GATE Physics Classical Mechanics Syllabus

Define the impact parameter $b$ and relate it to the scattering angle for Coulomb scattering.

The impact parameter $b$ is the perpendicular distance from the target of the incident particle's initial straight-line trajectory. For Coulomb scattering: $$b = \frac{Z_{1}Z_{2}e^{2}}{2E}\cot\left(\frac{\theta}{2}\right).$$

Give the general relation between differential cross-section, impact parameter, and scattering angle.

$$\frac{d\sigma}{d\Omega} = \frac{b}{\sin\theta}\left|\frac{db}{d\theta}\right|.$$ The absolute value accounts for the typical decrease of $b$ with increasing $\theta$.

Why does the total Rutherford cross-section diverge?

Because the Coulomb potential has infinite range ($\sim 1/r$), even particles with very large impact parameter are deflected slightly. Integrating $\dfrac{d\sigma}{d\Omega}\propto\sin^{-4}(\theta/2)$ over all angles diverges as $\theta\to 0$, so $\sigma_{\text{tot}}\to\infty$.

Set up the equations of motion for two coupled oscillators (e.g. two equal masses $m$, spring constant $k$, coupling spring $k'$).

With displacements $x_1, x_2$: $$m\ddot{x}_{1} = -k x_{1} - k'(x_{1}-x_{2}),$$ $$m\ddot{x}_{2} = -k x_{2} - k'(x_{2}-x_{1}).$$ These are coupled through the $k'$ term.

What are normal modes and normal coordinates?

Normal modes are special collective oscillation patterns in which all parts of the system move sinusoidally with the same single frequency and fixed phase relations. Normal coordinates are linear combinations of the original coordinates that decouple the equations, each oscillating independently at its own normal frequency.

Find the normal-mode frequencies of two identical masses $m$ coupled by springs $k$ (to walls) and $k'$ (between them).

Symmetric mode ($x_1=x_2$): $\omega_{+} = \sqrt{\dfrac{k}{m}}$. Antisymmetric mode ($x_1=-x_2$): $\omega_{-}=\sqrt{\dfrac{k+2k'}{m}}$. The normal coordinates are $\eta_{\pm}=\dfrac{1}{\sqrt2}(x_1\pm x_2)$.

What is the general eigenvalue equation for finding normal-mode frequencies?

From $T = \tfrac12 \dot{\eta}^{T} \mathbf{T} \dot{\eta}$ and $V = \tfrac12 \eta^{T}\mathbf{V}\eta$, the normal frequencies satisfy the generalized eigenvalue problem: $$\det\left(\mathbf{V} - \omega^{2}\mathbf{T}\right) = 0,$$ where $\mathbf{V}$ is the stiffness matrix and $\mathbf{T}$ the mass/kinetic-energy matrix.

How many normal modes does a system have, and what is a degenerate mode?

A system with $n$ degrees of freedom has $n$ normal modes (and $n$ normal frequencies). A degenerate mode occurs when two or more distinct normal modes share the same frequency $\omega$; any linear combination of degenerate modes is also a valid normal mode.

Define the moment of inertia tensor $I_{ij}$.

$$I_{ij} = \sum_{\alpha} m_{\alpha}\left(r_{\alpha}^{2}\delta_{ij} - x_{\alpha,i}\,x_{\alpha,j}\right)$$ (or the integral $\int \rho\,(r^2\delta_{ij}-x_i x_j)\,dV$). The diagonal elements are moments of inertia; the off-diagonal elements are products of inertia.

Express the angular momentum $\vec{L}$ and rotational kinetic energy $T$ in terms of the inertia tensor.

$$\vec{L} = \mathbf{I}\,\vec{\omega}, \qquad T = \frac{1}{2}\,\vec{\omega}\cdot\mathbf{I}\,\vec{\omega} = \frac{1}{2}\sum_{ij} I_{ij}\,\omega_{i}\omega_{j}.$$ In general $\vec{L}$ is not parallel to $\vec{\omega}$.

What are principal axes and principal moments of inertia?

Principal axes are the body-fixed axes for which the inertia tensor is diagonal (products of inertia vanish). The diagonal entries $I_1, I_2, I_3$ are the principal moments of inertia, the eigenvalues of $\mathbf{I}$. Along a principal axis, $\vec{L} = I_i\,\vec{\omega}$ is parallel to $\vec{\omega}$.

State the parallel axis (Steiner) theorem.

$$I = I_{\text{cm}} + M d^{2},$$ where $I_{\text{cm}}$ is the moment of inertia about an axis through the center of mass, $M$ the total mass, and $d$ the perpendicular distance between the parallel axis and the CM axis.

Classify rigid bodies as spherical, symmetric, and asymmetric tops by their principal moments.

Spherical top: $I_1 = I_2 = I_3$ (e.g. cube, sphere). Symmetric top: two equal, one different, $I_1 = I_2 \neq I_3$ (e.g. cone, disk). Asymmetric top: all different, $I_1 \neq I_2 \neq I_3$.

Define an orthogonal transformation and give its defining matrix condition.

A linear transformation $\vec{x}' = \mathbf{A}\vec{x}$ that preserves lengths and angles. The matrix satisfies $$\mathbf{A}^{T}\mathbf{A} = \mathbf{A}\mathbf{A}^{T} = \mathbf{1}, \quad\text{i.e.}\quad \mathbf{A}^{T}=\mathbf{A}^{-1},$$ equivalently $\sum_k A_{ik}A_{jk}=\delta_{ij}$.

What distinguishes a proper rotation from an improper orthogonal transformation?

For any orthogonal matrix $\det\mathbf{A}=\pm 1$. Proper rotations have $\det\mathbf{A}=+1$ (continuously connected to identity). Improper transformations have $\det\mathbf{A}=-1$ and include a reflection/inversion (orientation-reversing).

Define the three Euler angles and the rotation sequence they represent.

The Euler angles $(\phi, \theta, \psi)$ describe rotation from space to body axes via the $z$-$x'$-$z''$ convention: (1) rotate by $\phi$ about the $z$-axis, (2) rotate by $\theta$ about the new line of nodes ($x'$-axis), (3) rotate by $\psi$ about the new $z''$ (body symmetry) axis. The full matrix is $\mathbf{A} = \mathbf{B}\mathbf{C}\mathbf{D}$.

In the Euler-angle description of a symmetric top, what motions do $\dot\phi$, $\dot\theta$, and $\dot\psi$ represent?

$\dot\phi$ is precession (rotation of the body axis about the vertical space axis), $\dot\theta$ is nutation (the nodding/bobbing of the body axis), and $\dot\psi$ is spin (rotation of the body about its own symmetry axis).

Write Euler's equations of motion for a rigid body about its principal axes.

$$I_{1}\dot{\omega}_{1} - (I_{2}-I_{3})\omega_{2}\omega_{3} = N_{1},$$ $$I_{2}\dot{\omega}_{2} - (I_{3}-I_{1})\omega_{3}\omega_{1} = N_{2},$$ $$I_{3}\dot{\omega}_{3} - (I_{1}-I_{2})\omega_{1}\omega_{2} = N_{3}.$$

Describe torque-free motion of a symmetric top and find its body-frame precession rate.

With $N=0$ and $I_1=I_2\neq I_3$, Euler's equations give $\omega_3=$ const, and $\omega_1,\omega_2$ rotate at frequency $$\Omega = \frac{I_{3}-I_{1}}{I_{1}}\,\omega_{3}.$$ The angular velocity vector precesses about the symmetry axis in the body frame (free precession), as for the Earth's Chandler wobble.

For torque-free symmetric-top motion, what are the space cone and body cone?

$\vec{L}$ is fixed in space. $\vec{\omega}$ precesses about $\vec{L}$ tracing the space cone, while $\vec{\omega}$ also precesses about the body's symmetry axis tracing the body cone. The motion is the body cone rolling without slipping on the fixed space cone.

State Liouville's theorem.

The density of representative points (phase-space volume) of an ensemble of systems is conserved along the flow generated by Hamilton's equations: $$\frac{d\rho}{dt} = \frac{\partial\rho}{\partial t} + \{\rho, H\} = 0.$$ Phase-space volume is incompressible under Hamiltonian (canonical) evolution.

What is the physical/geometric significance of Liouville's theorem?

The volume element $\prod_i dq_i\,dp_i$ in phase space is invariant under canonical (Hamiltonian) time evolution; the phase fluid flows like an incompressible fluid. This is foundational for statistical mechanics (justifies the microcanonical measure) and implies phase-space area conservation.

Define a canonical transformation and the condition involving the Poisson bracket / generating function.

A transformation $(q,p)\to(Q,P)$ is canonical if it preserves the form of Hamilton's equations, equivalently if it preserves the fundamental Poisson brackets: $$\{Q_i,Q_j\}=0,\quad\{P_i,P_j\}=0,\quad\{Q_i,P_j\}=\delta_{ij}.$$ It can be derived from a generating function $F$ satisfying $p\,dq - P\,dQ = dF$ (up to a Hamiltonian change).

List the four types of generating functions and the relations from $F_1(q,Q,t)$.

The four types are $F_1(q,Q)$, $F_2(q,P)$, $F_3(p,Q)$, $F_4(p,P)$, related by Legendre transforms. For $F_1$: $$p_i = \frac{\partial F_1}{\partial q_i}, \quad P_i = -\frac{\partial F_1}{\partial Q_i}, \quad K = H + \frac{\partial F_1}{\partial t}.$$

Give the generating-function relations for the type-2 generator $F_2(q,P,t)$.

$$p_i = \frac{\partial F_2}{\partial q_i}, \quad Q_i = \frac{\partial F_2}{\partial P_i}, \quad K = H + \frac{\partial F_2}{\partial t}.$$ The identity transformation corresponds to $F_2 = \sum_i q_i P_i$.

Define action-angle variables and give the action integral.

For a periodic (separable) system, the action variable for each degree of freedom is $$J_i = \oint p_i\, dq_i,$$ integrated over one full cycle. Its conjugate angle variable $w_i$ increases linearly with time. The $J_i$ are constants of motion; $w_i$ advances by $1$ (or $2\pi$) per period.

How is the oscillation frequency obtained from action-angle variables?

In action-angle variables the transformed Hamiltonian depends only on the actions, $H=H(J)$. The angle evolves as $\dot{w}_i = \dfrac{\partial H}{\partial J_i} = \nu_i$ (constant), so the frequency of the $i$-th periodic motion is $$\nu_i = \frac{\partial H}{\partial J_i}.$$

Define the Poisson bracket of two functions $u$ and $v$.

$$\{u,v\} = \sum_{i}\left(\frac{\partial u}{\partial q_{i}}\frac{\partial v}{\partial p_{i}} - \frac{\partial u}{\partial p_{i}}\frac{\partial v}{\partial q_{i}}\right).$$

Write the time-evolution of a dynamical variable $u(q,p,t)$ in terms of Poisson brackets.

$$\frac{du}{dt} = \{u, H\} + \frac{\partial u}{\partial t}.$$ If $u$ has no explicit time dependence and $\{u,H\}=0$, then $u$ is a constant of motion.

Give the fundamental Poisson brackets and list the algebraic properties of the bracket.

Fundamental brackets: $\{q_i,q_j\}=0$, $\{p_i,p_j\}=0$, $\{q_i,p_j\}=\delta_{ij}$. Properties: antisymmetry $\{u,v\}=-\{v,u\}$, linearity, the product (Leibniz) rule $\{u,vw\}=\{u,v\}w+v\{u,w\}$, and the Jacobi identity $\{u,\{v,w\}\}+\{v,\{w,u\}\}+\{w,\{u,v\}\}=0$.

Write the Hamilton-Jacobi equation and identify Hamilton's principal function.

$$H\!\left(q_1,\dots,q_n,\frac{\partial S}{\partial q_1},\dots,\frac{\partial S}{\partial q_n}, t\right) + \frac{\partial S}{\partial t} = 0,$$ where $S(q,\alpha,t)$ is Hamilton's principal function, the generator of a canonical transformation to constant coordinates and momenta. The momenta are $p_i = \partial S/\partial q_i$.

For a conservative system, how does the Hamilton-Jacobi equation separate using Hamilton's characteristic function $W$?

Write $S(q,\alpha,t) = W(q,\alpha) - E t$. Then the time-independent (restricted) Hamilton-Jacobi equation becomes $$H\!\left(q_i, \frac{\partial W}{\partial q_i}\right) = E = \alpha_1.$$ $W$ is Hamilton's characteristic function; $\partial S/\partial t=-E$ since $H$ is conserved.

What is the physical interpretation of Hamilton's principal function $S$, and its total time derivative?

$S$ equals the action integral evaluated along the actual trajectory: $S=\int L\,dt$. Its total time derivative is the Lagrangian: $$\frac{dS}{dt} = L = \sum_i p_i\dot{q}_i - H.$$ Thus $S$ acts as the generating function $F_2$ that maps the evolved motion onto constant initial conditions.