Hamilton’s equations

At a Glance

• Frequency: 10 sub-parts across 8 of 13 years (2014, 2015, 2016, 2018, 2019, 2020, 2023, 2025)
• Priority tier: T1
• Marks (count): 10 (6), 15 (2), 20 (2)
• Average solve time: ~10 min
• Difficulty mix: medium 6, easy 3, hard 1
• Section: B | Dominant type: derivation

Why This Chapter Matters

Hamilton’s equations appear in 8 of the last 13 years and are the most commonly 10-mark questions in Paper 2 Section B — meaning they reward quick, clean execution. The method is a fixed procedure: write the Lagrangian, form momenta, do the Legendre transform to get $H$, write the four canonical equations. The same five-step skeleton works for pendulums, rolling bodies, central-force problems, and abstract algebraic Hamiltonians. A student who practises this once per physical setup can reliably collect 10 marks per appearance.

Minimum Theory

Hamiltonian. Given a Lagrangian $L(q,\dot q,t)$, the generalised momentum is $p_k=\partial L/\partial\dot q_k$. The Hamiltonian is $H(q,p,t)=\sum_k p_k\dot q_k-L,$ where $\dot q_k$ is expressed in terms of $p_k$ by inverting $p_k=\partial L/\partial\dot q_k$.

For natural systems (holonomic time-independent constraints, $V$ independent of velocities): $H=T+V$ expressed in $(q,p)$ — the total mechanical energy.

Hamilton’s canonical equations: $\dot q_k=\frac{\partial H}{\partial p_k},\qquad \dot p_k=-\frac{\partial H}{\partial q_k}.$

Conservation laws. (1) If $\partial H/\partial t=0$ (no explicit time dependence), then $dH/dt=0$ — energy conserved. (2) If $\partial H/\partial q_k=0$ ($q_k$ cyclic), then $\dot p_k=0$ — momentum $p_k$ conserved.

Legendre transform for cross-coupled $L$. When $L$ has cross terms $\dot x\dot y$, compute both momenta $p_x=\partial L/\partial\dot x$ and $p_y=\partial L/\partial\dot y$, solve for $\dot x,\dot y$, substitute into $H=p_x\dot x+p_y\dot y-L$.

Question Archetypes

Three patterns cover all Hamilton’s equations questions.

hamilton-eom (8 question(s); 2014, 2015, 2018, 2019, 2020, 2023, 2025)

Recognition Cues

• “Find the Hamiltonian and Hamilton’s equations of motion for [physical system].”
• Or: given $H$ in $(q,p)$, write $\dot q=\partial H/\partial p$ and $\dot p=-\partial H/\partial q$ directly.
• “Show that $H$ is a constant of motion” — use $dH/dt=\frac{\partial H}{\partial q}\dot q+\frac{\partial H}{\partial p}\dot p+\frac{\partial H}{\partial t}=0$ via Hamilton’s equations.

Solution Template

1. Write $T$ (fold in any holonomic rolling/constraint before forming momenta).
2. Write $V$; form $L=T-V$.
3. Compute $p_k=\partial L/\partial\dot q_k$; solve for $\dot q_k$ in terms of $p_k$.
4. Form $H=T+V$ in $(q,p)$ (for natural systems) or use $H=\sum p_k\dot q_k-L$.
5. Write the $2n$ canonical equations. Identify any cyclic coordinates (conserved momenta).
6. Combine $\dot p=f(q)$ and $p=g\dot q$ to get second-order EOM if asked.

Worked Example(s)

2014 Paper 2, 2014-P2-Q5e (10 marks)

Find the equation of motion of a compound pendulum via Hamilton’s equations.

Setup. Mass $M$, moment of inertia $I$ about pivot, CG distance $h$.

$T=\frac12I\dot\theta^2$, $V=-Mgh\cos\theta$. $p=I\dot\theta\Rightarrow\dot\theta=p/I$.

$H=\frac{p^2}{2I}-Mgh\cos\theta.$

Hamilton’s equations: $\dot\theta=p/I$, $\dot p=-Mgh\sin\theta$.

Combine: $I\ddot\theta=-Mgh\sin\theta$.

$\boxed{\;\ddot\theta+\frac{Mgh}{I}\sin\theta=0;\quad T=2\pi\sqrt{\frac{I}{Mgh}}.\;}$

2015 Paper 2, 2015-P2-Q6b (15 marks)

Solve the plane pendulum via Hamiltonian; show $H$ is conserved.

$T=\frac12ml^2\dot\theta^2$, $V=-mgl\cos\theta$. $p_\theta=ml^2\dot\theta$.

$H=\frac{p_\theta^2}{2ml^2}-mgl\cos\theta.$

$H$ conserved: $dH/dt=\frac{\partial H}{\partial\theta}\dot\theta+\frac{\partial H}{\partial p_\theta}\dot p_\theta$. Substituting $\dot\theta=\partial H/\partial p_\theta$ and $\dot p_\theta=-\partial H/\partial\theta$: the two terms cancel.

$\boxed{\;\frac{dH}{dt}=0\quad(\partial H/\partial t=0).\;}$

2019 Paper 2, 2019-P2-Q7a (15 marks)

Hamilton’s equations for a sphere rolling down a rough incline; find acceleration.

Fold the constraint $\dot\phi=\dot x/r$ into $T$: $T=\frac12M\dot x^2+\frac12(\frac25Mr^2)(\dot x/r)^2=\frac{7}{10}M\dot x^2$.

$p=\frac75M\dot x$, so $H=\frac{5p^2}{14M}-Mgx\sin\theta$.

$\dot x=5p/(7M)$, $\dot p=Mg\sin\theta$.

$\ddot x=\frac{5\dot p}{7M}=\frac57g\sin\theta.\qquad\boxed{\;\ddot x=\frac57g\sin\theta.\;}$

2020 Paper 2, 2020-P2-Q6c (20 marks)

Hamiltonian for mass $m$ on cylinder $x^2+y^2=R^2$ under force $\vec F=-k\vec r$.

Cylindrical coords $(\theta,z)$: $T=\frac12m(R^2\dot\theta^2+\dot z^2)$, $V=\frac12k(R^2+z^2)$.

$p_\theta=mR^2\dot\theta$, $p_z=m\dot z$.

$H=\frac{p_\theta^2}{2mR^2}+\frac{p_z^2}{2m}+\frac12k(R^2+z^2).$

$\theta$ is cyclic: $\dot p_\theta=0\Rightarrow p_\theta=$ const (uniform rotation). Axial: $\dot p_z=-kz\Rightarrow m\ddot z=-kz$ (SHM, $\omega=\sqrt{k/m}$).

$\boxed{\;\text{Helical oscillation: uniform rotation }+\text{ axial SHM at }\omega=\sqrt{k/m}.\;}$

2018 Paper 2, 2018-P2-Q7c (20 marks)

$H=p_1q_1-aq_1^2+bq_2^2-p_2q_2$. Solve Hamilton’s equations; show $(p_2-bq_2)/q_1=$ const.

$\dot q_1=q_1$, $\dot q_2=-q_2$, $\dot p_1=2aq_1-p_1$, $\dot p_2=p_2-2bq_2$.

$q_1=Ae^t$, $q_2=Be^{-t}$; $p_2=Ce^t+bBe^{-t}$.

$p_2-bq_2=Ce^t\;\Longrightarrow\;\frac{p_2-bq_2}{q_1}=\frac{Ce^t}{Ae^t}=\frac CA=\text{const}.$

Shortcut: $\dot w=w$ where $w=p_2-bq_2$ (same as $\dot q_1=q_1$) ⇒ $w/q_1$ is a constant of motion.

$\boxed{\;\frac{p_2-bq_2}{q_1}=\text{const}.\;}$

2023 Paper 2, 2023-P2-Q5d (10 marks)

Planet of mass $m$; $T=\frac12m(\dot r^2+r^2\dot\theta^2)$, $V=GMm(1/2a-1/r)$. Find $H$ and Hamilton’s equations.

$p_r=m\dot r$, $p_\theta=mr^2\dot\theta$.

$H=\frac{p_r^2}{2m}+\frac{p_\theta^2}{2mr^2}+GMm\!\left(\frac{1}{2a}-\frac1r\right).$

$\dot p_\theta=0$ ($\theta$ cyclic — angular momentum conserved). $\dot p_r=p_\theta^2/(mr^3)-GMm/r^2$.

$\boxed{\;\dot r=\frac{p_r}m,\quad\dot\theta=\frac{p_\theta}{mr^2},\quad\dot p_r=\frac{p_\theta^2}{mr^3}-\frac{GMm}{r^2},\quad\dot p_\theta=0.\;}$

2025 Paper 2, 2025-P2-Q8c-i (10 marks)

$V=-k\cos\theta/r^2$. Find $H$ and Hamilton’s equations in spherical polars $(r,\theta,\phi)$.

$T=\frac12m(\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\dot\phi^2)$. Momenta: $p_r=m\dot r$, $p_\theta=mr^2\dot\theta$, $p_\phi=mr^2\sin^2\theta\dot\phi$.

$H=\frac{p_r^2}{2m}+\frac{p_\theta^2}{2mr^2}+\frac{p_\phi^2}{2mr^2\sin^2\theta}-\frac{k\cos\theta}{r^2}.$

$\phi$ is cyclic: $\dot p_\phi=0$. Key derivatives: $\dot p_r=p_\theta^2/(mr^3)+p_\phi^2/(mr^3\sin^2\theta)-2k\cos\theta/r^3$; $\dot p_\theta=p_\phi^2\cos\theta/(mr^2\sin^3\theta)-k\sin\theta/r^2$.

2025 Paper 2, 2025-P2-Q8c-ii (10 marks)

$L=m\dot x\dot y-m\omega_0^2xy$. Find $H$ and Hamilton’s equations; identify the system.

Cross-coupled momenta: $p_x=m\dot y$, $p_y=m\dot x$. Invert: $\dot x=p_y/m$, $\dot y=p_x/m$.

$H=p_x\dot x+p_y\dot y-L=p_xp_y/m+m\omega_0^2xy-m\dot x\dot y+m\omega_0^2xy$. After substituting $m\dot x\dot y=p_xp_y/m$:

$H=\frac{p_xp_y}{m}+m\omega_0^2xy.$

$\dot p_x=-m\omega_0^2y$, $\dot p_y=-m\omega_0^2x$. Combining: $\ddot x=-\omega_0^2x$, $\ddot y=-\omega_0^2y$.

$\boxed{\;\text{Two independent SHMs at }\omega_0.\;}$

Common Traps

• Fold the rolling constraint before forming momenta. For the sphere: effective mass is $\frac75M$, so $p=\frac75M\dot x$ and $H=\frac{5p^2}{14M}-Mgx\sin\theta$. Computing momenta without folding gives the wrong $H$.
• Sign in $\dot p_k=-\partial H/\partial q_k$: for $H=p_1q_1-\ldots$, $\partial H/\partial q_1=p_1$ so $\dot p_1=-(p_1-2aq_1)=2aq_1-p_1$. One sign error cascades through the entire solution.
• $\partial H/\partial r$ for a term $p_\theta^2/(2mr^2)$: $\partial/\partial r=-p_\theta^2/(mr^3)$, so $\dot p_r=+p_\theta^2/(mr^3)+\ldots$ (centrifugal).
• $H$ conserved requires $\partial H/\partial t=0$ — write this explicitly when asked to “show $H$ is a constant of motion.”
• For cross-coupled $L=m\dot x\dot y$: momenta $p_x=m\dot y$ and $p_y=m\dot x$ — the subscripts are swapped. Don’t write $p_x=m\dot x$.

hamilton-jacobi (1 question(s); 2016)

Recognition Cues

• “Find Hamilton’s principal/characteristic function $S$.”
• Free particle from origin to $(x,y,z)$ in time $\tau$.

Solution Template

Hamilton’s principal function $S=$ action along the actual trajectory. For a free particle with constant velocity $\dot x_i=x_i/\tau$: integrate $L=T$ over $[0,\tau]$.

Worked Example(s)

2016 Paper 2, 2016-P2-Q5c (10 marks)

Free particle, mass $m$, from origin to $(x,y,z)$ in time $\tau$. Find Hamilton’s characteristic function $S$.

Constant-velocity motion: $\dot x_i=x_i/\tau$. Integrand constant in time: $S=\int_0^\tau\tfrac12m\frac{r^2}{\tau^2}\,dt=\frac{mr^2}{2\tau},\quad r^2=x^2+y^2+z^2.$

$\boxed{\;S=\frac{m(x^2+y^2+z^2)}{2\tau}.\;}$

Verify: $\partial S/\partial\tau=-mr^2/(2\tau^2)=-H$; $\frac1{2m}|\nabla S|^2=\frac{m r^2}{2\tau^2}=H$ ✓.

lagrangian-from-hamiltonian (1 question(s); 2015)

Recognition Cues

• Given $H(q,p,t)$; find $L$.
• Use $\dot q=\partial H/\partial p$ to invert $p=p(q,\dot q,t)$, then $L=p\dot q-H$.

Worked Example(s)

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

$H=\frac{p^2}{2\alpha}-bqpe^{-\alpha t}+\frac{b\alpha}{2}q^2e^{-\alpha t}(\alpha+be^{-\alpha t})+\frac k2q^2$. Find $L$.

$\dot q=\partial H/\partial p=p/\alpha-bqe^{-\alpha t}\Rightarrow p=\alpha(\dot q+bqe^{-\alpha t})$.

Substitute into $L=p\dot q-H$; the $bqe^{-\alpha t}$ cross terms and $e^{-2\alpha t}$ terms all cancel:

$\boxed{\;L=\frac\alpha2\dot q^2+\alpha bq\dot q e^{-\alpha t}-\frac{b\alpha^2q^2e^{-\alpha t}}{2}-\frac{kq^2}{2}.\;}$

(Adding $dF/dt$ with $F=-\alpha bq^2e^{-\alpha t}/2$ reduces this to the SHM Lagrangian $\frac\alpha2\dot q^2-\frac k2q^2$ — see MF-03 gauge-lagrangian.)

Marks-Aware Writing

10-mark questions (2014, 2016, 2023, 2025): Write $T$ and $V$ (one line each), form $p_k$ and $H$ (two lines), state all canonical equations (four lines), identify cyclic coordinates. Box the answer.

15-mark questions (2015, 2019): For the rolling sphere: show the constraint fold explicitly ($I/Mr^2$ factor), state the effective mass in $H$, derive the acceleration. For the pendulum: also write the conservation proof $dH/dt=0$ step by step.

20-mark questions (2018, 2020): Solve all four ODEs (show integrating factor for the $p$ equations); for the invariant proof show either the explicit cancellation or the shortcut $\dot w/w=\dot q_1/q_1$.

Practice Set