Analytical Mechanics

Q1. (4 marks)

• A holonomic constraint is one expressible as an equation relating only the coordinates (and possibly time): $f(q_1,\dots,q_n,t)=0$. (2)
• The constraint $x^2+y^2=a^2$ contains no explicit time dependence, so it is scleronomic. (2) (If the radius varied with time, e.g. $a(t)$, it would be rheonomic.)

Q2. (6 marks)

• A point on a sphere has 2 degrees of freedom. (2)
• Suitable generalized coordinates: polar angle $\theta$ and azimuthal angle $\phi$. (1)
• Position: velocity components give $T = \tfrac{1}{2}mR^2\left(\dot\theta^2 + \sin^2\theta\,\dot\phi^2\right).$ (3)

Q3. (5 marks)

• Euler–Lagrange equation: $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i}\right) - \frac{\partial L}{\partial q_i} = 0.$ (2)
• A cyclic coordinate is one that does not appear explicitly in $L$: $\partial L/\partial q_i = 0$. (1.5)
• Then $\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot q_i}\right)=0$, so the conjugate momentum $p_i = \partial L/\partial\dot q_i$ is conserved. (1.5)

Q4. (6 marks)

• $T=\tfrac12 m l^2\dot\theta^2$, $V=-mgl\cos\theta$, so $L = \tfrac12 m l^2 \dot\theta^2 + mgl\cos\theta.$ (2)
• $\dfrac{\partial L}{\partial \dot\theta}=ml^2\dot\theta$, $\dfrac{d}{dt}(\cdot)=ml^2\ddot\theta$; $\dfrac{\partial L}{\partial\theta}=-mgl\sin\theta$. (2)
• Euler–Lagrange gives: $ml^2\ddot\theta + mgl\sin\theta = 0 \implies \ddot\theta + \frac{g}{l}\sin\theta=0.$ (2)

Q5. (6 marks)

• Generalized momentum: $p_i = \partial L/\partial\dot q_i$. (1)
• Hamiltonian: $H = \sum_i p_i\dot q_i - L$. (1)
• For $L=\tfrac12 m\dot x^2 - V(x)$: $p=m\dot x \Rightarrow \dot x = p/m$, so $H = \frac{p^2}{2m} + V(x).$ (2)
• Hamilton's equations: $\dot x = \frac{\partial H}{\partial p} = \frac{p}{m}, \qquad \dot p = -\frac{\partial H}{\partial x} = -\frac{dV}{dx}.$ (2)

Q6. (5 marks)

• Definition: $\{f,g\} = \sum_i\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right).$ (2)
• Fundamental brackets: $\{q_i,p_j\}=\delta_{ij}, \quad \{q_i,q_j\}=0, \quad \{p_i,p_j\}=0.$ (3, one each)

Q7. (4 marks)

• Liouville's theorem: The phase-space density $\rho$ of a system of representative points is conserved along the flow: $d\rho/dt = 0$; equivalently the phase-space volume occupied by a set of trajectories is invariant in time. (2)
• Implication: Hamiltonian flow is incompressible — an initial region of phase space may deform but preserves its volume. (2)

Q8. (6 marks)

• Equations of motion: $m\ddot x_1 = -k x_1 + k(x_2 - x_1) = -2k x_1 + k x_2,$ $m\ddot x_2 = -k x_2 - k(x_2 - x_1) = k x_1 - 2k x_2.$ (2)
• Assume $x_j = A_j e^{i\omega t}$; characteristic equation from matrix $\begin{pmatrix}2k-m\omega^2 & -k\\ -k & 2k-m\omega^2\end{pmatrix}=0$: $(2k-m\omega^2)^2 = k^2.$ (2)
• Solutions: $\omega_1 = \sqrt{\frac{k}{m}} \ \text{(symmetric)}, \qquad \omega_2 = \sqrt{\frac{3k}{m}} \ \text{(antisymmetric)}.$ (2)

Q9. (4 marks)

• Noether's theorem: Every continuous symmetry of the action (Lagrangian) corresponds to a conserved quantity. (2)
• (a) Time-translation symmetry $\Rightarrow$ conservation of energy. (1)
• (b) Spatial-translation symmetry $\Rightarrow$ conservation of linear momentum. (1)

Q10. (4 marks)

• Euler's equations (torque-free, $\mathbf{N}=0$): $I_1\dot\omega_1 = (I_2 - I_3)\omega_2\omega_3,$ $I_2\dot\omega_2 = (I_3 - I_1)\omega_3\omega_1,$ $I_3\dot\omega_3 = (I_1 - I_2)\omega_1\omega_2.$ (3)
• Conserved: total angular momentum (vector, in space frame) and rotational kinetic energy. (1)

[
  {"claim":"Pendulum EOM: from L, EL gives ml^2 thetadd + mgl sin(theta)=0",
   "code":"m,l,g,t=symbols('m l g t',positive=True); th=Function('theta'); L=Rational(1,2)*m*l**2*th(t).diff(t)**2+m*g*l*cos(th(t)); EL=diff(diff(L,th(t).diff(t)),t)-diff(L,th(t)); result=simplify(EL-(m*l**2*th(t).diff(t,2)+m*g*l*sin(th(t))))==0"},
  {"claim":"1D Hamiltonian H=p^2/2m+V from Legendre transform",
   "code":"m,x,p,v=symbols('m x p v'); V=Function('V'); L=Rational(1,2)*m*v**2-V(x); pexpr=diff(L,v); vsol=solve(Eq(p,pexpr),v)[0]; H=p*vsol-L.subs(v,vsol); result=simplify(H-(p**2/(2*m)+V(x)))==0"},
  {"claim":"Coupled spring normal frequencies squared are k/m and 3k/m",
   "code":"k,m,w=symbols('k m w',positive=True); M=Matrix([[2*k-m*w**2,-k],[-k,2*k-m*w**2]]); sols=solve(M.det(),w**2); result=set([simplify(s) for s in sols])==set([k/m,3*k/m])"},
  {"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"}
]