Classical Mechanics (Lagrangian-Hamiltonian formalisms & Small Oscillations) – CSIR-UGC NET/JRF/PhD Quick Notes

These condensed notes focus on material repeatedly tested in Part B (conceptual) and Part C (problem-solving) of the Physical Sciences paper.

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1. Variational Foundations

• Principle of least action:
  $\delta S=0,\quad S=\displaystyle\int_{t_1}^{t_2}L(q_i,\dot q_i,t)\,dt$

• Euler-Lagrange:
  $\displaystyle\frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot q_i}\right)-\frac{\partial L}{\partial q_i}=0$

• Cyclic (ignorable) coordinate ⇒ conjugate momentum conserved: $\partial L/\partial q_k=0 \implies p_k=\text{const.}$

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2. From Lagrange to Hamilton

1. Define canonical momentum $p_i=\partial L/\partial\dot q_i$

2. Legendre transform: $H(q,p,t)=\sum p_i\dot q_i-L$

3. Hamilton’s equations:
   $\dot q_i=\partial H/\partial p_i,\quad \dot p_i=-\partial H/\partial q_i$

Poisson bracket of any two functions $A,B$:
$\{A,B\}=\sum_i\!\bigl(\partial A/\partial q_i\,\partial B/\partial p_i-\partial A/\partial p_i\,\partial B/\partial q_i\bigr)$

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3. Small-Oscillation Formalism

For equilibrium at $q_i^{(0)}$:

1. Expand $L=T-V$ to second order in $\eta_i=q_i-q_i^{(0)}$.

2. Obtain matrix form:

   V=\tfrac12\sum K_{ij}\eta_i\eta_j $$

3. Equations: $\mathbf M\ddot{\boldsymbol\eta}+\mathbf K\boldsymbol\eta=0$

4. Normal-mode frequencies from det$\,|K-\omega^2 M|=0$.

Example: double pendulum small angles → 2×2 eigenvalue problem yields two normal ω’s.

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4. Quick-Reference Identities

Topic	Key Result
Generalized energy	$E=\sum \dot q_i\,p_i-L;\;$ if $L$ is time-independent → $E=H$ conserved
Noether theorem	Continuous symmetry ⇔ conserved quantity (e.g. spatial translation → momentum)
Legendre transform trick	For 1-D systems often faster to write $H=\tfrac{p^2}{2m}+V(q)$ directly
Small oscillation of 1-D potential $V(x)$	Around stable point $x_0$: $\omega=\sqrt{\,V''(x_0)/m}$

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5. Illustrative Mini-Problems

1. Simple Harmonic Oscillator
   Lagrangian $L=\frac12m\dot x^2-\frac12kx^2$
   → $p=m\dot x,\;H=p^2/2m+kx^2/2$
   → frequency $\omega=\sqrt{k/m}$

2. Bead on Rotating Hoop (small θ about bottom)
   Effective potential near equilibrium $V_\text{eff}≈\frac12(mg/R- mΩ^2)\theta^2$
   → stable only if $Ω^2<g/R$; then $\omega=\sqrt{g/R-Ω^2}$.

3. Coupled Mass–Spring (2 masses, spring k)
   Matrices $M=m\mathbf I,\;K=k\begin{pmatrix}2&-1\\-1&2\end{pmatrix}$
   → eigen-ω: $\omega_1=\sqrt{k/m},\; \omega_2=\sqrt{3k/m}$.

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6. Common Exam-Level Pitfalls

• Forgetting to include generalized coordinates of the rigid body: need three Euler angles plus possible translations (total 6 DOF).

• Mixing up small-angle linearization: keep only up to $\theta^2$ in V and drop higher-order terms in T too.

• Ignoring sign of $V''(x_0)$: positive ⇒ stable equilibrium; negative ⇒ unstable (imaginary ω).

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7. Rapid Revision Sheet

1. Write down Euler-Lagrange, canonical momenta, Hamilton equations.

2. Memorize SHM frequency formula in 1-D: $\omega=\sqrt{V''/m}$.

3. Remember matrices $M, K$ for small-oscillation multi-DOF.

4. For cyclic coordinate: $p$ conserved, reduces DOF.

5. Poisson-bracket properties: antisymmetry, bilinearity, Jacobi identity.