CSIR NET Physical Sciences 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 CSIR NET Physical Sciences, not a summary of it.

1. 1

   Newton’s Laws

   3 topics

   • Dynamical Systems
   • Phase Space Dynamics
   • Stability Analysis
2. 2

   Central Force Motions

   1 topic

   • Two-Body Collisions
     • Scattering in Laboratory Frame
     • Scattering in Centre of Mass Frame
3. 3

   Rigid Body Dynamics

   1 topic

   • Moment of Inertia Tensor
4. 4

   Non-Inertial Frames and Pseudoforces

   5 topics

   • Variational Principle
   • Generalized Coordinates
   • Lagrangian and Hamiltonian Formalism
     • Equations of Motion
   • Conservation Laws and Cyclic Coordinates
   • Periodic Motion
     • Small Oscillations
     • Normal Modes
5. 5

   Special Theory of Relativity

   3 topics

   • Lorentz Transformations
   • Relativistic Kinematics
   • Mass–Energy Equivalence

Classical Mechanics flashcards for CSIR NET Physical Sciences

18 of 50 cards from the Classical Mechanics deck — real questions with worked answers.

1. What is the dimension of the phase space for a mechanical system with $N$ degrees of freedom?

   The phase space has $2N$ dimensions, spanned by the $N$ generalized coordinates $q_i$ and their $N$ conjugate momenta $p_i$.

2. Define a 'phase point' and a 'phase trajectory' in phase space.

   A phase point is a single point $(q_1,\dots,q_N,p_1,\dots,p_N)$ representing the complete instantaneous state of the system. As the system evolves in time it traces out a curve called the phase trajectory (phase path).

3. Write Hamilton's canonical equations of motion that govern flow in phase space.

   $$\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}$$

4. Why can two distinct phase trajectories never intersect in phase space?

   Because Hamilton's equations are first-order and give a unique velocity vector $(\dot{q},\dot{p})$ at each phase point. If trajectories crossed, the future evolution from the crossing point would not be uniquely determined, violating determinism.

5. What is the phase trajectory of a one-dimensional simple harmonic oscillator, and what shape does it have?

   It is an ellipse in the $(x,p)$ plane given by $\frac{p^{2}}{2m} + \frac{1}{2}m\omega^{2}x^{2} = E$, since energy is conserved. Trajectories of different energies are nested ellipses.

6. State Liouville's theorem.

   The phase space density $\rho$ of an ensemble of systems is conserved along the flow: the volume of any region of phase space remains constant in time as it evolves under Hamiltonian dynamics. The phase fluid is incompressible.

7. Express Liouville's theorem mathematically using the total time derivative of the density.

   $$\frac{d\rho}{dt} = \frac{\partial \rho}{\partial t} + \{\rho, H\} = 0$$ where $\{\rho,H\}$ is the Poisson bracket.

8. Show that the phase-space flow velocity field is divergence-free for a Hamiltonian system.

   $$\sum_i\left(\frac{\partial \dot{q}_i}{\partial q_i} + \frac{\partial \dot{p}_i}{\partial p_i}\right) = \sum_i\left(\frac{\partial^{2}H}{\partial q_i\partial p_i} - \frac{\partial^{2}H}{\partial p_i\partial q_i}\right) = 0$$ The vanishing divergence implies incompressible flow.

9. Define the Poisson bracket of two phase-space functions $f$ and $g$.

   $$\{f,g\} = \sum_i\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)$$

10. What are the fundamental Poisson brackets among canonical coordinates and momenta?

    $$\{q_i,q_j\}=0,\qquad \{p_i,p_j\}=0,\qquad \{q_i,p_j\}=\delta_{ij}$$

11. Write the equation of motion of any dynamical variable $f(q,p,t)$ in Poisson-bracket form.

    $$\frac{df}{dt} = \{f,H\} + \frac{\partial f}{\partial t}$$

12. What is the condition (in Poisson-bracket form) for a quantity $f$ with no explicit time dependence to be a constant of motion?

    $f$ is conserved if and only if its Poisson bracket with the Hamiltonian vanishes: $\{f,H\}=0$.

13. Define a fixed point (equilibrium point) of a dynamical system $\dot{\vec{x}} = \vec{F}(\vec{x})$.

    A fixed point $\vec{x}^{*}$ is a point where the flow vanishes: $\vec{F}(\vec{x}^{*}) = 0$. A system placed exactly at $\vec{x}^{*}$ remains there for all time.

14. What is the general procedure for linear stability analysis about a fixed point $\vec{x}^{*}$?

    Write $\vec{x} = \vec{x}^{*} + \delta\vec{x}$, substitute into $\dot{\vec{x}}=\vec{F}(\vec{x})$, and keep only first-order terms to obtain $\dot{\delta\vec{x}} = J\,\delta\vec{x}$, where $J$ is the Jacobian matrix evaluated at $\vec{x}^{*}$. The eigenvalues of $J$ determine stability.

15. Write the Jacobian (stability) matrix for a two-dimensional system $\dot{x}=f(x,y)$, $\dot{y}=g(x,y)$.

    $$J = \begin{pmatrix} \dfrac{\partial f}{\partial x} & \dfrac{\partial f}{\partial y} \\[2mm] \dfrac{\partial g}{\partial x} & \dfrac{\partial g}{\partial y} \end{pmatrix}_{(x^{*},y^{*})}$$

16. In terms of the eigenvalues $\lambda$ of the Jacobian, when is a fixed point linearly stable versus unstable?

    Stable if all eigenvalues have negative real part ($\mathrm{Re}\,\lambda < 0$); unstable if any eigenvalue has positive real part ($\mathrm{Re}\,\lambda > 0$). Purely imaginary eigenvalues give marginal (borderline) cases.

17. For a 2D linear system, express the eigenvalues in terms of the trace $\tau$ and determinant $\Delta$ of the Jacobian.

    $$\lambda_{1,2} = \frac{\tau \pm \sqrt{\tau^{2} - 4\Delta}}{2}, \qquad \tau = \lambda_1+\lambda_2,\quad \Delta = \lambda_1\lambda_2$$

18. Classify the fixed point of a 2D system when $\Delta < 0$.

    It is a saddle point (unstable). The eigenvalues are real with opposite signs, so the fixed point attracts along one direction and repels along another.

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