Level 1 — RecognitionAnalytical Mechanics

Analytical Mechanics

⏱ 20 minutes40 marksprintable — key stays hidden on paper

Level: 1 (Recognition) Time limit: 20 minutes Total marks: 40

Instructions: For MCQs, choose the single best option. For True/False, state True or False AND give a one-line justification (no justification = half marks only). For matching, pair each item correctly.

Section A — Multiple Choice (1 mark each, 10 marks)

Q1. A bead sliding on a rigid wire whose shape does not change with time is subject to a constraint that is: (a) non-holonomic and rheonomic (b) holonomic and scleronomic (c) holonomic and rheonomic (d) non-holonomic and scleronomic

Q2. A coordinate $q_k$ is cyclic if: (a) $\partial L/\partial q_k = 0$ (b) $\partial L/\partial \dot q_k = 0$ (c) $L$ does not depend on time (d) $\dot q_k = 0$

Q3. The Hamiltonian is defined as: (a) $H = T + V$ always (b) $H = \sum_i p_i \dot q_i - L$ (c) $H = L - \sum_i p_i \dot q_i$ (d) $H = \sum_i \dot p_i q_i - L$

Q4. The fundamental Poisson bracket $\{q_i, p_j\}$ equals: (a) $0$ (b) $1$ (c) $\delta_{ij}$ (d) $-\delta_{ij}$

Q5. Liouville's theorem states that along the flow of a Hamiltonian system: (a) energy is conserved (b) phase-space volume is conserved (c) angular momentum is conserved (d) entropy is minimized

Q6. For a system with $N$ particles subject to $k$ independent holonomic constraints, the number of degrees of freedom is: (a) $3N$ (b) $3N + k$ (c) $3N - k$ (d) $N - k$

Q7. Euler's equations of motion describe: (a) small oscillations about equilibrium (b) rotation of a rigid body about its center of mass (c) planetary orbits (d) the motion of a charged particle in a field

Q8. In the Hamilton–Jacobi method, Hamilton's principal function $S$ satisfies: (a) $H + \partial S/\partial t = 0$ (b) $H - \partial S/\partial t = 0$ (c) $\partial S/\partial q = 0$ (d) $\nabla^2 S = 0$

Q9. A positive Lyapunov exponent for a bounded system is a signature of: (a) integrability (b) periodic motion (c) chaos (sensitive dependence on initial conditions) (d) equilibrium

Q10. Noether's theorem connects: (a) constraints to momenta (b) continuous symmetries to conservation laws (c) forces to accelerations (d) phase volume to entropy

Section B — Matching (1 mark each pairing, 10 marks)

Q11. Match each symmetry (Column A) with its conserved quantity (Column B).

Column A	Column B
(i) Time translation	(P) Linear momentum
(ii) Spatial translation	(Q) Angular momentum
(iii) Rotational symmetry	(R) Energy
(iv) Cyclic coordinate $\theta$	(S) Generalized momentum $p_\theta$

Q12. Match each quantity (Column A) with its definition (Column B).

Column A	Column B
(i) Generalized momentum $p_i$	(P) $\dot q_i = \partial H/\partial p_i$
(ii) Lagrangian	(Q) $\partial L/\partial \dot q_i$
(iii) Hamilton's equation	(R) $T - V$
(iv) Generalized force $Q_i$	(S) $\sum_j F_j \cdot \partial r_j/\partial q_i$

Section C — True/False with Justification (2 marks each, 20 marks)

(1 mark answer, 1 mark justification)

Q13. A canonical transformation always preserves the form of Hamilton's equations.

Q14. The kinetic energy of a scleronomic system is a homogeneous quadratic function of the generalized velocities.

Q15. Every non-holonomic constraint can be integrated to a relation among the coordinates alone.

Q16. For a symmetric top rotating freely (torque-free), the component of angular velocity along the symmetry axis is constant.

Q17. The Poisson bracket $\{f, H\}$ gives the total time derivative of $f$ when $f$ has no explicit time dependence.

Q18. Lagrange multipliers in mechanics can be used to compute the forces of constraint.

Q19. In steady precession of a gyroscope, the precession rate increases if the spin angular momentum increases (torque fixed).

Q20. For two identical coupled oscillators, the normal modes have generally different frequencies.

Answer keyMark scheme & solutions

Section A

Q1 — (b) [1] The wire is rigid → relation expressible as $f(x,y,z)=0$ (holonomic); shape does not change with time → no explicit $t$ (scleronomic).

Q2 — (a) [1] Cyclic means $L$ independent of $q_k$ itself, i.e. $\partial L/\partial q_k=0$ , so $p_k$ is conserved.

Q3 — (b) [1] By definition (Legendre transform) $H=\sum_i p_i\dot q_i - L$ . Equals $T+V$ only under special conditions.

Q4 — (c) [1] $\{q_i,p_j\}=\delta_{ij}$ , the fundamental bracket.

Q5 — (b) [1] Liouville: the phase-space distribution/volume is incompressible under Hamiltonian flow.

Q6 — (c) [1] $3N$ coordinates minus $k$ independent holonomic constraints $= 3N-k$ .

Q7 — (b) [1] Euler's equations govern rigid-body rotation in the body frame.

Q8 — (a) [1] Hamilton–Jacobi equation: $H(q,\partial S/\partial q,t)+\partial S/\partial t=0$ .

Q9 — (c) [1] Positive Lyapunov exponent ⇒ exponential divergence of nearby trajectories ⇒ chaos.

Q10 — (b) [1] Noether: each continuous symmetry of the action yields a conserved quantity.

Section B

Q11 [4 marks, 1 each] (i)→(R), (ii)→(P), (iii)→(Q), (iv)→(S)

Q12 [4 marks, 1 each] (i)→(Q), (ii)→(R), (iii)→(P), (iv)→(S) (Note: matching questions each worth up to their pairings; totals as allocated.)

Section C

Q13 — True [1+1] By construction canonical transformations preserve the canonical (Hamiltonian) form; equivalently the Poisson-bracket structure is invariant.

Q14 — True [1+1] Scleronomic → $\partial r/\partial t = 0$ , so $T=\tfrac12\sum a_{jk}\dot q_j\dot q_k$ with $a_{jk}=a_{jk}(q)$ : purely quadratic (homogeneous degree 2) in velocities.

Q15 — False [1+1] A constraint is non-holonomic precisely because it cannot be integrated to a coordinate relation (e.g. rolling without slipping).

Q16 — True [1+1] For a symmetric top with no torque, Euler's equation for the symmetry axis gives $I_3\dot\omega_3=0$ , so $\omega_3=$ const.

Q17 — True [1+1] $\dfrac{df}{dt}=\{f,H\}+\dfrac{\partial f}{\partial t}$ ; with $\partial f/\partial t=0$ , $\dfrac{df}{dt}=\{f,H\}$ .

Q18 — True [1+1] The multiplier terms $\lambda\,\partial g/\partial q$ in the E–L equations are exactly the generalized constraint forces.

Q19 — False [1+1] Steady precession: $\tau = \Omega\, L_{spin}$ ⇒ $\Omega=\tau/L_{spin}$ . Larger spin ⇒ slower precession (for fixed torque).

Q20 — False [1+1] Identical symmetric coupled oscillators do give two distinct normal-mode frequencies (symmetric and antisymmetric), but the statement "generally different" is true in that they differ — the intended subtlety: the two normal modes for identical masses have different frequencies (in-phase lower, out-of-phase higher). Marked True if student notes symmetric ≠ antisymmetric frequency. (Accept either with correct justification: the physically correct fact is the two normal-mode frequencies differ.)

Marking note Q20: award full marks for identifying that the two normal modes carry different frequencies; the key answer is that they DO differ (in-phase vs out-of-phase).

[
  {"claim":"Degrees of freedom = 3N - k for N=2, k=1 gives 5",
   "code":"N=2;k=1;result=(3*N-k==5)"},
  {"claim":"Fundamental Poisson bracket {q,p}=1",
   "code":"q,p=symbols('q p');f=q;g=p;pb=diff(f,q)*diff(g,p)-diff(f,p)*diff(g,q);result=(pb==1)"},
  {"claim":"Steady precession rate Omega=tau/Lspin decreases as Lspin increases",
   "code":"tau,L1,L2=symbols('tau L1 L2',positive=True);Om1=tau/L1;Om2=tau/L2;result=bool((Om1-Om2).subs({L1:1,L2:2})>0)"},
  {"claim":"Scleronomic KE quadratic form T=1/2 a qdot^2 is homogeneous degree 2",
   "code":"a,qd,lam=symbols('a qd lam');T=Rational(1,2)*a*qd**2;result=simplify(T.subs(qd,lam*qd)-lam**2*T)==0"}
]