GATE Physics Classical Mechanics Syllabus

Classical Mechanics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Classical Mechanics in GATE Physics, not a summary of it.

1. 1

   Lagrangian Formulation

   5 topics

   • D'Alembert's Principle
   • Euler-Lagrange Equation
   • Hamilton's Principle
   • Calculus of Variations
   • Symmetry and Conservation Laws
2. 2

   Central Force Motion

   2 topics

   • Kepler Problem
   • Rutherford Scattering
3. 3

   Small Oscillations

   2 topics

   • Coupled Oscillations
   • Normal Modes
4. 4

   Rigid Body Dynamics

   4 topics

   • Inertia Tensor
   • Orthogonal Transformations
   • Euler Angles
   • Torque Free Motion of a Symmetric Top
5. 5

   Hamiltonian and Hamilton's Equations of Motion

   5 topics

   • Liouville's Theorem
   • Canonical Transformations
   • Action-Angle Variables
   • Poisson Brackets
   • Hamilton-Jacobi Equation

Classical Mechanics flashcards for GATE Physics

18 of 53 cards from the Classical Mechanics deck — real questions with worked answers.

1. State D'Alembert's Principle in its standard form.

   It states that the sum of the differences between the applied forces and the inertial forces (time-derivatives of momentum) does zero virtual work along any virtual displacement consistent with the constraints: $$\sum_{i}\left(\vec{F}_{i}^{(a)} - \dot{\vec{p}}_{i}\right)\cdot \delta \vec{r}_{i} = 0$$ It eliminates the (unknown) constraint forces, which do no virtual work.

2. What is a virtual displacement $\delta \vec{r}$, and how does it differ from a real displacement $d\vec{r}$?

   A virtual displacement $\delta \vec{r}$ is an infinitesimal, instantaneous (at fixed time, $\delta t = 0$) change in the configuration consistent with the constraints. A real displacement $d\vec{r}$ occurs over an actual time interval $dt$ and includes explicit time variation. For workless (ideal) constraints, constraint forces satisfy $\sum_i \vec{f}_i^{(c)}\cdot\delta\vec{r}_i = 0$.

3. Write the Euler-Lagrange equations of motion for a system with generalized coordinates $q_j$ and Lagrangian $L$.

   $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right) - \frac{\partial L}{\partial q_{j}} = 0, \qquad j = 1,\dots,n$$ where $L = T - V$ for monogenic (potential-derivable) forces.

4. Define the Lagrangian $L$ and the generalized (canonical) momentum $p_j$.

   The Lagrangian is $L = T - V$, the difference between kinetic and potential energy expressed in generalized coordinates and velocities. The generalized momentum conjugate to $q_j$ is $$p_{j} = \frac{\partial L}{\partial \dot{q}_{j}}.$$

5. What is a cyclic (ignorable) coordinate, and what does it imply?

   A coordinate $q_j$ is cyclic if it does not appear explicitly in the Lagrangian, i.e. $\dfrac{\partial L}{\partial q_{j}} = 0$. Then the Euler-Lagrange equation gives $\dfrac{d}{dt}\dfrac{\partial L}{\partial \dot q_j}=0$, so the conjugate momentum $p_{j} = \dfrac{\partial L}{\partial \dot{q}_{j}}$ is conserved.

6. State Hamilton's Principle (the principle of stationary action).

   The actual path taken by a system between times $t_1$ and $t_2$ makes the action integral stationary: $$\delta S = \delta \int_{t_{1}}^{t_{2}} L(q,\dot{q},t)\, dt = 0$$ for variations with fixed endpoints $\delta q(t_1)=\delta q(t_2)=0$. The Euler-Lagrange equations follow from this.

7. What is the fundamental problem of the calculus of variations, and what is the resulting equation?

   It seeks the function $y(x)$ that makes the functional $J = \int_{x_1}^{x_2} f(y, y', x)\, dx$ stationary. The extremal satisfies the Euler equation: $$\frac{\partial f}{\partial y} - \frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right) = 0.$$

8. State the Beltrami identity (the special case of the Euler equation when $f$ has no explicit $x$ dependence).

   If $f = f(y, y')$ does not depend explicitly on $x$, then a first integral exists: $$f - y'\frac{\partial f}{\partial y'} = \text{constant}.$$ This is the form used to solve the brachistochrone problem.

9. What curve is the solution to the brachistochrone problem?

   A cycloid. It is the curve of fastest descent between two points under gravity, obtained by applying the Beltrami identity to $f = \sqrt{\dfrac{1+y'^{2}}{2gy}}$.

10. How are constraints handled in variational problems using Lagrange multipliers?

    For a constraint $g(y,x)=0$ (or isoperimetric constraint), one extremizes $\int (f + \lambda g)\,dx$, giving $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} + \lambda\left(\frac{\partial g}{\partial y} - \frac{d}{dx}\frac{\partial g}{\partial y'}\right)=0.$$ The multiplier $\lambda$ relates to the (generalized) force of constraint.

11. State Noether's theorem connecting symmetry and conservation laws.

    For every continuous symmetry of the action (a transformation leaving the Lagrangian invariant up to a total time derivative), there exists a corresponding conserved quantity. Symmetries of the action imply conservation laws.

12. Which conservation law corresponds to each of: time-translation, space-translation, and rotational symmetry?

    Time-translation invariance $\Rightarrow$ conservation of energy (the Hamiltonian/Jacobi integral). Spatial-translation invariance $\Rightarrow$ conservation of linear momentum. Rotational invariance $\Rightarrow$ conservation of angular momentum.

13. Define the energy (Jacobi) function $h$ and state when it equals the total energy.

    $$h = \sum_{j}\dot{q}_{j}\frac{\partial L}{\partial \dot{q}_{j}} - L.$$ If $L$ has no explicit time dependence, $h$ is conserved. If additionally the constraints are scleronomic (time-independent) and $V$ is velocity-independent, then $h = T + V = E$, the total energy.

14. Reduce the two-body central-force (Kepler) problem to an effective one-body problem.

    Introduce the reduced mass $\mu = \dfrac{m_1 m_2}{m_1+m_2}$ and relative coordinate $\vec{r}=\vec{r}_1-\vec{r}_2$. The CM moves uniformly; the relative motion obeys a one-body equation for a particle of mass $\mu$ in potential $V(r)$. Motion is planar because $\vec{L}$ is conserved.

15. Write the effective potential for the Kepler problem and identify its terms.

    $$V_{\text{eff}}(r) = V(r) + \frac{L^{2}}{2\mu r^{2}} = -\frac{k}{r} + \frac{L^{2}}{2\mu r^{2}}$$ for an attractive inverse-square force $\vec{F}=-\dfrac{k}{r^2}\hat r$. The first term is the gravitational/Coulomb potential; the second is the (repulsive) centrifugal barrier.

16. Give the orbit equation and eccentricity for the Kepler problem.

    The orbit is a conic section: $$\frac{1}{r} = \frac{\mu k}{L^{2}}\left(1 + e\cos\theta\right), \qquad e = \sqrt{1 + \frac{2EL^{2}}{\mu k^{2}}}.$$ The orbit shape depends on $e$ (and hence energy $E$).

17. Classify Kepler orbits by eccentricity $e$ and energy $E$.

    Circle: $e=0$, $E = -\dfrac{\mu k^2}{2L^2}$ (minimum). Ellipse: $0<e<1$, $E<0$. Parabola: $e=1$, $E=0$. Hyperbola: $e>1$, $E>0$.

18. State Kepler's three laws of planetary motion.

    1. (Law of orbits) Planets move in ellipses with the Sun at one focus. 2. (Law of areas) The radius vector sweeps equal areas in equal times ($\dot{A} = \dfrac{L}{2\mu}=$ const), expressing angular-momentum conservation. 3. (Law of periods) $T^{2}\propto a^{3}$, where $a$ is the semi-major axis.

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