constraints at a fixed instant of time. It is not the actual displacement during a time interval.

Generalized coordinates

If a system has $n$ independent degrees of freedom, write

\[\mathbf r_i=\mathbf r_i(q_1,q_2,\dots,q_n,t).\]

The virtual displacement is

\[\delta\mathbf r_i=\sum_j\frac{\partial \mathbf r_i}{\partial q_j}\delta q_j.\]

The generalized force is

\[Q_j=\sum_i\mathbf F_i\cdot\frac{\partial\mathbf r_i}{\partial q_j}.\]

Generalized coordinates need not be lengths. They may be angles, distances along a curve, or any independent variables that specify the configuration. The advantage is that the number of equations becomes equal to the number of degrees of freedom.

Lagrange’s equation

Using D’Alembert’s principle and the kinetic energy

\[T=\frac12\sum_i m_i\dot{\mathbf r}_i^2,\]

one obtains

\[\boxed{ \frac{d}{dt}\left(\frac{\partial T}{\partial\dot q_j} \right)-\frac{\partial T}{\partial q_j}=Q_j. }\]

If the forces are conservative,

\[Q_j=-\frac{\partial V}{\partial q_j}.\]

With

\[L=T-V,\]

Lagrange’s equation becomes

\[\boxed{ \frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_j} \right)-\frac{\partial L}{\partial q_j}=0. }\]

Cyclic coordinate

If a coordinate $q_k$ does not appear explicitly in $L$, then

\[\frac{\partial L}{\partial q_k}=0.\]

Therefore

\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_k} \right)=0,\]

so the conjugate momentum

\[p_k=\frac{\partial L}{\partial\dot q_k}\]

is conserved.

Example: simple pendulum

For a pendulum of length $l$ and mass $m$, choose $q=\theta$.

\[T=\frac12ml^2\dot\theta^2, \qquad V=mgl(1-\cos\theta).\]

The Lagrangian is

\[L=\frac12ml^2\dot\theta^2-mgl(1-\cos\theta).\]

Lagrange’s equation gives

\[ml^2\ddot\theta+mgl\sin\theta=0,\]

or

\[\boxed{ \ddot\theta+\frac{g}{l}\sin\theta=0. }\]

For small oscillations,

\[\sin\theta\simeq\theta,\]

so

\[\ddot\theta+\frac{g}{l}\theta=0.\]

Example: particle in a central potential

For motion in a plane under $V(r)$,

\[L=\frac12m(\dot r^2+r^2\dot\theta^2)-V(r).\]

Since $\theta$ is cyclic,

\[p_\theta=\frac{\partial L}{\partial\dot\theta}=mr^2\dot\theta\]

is conserved. This is angular momentum conservation.

Particle in a uniform gravitational field

Let a particle move vertically under gravity. Choose the upward coordinate $y$. Then

\[T=\frac12m\dot y^2, \qquad V=mgy.\]

The Lagrangian is

\[L=\frac12m\dot y^2-mgy.\]

Now

\[\frac{\partial L}{\partial \dot y}=m\dot y, \qquad \frac{d}{dt}\left(\frac{\partial L}{\partial \dot y}\right)=m\ddot y,\]

and

\[\frac{\partial L}{\partial y}=-mg.\]

Therefore Lagrange’s equation gives

\[m\ddot y+mg=0,\]

or

\[\boxed{\ddot y=-g.}\]

This is the usual equation of vertical motion obtained without separately writing Newton’s vector equation.

Derivation points

In a derivation of Lagrange’s equation, the important steps are:

1. write D’Alembert’s principle in virtual-work form;
2. express $\delta\mathbf r_i$ in terms of $\delta q_j$;
3. introduce generalized forces $Q_j$;
4. use the kinetic-energy identity;
5. use the independence of $\delta q_j$ to obtain the equations.

Main points

• D’Alembert’s principle removes ideal constraint forces from the virtual-work equation.
• Generalized coordinates describe only independent motion.
• Lagrange’s equation is $\frac{d}{dt}(\partial L/\partial\dot q)-\partial L/\partial q=0$.
• A cyclic coordinate gives a conserved conjugate momentum.
• Simple applications include the pendulum and central-force motion.

Practice questions

1. State D’Alembert’s principle.
2. Define generalized force $Q_j$.
3. Derive Lagrange’s equation from D’Alembert’s principle.
4. Obtain the equation of motion of a simple pendulum.
5. Show that angular momentum is conserved for a central potential.