Lagrange’s equations

At a Glance

• Frequency: 9 sub-parts across 9 of 13 years (2013, 2015, 2016, 2017, 2018, 2021, 2022, 2023, 2025)
• Priority tier: T1
• Marks (count): 10 (3), 15 (3), 20 (3)
• Average solve time: ~15 min
• Difficulty mix: hard 4, medium 3, easy 2
• Section: B | Dominant type: derivation

Why This Chapter Matters

Lagrange’s equations appear in 9 of the last 13 years, with an even split across 10-, 15-, and 20-mark questions. The method is systematic: once you write down $T$ and $V$, the rest is calculus. UPSC favours four physical setups — rolling bodies, pulley systems, central-force orbits, and coupled oscillators — and one abstract setup (a given Lagrangian to differentiate). Mastering the procedure means these 9 appearances become largely mechanical.

Minimum Theory

Euler–Lagrange equations. For a system with generalised coordinates $q_1,\ldots,q_n$ and Lagrangian $L=T-V$: $\frac{d}{dt}\frac{\partial L}{\partial\dot q_k}-\frac{\partial L}{\partial q_k}=Q_k,\qquad k=1,\ldots,n,$ where $Q_k$ is the generalised force for $q_k$ (zero if no non-conservative forces, or if $V$ is complete). For a force $\mathbf F$ applied at point $\mathbf r$: $Q_k=\mathbf F\cdot\partial\mathbf r/\partial q_k$.

Steps to form the Lagrangian:

1. Choose generalised coordinates; apply any holonomic constraints to reduce the number of DOFs.
2. Write $T$ (kinetic energy) in terms of $\dot q_k$. For rigid bodies: $T=\frac12 I_\text{pivot}\dot\theta^2$ (rotation about fixed axis) or $T=\frac12 mv_{CM}^2+\frac12 I_{CM}\dot\theta^2$ (König’s theorem).
3. Write $V$ in terms of $q_k$.
4. Form $L=T-V$ and differentiate.

Normal modes. For a system near equilibrium, linearise $T$ and $V$ to get $T=\frac12\dot{\mathbf q}^TM\dot{\mathbf q}$ and $V=\frac12{\mathbf q}^TK\mathbf q$. Normal-mode frequencies satisfy $\det(K-\omega^2 M)=0$.

Question Archetypes

Three patterns cover all Lagrange’s equations questions.

lagrange-eom (6 question(s); 2016, 2017, 2018, 2021, 2022, 2025)

Recognition Cues

• A physical system with one or two degrees of freedom: rolling body, Atwood machine, central force, bead on curve.
• “Assign generalized coordinates, write Lagrange’s equations, find the [acceleration / velocity at the bottom / equation of motion].”
• Or a given Lagrangian in $(x,y,\dot x,\dot y)$: just differentiate.

Solution Template

1. Identify DOFs; apply holonomic constraints to reduce (e.g., rolling: $s=r\psi$).
2. Write $T$ (use $I=Mr^2$ for hoop, $I=\frac12Mr^2$ for disc, $I=\frac25Mr^2$ for sphere).
3. Write $V$; form $L=T-V$.
4. For each $q_k$: compute $\partial L/\partial\dot q_k$, differentiate w.r.t. $t$, subtract $\partial L/\partial q_k$, set equal to $Q_k$.
5. Solve the resulting ODE(s) for the asked quantity.

Worked Example(s)

2021 Paper 2, 2021-P2-Q6c (15 marks)

Derive Lagrangian equations for two masses $m_1,m_2$ on a smooth pulley (Atwood machine).

One DOF. Let $x$ = downward displacement of $m_1$; $m_2$ rises by $x$. Inextensibility gives one generalised coordinate.

$T=\tfrac12(m_1+m_2)\dot x^2,\qquad V=(m_2-m_1)gx.$ $L=\tfrac12(m_1+m_2)\dot x^2+(m_1-m_2)gx.$ EL: $(m_1+m_2)\ddot x=(m_1-m_2)g$. $\boxed{\;\ddot x=\frac{(m_1-m_2)g}{m_1+m_2}.\;}$

2016 Paper 2, 2016-P2-Q8b (15 marks)

Hoop of radius $r$ rolls without slipping down incline of length $l$, angle $\phi$. Write Lagrange’s equations and find velocity at bottom.

Rolling constraint: $\dot s=r\dot\psi$ (holonomic; reduces to one DOF $s$). Hoop: $I=Mr^2$, so total $T=\frac12M\dot s^2+\frac12(Mr^2)(\dot s/r)^2=M\dot s^2$.

$L=M\dot s^2-Mg(l-s)\sin\phi.$ EL: $2M\ddot s=Mg\sin\phi\Rightarrow\ddot s=\tfrac12 g\sin\phi$.

Kinematics ($v^2=2al$ from rest): $v^2=2\cdot\tfrac12 g\sin\phi\cdot l=gl\sin\phi$.

$\boxed{\;v=\sqrt{gl\sin\phi}.\;}$ (Independent of $M$ and $r$ — good sanity check.)

2022 Paper 2, 2022-P2-Q5d (10 marks)

Particle under central force $F=-k/r^2$. Find Lagrangian and equations of motion.

In polar $(r,\theta)$: $T=\frac12m(\dot r^2+r^2\dot\theta^2)$, $V=-k/r$.

$L=\tfrac12m(\dot r^2+r^2\dot\theta^2)+\frac{k}{r}.$ EL for $r$: $m\ddot r=mr\dot\theta^2-k/r^2$. EL for $\theta$: $\frac{d}{dt}(mr^2\dot\theta)=0$ ($\theta$ cyclic — angular momentum conserved).

$\boxed{\;m\ddot r=mr\dot\theta^2-\frac{k}{r^2},\quad mr^2\dot\theta=\text{const}.\;}$

2025 Paper 2, 2025-P2-Q5d (10 marks)

Bead on frictionless cycloid $x=a(\theta-\sin\theta)$, $y=a(1+\cos\theta)$. Find $L$; show EOM is $\ddot u+(g/4a)u=0$, $u=\cos(\theta/2)$.

Speed. $\dot x^2+\dot y^2=a^2\dot\theta^2\cdot2(1-\cos\theta)=4a^2\sin^2(\theta/2)\dot\theta^2$. Using $1-\cos\theta=2\sin^2(\theta/2)$.

$T=2ma^2\sin^2\!\left(\tfrac\theta2\right)\dot\theta^2,\qquad V=mga(1+\cos\theta)=2mga\cos^2\!\left(\tfrac\theta2\right).$

$\boxed{\;L=2ma^2\sin^2\!\left(\tfrac\theta2\right)\dot\theta^2-2mga\cos^2\!\left(\tfrac\theta2\right).\;}$

EL equation (after dividing by $2ma\sin(\theta/2)$): $2a\sin\tfrac\theta2\ddot\theta+a\cos\tfrac\theta2\dot\theta^2-g\cos\tfrac\theta2=0.$

With $u=\cos(\theta/2)$: $\dot u=-\frac12\sin(\theta/2)\dot\theta$, $\ddot u=-\frac12\sin(\theta/2)\ddot\theta-\frac14\cos(\theta/2)\dot\theta^2$. Multiplying the EL equation by $-\frac14$: $a\ddot u+\frac{g}{4}u=0$.

$\boxed{\;\frac{d^2u}{dt^2}+\frac{g}{4a}u=0\quad(\text{SHM, period }4\pi\sqrt{a/g}).\;}$

2018 Paper 2, 2018-P2-Q6c (20 marks)

$L=\frac12m(a\dot x^2+2b\dot x\dot y+c\dot y^2)-\frac12k(ax^2+2bxy+cy^2)$, $b^2\ne ac$. Write EL equations; identify system.

With $M=\begin{pmatrix}a&b\\b&c\end{pmatrix}$, both $T$ and $V$ share the same matrix: $mM\ddot{\mathbf r}=-kM\mathbf r$. Since $\det M=ac-b^2\ne0$, multiply by $M^{-1}$:

$m\ddot x+kx=0,\qquad m\ddot y+ky=0.$

$\boxed{\;\text{Two independent SHMs with }\ \omega=\sqrt{k/m}.\;}$

2017 Paper 2, 2017-P2-Q6c (20 marks)

Two uniform rods $AB$, $AC$ (each mass $m$, length $2a$) hinged at $A$, moving on a horizontal plane. Prove $T=m[\dot\xi^2+\dot\eta^2+(\frac13+\sin^2\phi)a^2\dot\theta^2+(\frac13+\cos^2\phi)a^2\dot\phi^2]$; derive Lagrange’s equations if force $(X,Y)$ acts at $A$.

König’s theorem for each rod (translation of $G_i$ + rotation about $G_i$, $I_{G_i}=ma^2/3$). Express the hinge $A$ through the system CM: $\xi=x_A+a\cos\phi\cos\theta$, etc. After substituting and collecting trig identities ($\cos(\theta\pm\phi)$ sums), cross terms cancel giving the stated $T$.

Generalised forces from $(X,Y)$ at $A$: $Q_\xi=X$, $Q_\eta=Y$, $Q_\theta=a\cos\phi(X\sin\theta-Y\cos\theta)$, $Q_\phi=a\sin\phi(X\cos\theta+Y\sin\theta)$.

EL equations: $2m\ddot\xi=X,\qquad 2m\ddot\eta=Y,$ $\tfrac{2ma^2}{3}[(1+3\sin^2\phi)\ddot\theta+3\sin2\phi\,\dot\theta\dot\phi]=Q_\theta,$ $\tfrac{2ma^2}{3}[(1+3\cos^2\phi)\ddot\phi-\tfrac32\sin2\phi(\dot\theta^2+\dot\phi^2)]=Q_\phi.$

Common Traps

• Rolling constraint first. For the hoop: $I=Mr^2$ makes total KE $=M\dot s^2$ (double the translational). A disc ($I=\frac12Mr^2$) gives $a=\frac23g\sin\phi$; a sphere ($I=\frac25Mr^2$) gives $a=\frac57g\sin\phi$. Don’t mix them.
• For the given-Lagrangian problem (2018): the condition $b^2\ne ac$ makes $M$ invertible — don’t attempt a generalised eigenvalue problem; the normal modes are just $x$ and $y$ themselves.
• For the hinged-rods problem (2017): the force acts at $A$, not at the CM — compute $Q_k=\mathbf F\cdot\partial\mathbf r_A/\partial q_k$.
• For the cycloid (2025): use $1-\cos\theta=2\sin^2(\theta/2)$ to simplify $T$, then recognise the EL equation can be written in terms of $u=\cos(\theta/2)$ — the tautochrone.

normal-modes (2 question(s); 2013, 2023)

Recognition Cues

• “Show that the periods are $2\pi/n$ where $n^2=\ldots$” — multiple natural frequencies required.
• Two coupled masses or rods; small-angle linearisation.
• Or: given $L=\frac12m(\dot x^2+\dot y^2)-\frac12m(\omega_1^2x^2+\omega_2^2y^2)+kxy$; find a rotation that decouples it.

Solution Template

1. Choose generalised coordinates; write full (nonlinear) $T$ and $V$.
2. Linearise: $\cos(\theta_1-\theta_2)\approx1$, $\cos\theta\approx1-\theta^2/2$; keep only quadratic terms.
3. Read off mass matrix $M$ and stiffness matrix $K$ from $T=\frac12\dot{\mathbf q}^TM\dot{\mathbf q}$, $V=\frac12\mathbf q^TK\mathbf q$.
4. Solve $\det(K-n^2M)=0$ for the normal frequencies. Use the quadratic formula; factor the discriminant.
5. Check: all $n^2>0$ (stable equilibrium).

Worked Example(s)

2013 Paper 2, 2013-P2-Q8a (15 marks)

Two equal rods $AB$, $BC$ (mass $m$, length $l$) jointed at $B$, suspended from $A$. Show $n^2=(3\pm6/\sqrt7)g/l$.

Generalised coordinates: $\theta_1$ (angle of $AB$), $\theta_2$ (angle of $BC$).

Full $T$ (König for $BC$: translation of $G_2$ + spin $\dot\theta_2$ about $G_2$): $T=\frac{2ml^2}{3}\dot\theta_1^2+\frac{ml^2}{6}\dot\theta_2^2+\frac{ml^2}{2}\cos(\theta_1-\theta_2)\dot\theta_1\dot\theta_2.$

Linearised ($\cos(\theta_1-\theta_2)\approx1$, $V\approx\frac{3mgl}{4}\theta_1^2+\frac{mgl}{4}\theta_2^2$): $M=ml^2\begin{pmatrix}4/3&1/2\\ 1/2&1/3\end{pmatrix},\qquad K=mgl\begin{pmatrix}3/2&0\\ 0&1/2\end{pmatrix}.$

$\det(K-n^2M)=0$ expands to $7l^2n^4-42gln^2+27g^2=0$. Discriminant $=12\sqrt7\,gl$: $n^2=\frac{42gl\pm12\sqrt7\,gl}{14l^2}=\frac{g}{l}\!\left(3\pm\frac6{\sqrt7}\right).$

$\boxed{\;n^2=\left(3\pm\frac6{\sqrt7}\right)\frac gl;\quad T=\frac{2\pi}n.\;}$

2023 Paper 2, 2023-P2-Q6c (20 marks)

$L=\frac12m(\dot x^2+\dot y^2)-\frac12m(\omega_1^2x^2+\omega_2^2y^2)+kxy$. Find $\theta$ so that the $(q_1,q_2)$ Lagrangian has no $q_1q_2$ cross term. Find the Lagrange equations independent of $\theta$.

Rotation invariance of $T$: $\dot x^2+\dot y^2=\dot q_1^2+\dot q_2^2$ (orthogonal rotation).

Cross-term coefficient in $L$ after rotation: $\tfrac12m(\omega_1^2-\omega_2^2)\sin2\theta+k\cos2\theta=0\;\Longrightarrow\;\tan2\theta=\frac{2k}{m(\omega_2^2-\omega_1^2)}.$

$\boxed{\;\theta=\tfrac12\arctan\!\frac{2k}{m(\omega_2^2-\omega_1^2)}.\;}$

Normal frequencies (eigenvalues of the potential matrix, independent of $\theta$): $\Omega_{1,2}^2=\frac{\omega_1^2+\omega_2^2}{2}\pm\sqrt{\!\left(\frac{\omega_1^2-\omega_2^2}{2}\right)^2+\frac{k^2}{m^2}}.$

EL equations: $\ddot q_1+\Omega_1^2q_1=0$, $\ddot q_2+\Omega_2^2q_2=0$.

Common Traps

• Rod $AB$ rotates about fixed pivot $A$: use $I_A=ml^2/3$. Rod $BC$ rotates about its own (moving) CG: use $I_{G_2}=ml^2/12$ plus translational KE. Confusing these gives wrong $M$.
• Discriminant $\sqrt{1008}=12\sqrt7$ — factor $1008=144\times7$ to see the simplification; otherwise the answer looks wrong.
• In the 2023 problem: $K=kM$ leads to a degenerate generalised eigenvalue problem when the matrices share the same eigenvectors — but here the cross-term rotation is needed first.

gauge-lagrangian (1 question(s); 2015)

Recognition Cues

• A Lagrangian with explicit $e^{-\alpha t}$ time dependence given; find an equivalent time-independent form.
• “Find an equivalent Lagrangian not explicitly dependent on time.”

Solution Template

Add $\dfrac{dF}{dt}$ for $F(q,t)$ chosen to absorb the time-dependent terms. The key identity: $\dfrac{d}{dt}(-\alpha b q^2 e^{-\alpha t}/2)=-\alpha bq\dot q e^{-\alpha t}+\alpha^2 bq^2 e^{-\alpha t}/2$, which cancels both offending terms in $L$.

Worked Example(s)

2015 Paper 2, 2015-P2-Q7c-ii (10 marks)

Starting from $L=\frac\alpha2\dot q^2+\alpha bq\dot q e^{-\alpha t}-\frac{b\alpha^2 q^2 e^{-\alpha t}}{2}-\frac{kq^2}{2}$, find an equivalent time-independent Lagrangian.

Add $\dfrac{dF}{dt}$ with $F=-\dfrac{\alpha bq^2 e^{-\alpha t}}{2}$: $\frac{dF}{dt}=-\alpha bq\dot q e^{-\alpha t}+\frac{\alpha^2 bq^2 e^{-\alpha t}}{2}.$ Both $e^{-\alpha t}$ terms in $L$ cancel: $L'=L+\frac{dF}{dt}=\frac\alpha2\dot q^2-\frac{kq^2}{2}.$

$\boxed{\;L'=\frac\alpha2\dot q^2-\frac{kq^2}{2}\quad(\text{SHM with mass }\alpha,\text{ spring constant }k).\;}$

Common Traps

• The time-dependent parts are “gauge artifacts” — the underlying dynamics is SHM with frequency $\sqrt{k/\alpha}$, independent of $b$.
• Spot $F$ by matching: $dF/dt$ must reproduce the unwanted terms with opposite signs. Here both the $q\dot q\,e^{-\alpha t}$ and $q^2 e^{-\alpha t}$ terms are cancelled by a single $F$.

Marks-Aware Writing

10-mark questions (2021, 2022, 2025-Q5d): Write $T$ and $V$ (two lines), form $L$, write EL for the single coordinate, state the result. For the given-$L$ variant (2022): just differentiate and identify.

15-mark questions (2013, 2016, 2021): State the constraint (holonomic); write $T$ with moment of inertia used explicitly; form $L$; EL equation in one line; solve by kinematics or energy. For normal modes (2013): show both linearised $T$ and $V$ matrices before writing the characteristic equation.

20-mark questions (2017, 2018, 2023): Full derivation expected. For 2017: prove the $T$ formula step by step (König’s theorem, write each rod’s KE, collect terms), state all four generalised forces, write all four EL equations. For 2023: show the rotation, compute the cross-term coefficient, set to zero, state $\Omega_{1,2}$.

Practice Set