Unit 8 of 10
Mechanics and fluid dynamics
Exactly 7 in both papers. Mostly Lagrangian and Hamiltonian ideas, with two or three fluid questions.
- 7questions expected next time
- 7asked in 2025
- 7asked in 2026
- 2ideas asked in both papers
Poisson brackets and the Hamiltonian from a Lagrangian are near-certain.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Poisson brackets and constants of motion
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dA/dt = [A, H] + ∂A/∂t, so A is conserved when that is 0. [qᵢ, pⱼ] = δᵢⱼ.
Small oscillations and normal modes
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Solve det(V − ω²T) = 0. The modes are orthogonal with respect to the mass matrix T.
Asked in 2026
7 questions. Tap one for the answer and working.
Q40Canonical transformation preserves symplectic structure: which quantity is invariantD
Answer: phase space area
Canonical transformations preserve the symplectic form, hence phase-space volume/area (Liouville, Poincare invariant).
Sure
Q46Non-orthogonal eigenvectors under dot product: what ensures orthogonality (mass matrix)A
Answer: Mass matrix
Small oscillations: V a = w^2 T a; eigenvectors are orthogonal w.r.t. the mass (kinetic) matrix T, not the plain dot product (that needs T = identity).
Fairly sure
could be read as "identity matrix" (C) if the question means standard dot product works only when T = I; chose mass matrix.
Q50{A,H}=0 implies what about AB
Answer: A conserved
dA/dt = {A,H} + dA/dt(explicit) = 0 for time-independent A.
Sure
Q55H=p^2/2m+V(q): condition on motion for action-angle variablesC
Answer: periodic, bounded
Action-angle variables exist for bounded periodic (libration/rotation) motion.
Sure
Q61Sphere in inviscid flow, add slight viscosity (Navier-Stokes): qualitative drag correctionB
Answer: becomes finite
Inviscid: d'Alembert paradox, drag 0. Small viscosity (Navier-Stokes) gives finite drag (Stokes drag 6 pi mu a U at low Re).
Sure
Q67Two-degree system with kinetic and potential coupling, after diagonalisation one eigenfrequency: meaningB
A zero eigenfrequency means zero restoring force for that normal mode: free (rigid-body) motion mode
Sure
Q71Torque-free rotation about intermediate axis (I1<I2<I3): long-term behaviourB
Rotation about intermediate axis is unstable (Euler eqns linearised give exp. growth, sqrt of positive product) -> exponential divergence (tumbling)
Sure
Asked in 2025
7 questions. Tap one for the answer and working.
Q67Continuity equation, spherical symmetryB
Spherical symmetry: d(rho)/dt + (1/r^2) d(rho q r^2)/dr = 0
Matches the official key
Q68DOF of rigid body with one point fixed (3)C
Rigid body with one point fixed: 3 degrees of freedom (Euler angles)
Matches the official key
Q69Which Poisson-bracket identity failsD
[q_r,p_s] = delta_rs, not 0; other three always hold (antisymmetry, Jacobi, [q,q]=0)
Matches the official key
Q70Continuity eq; incompressible ⇒ ∇·v=0C
A is the continuity equation (true). R is false: incompressible means div v = 0, not curl v = 0
Matches the official key
Q71Coupled-oscillator Lagrangian, normal modesD
p1=2q1'+q2', p2=2q2'+q1' so p1+p2=3(q1'+q2') (A ok); p1'=-2q1, p2'=-2q2 (B ok); adding EOM gives s''=-(2/3)s, s=q1+q2 (C ok); p1'/p2' = q1/q2, not q2/q1 (D false)
Matches the official key
Q72Relations between φ and ψ (Cauchy–Riemann)D
Cauchy-Riemann: phi_x = psi_y, phi_y = -psi_x, so a and d
Matches the official key
Q73Hamiltonian from given LagrangianC
p_x=2a x', p_y=2b y'; H = p_x^2/4a + p_y^2/4b + kxy
Matches the official key
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Hamilton's variational principle
- Sources, sinks and doublets
- Complex potential
- Sphere in a perfect fluid
- Euler's equation of motion
Full syllabus for this unit
Generalised coordinates, Lagrange and Hamilton equations, variational principle, rigid body (Euler), small oscillations, Poisson brackets, canonical transformations; continuity, Euler’s equation, 2-D flow, complex potential, sources, sinks, doublets, sphere in a fluid, vorticity, Navier–Stokes.