CSIR NET Physical Sciences Classical Mechanics Flashcards

24 sample cards from the Classical Mechanics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What is a limit cycle?

   A limit cycle is an isolated closed trajectory in phase space toward which (stable) or away from which (unstable) neighbouring trajectories spiral. It represents self-sustained periodic oscillation and occurs only in nonlinear systems.

2. How does a center differ from a stable limit cycle?

   A center is surrounded by a continuous family of closed orbits (typical of conservative systems) and its amplitude depends on initial conditions. A stable limit cycle is an isolated closed orbit (dissipative/nonlinear systems) whose amplitude is fixed and independent of initial conditions.

3. State the Poincaré–Bendixson theorem and one of its key consequences.

   In a 2D continuous dynamical system, a trajectory confined to a closed bounded region containing no fixed points must approach a periodic orbit (limit cycle). A consequence: chaos is impossible in autonomous continuous systems of dimension less than three.

4. What is the divergence condition for area-preserving versus contracting flow in phase space?

   If $\nabla\cdot\vec{F} = 0$ the flow conserves phase-space volume (conservative/Hamiltonian). If $\nabla\cdot\vec{F} < 0$ the flow contracts volume (dissipative), allowing attractors.

5. What is an attractor, and can a conservative (Hamiltonian) system possess one?

   An attractor is a set toward which trajectories converge as $t\to\infty$ (e.g. stable fixed point, limit cycle, strange attractor). A conservative Hamiltonian system cannot have attractors because Liouville's theorem forbids phase-volume contraction.

6. Write the damped harmonic oscillator equation and give its Jacobian eigenvalues at the origin.

   $$\ddot{x} + 2\gamma\dot{x} + \omega_0^{2}x = 0$$ Eigenvalues: $\lambda_{1,2} = -\gamma \pm \sqrt{\gamma^{2} - \omega_0^{2}}$.

7. Classify the origin of the damped oscillator for the underdamped, overdamped, and critically damped cases.

   Underdamped ($\gamma<\omega_0$): stable spiral (focus). Overdamped ($\gamma>\omega_0$): stable node. Critically damped ($\gamma=\omega_0$): stable degenerate (improper) node. Undamped ($\gamma=0$): center.

8. What does the phase portrait of an undamped simple pendulum look like, including its fixed points?

   Centers at $\theta = 2n\pi$ (stable, hanging down) surrounded by closed orbits (libration), and saddle points at $\theta = (2n+1)\pi$ (unstable, inverted). The saddles are connected by separatrices dividing oscillation from rotation.

9. Define a separatrix in phase space.

   A separatrix is the special trajectory (homoclinic/heteroclinic orbit passing through saddle points) that separates qualitatively different regions of motion in the phase portrait — for example, libration (bounded oscillation) from rotation (unbounded circulation) in a pendulum.

10. For the simple pendulum $\ddot{\theta} + \frac{g}{L}\sin\theta = 0$, find the Jacobian eigenvalues at the inverted equilibrium $\theta=\pi$.

    Linearizing about $\theta=\pi$ gives $\ddot{\phi} = \frac{g}{L}\phi$, so eigenvalues are $\lambda = \pm\sqrt{g/L}$ — real with opposite signs, confirming a saddle (unstable).

11. What is a bifurcation in a dynamical system?

    A bifurcation is a qualitative change in the topology of the phase portrait (e.g. creation, destruction, or change of stability of fixed points or limit cycles) as a control parameter is varied through a critical value.

12. Describe the saddle-node bifurcation.

    In a saddle-node (fold/tangent) bifurcation, as a parameter $r$ crosses a critical value, two fixed points (one stable, one unstable) collide and annihilate. Normal form: $\dot{x} = r + x^{2}$, with no fixed points for $r>0$ and two for $r<0$.

13. Describe the pitchfork bifurcation and give its supercritical normal form.

    A symmetric fixed point loses stability and gives rise to two new symmetric fixed points. Supercritical normal form: $\dot{x} = rx - x^{3}$. For $r<0$ only $x=0$ (stable); for $r>0$, $x=0$ becomes unstable and two stable points appear at $x=\pm\sqrt{r}$.

14. Describe the Hopf bifurcation.

    A Hopf bifurcation occurs when a pair of complex-conjugate eigenvalues of the Jacobian crosses the imaginary axis as a parameter varies. A stable fixed point becomes unstable and a limit cycle is born (supercritical) — the onset of self-sustained oscillations.

15. Write the transcritical bifurcation normal form and describe its behaviour.

    $$\dot{x} = rx - x^{2}$$ Two fixed points $x=0$ and $x=r$ exist for all $r$ and exchange their stability as $r$ passes through $0$ (they collide at $r=0$ but persist).

16. Define the Poincaré section (surface of section).

    A Poincaré section is a lower-dimensional surface transverse to the flow; the sequence of points where successive trajectory crossings pierce it forms a Poincaré map. It reduces a continuous flow to a discrete map, revealing periodic orbits (fixed points), tori (closed curves), and chaos (scattered points).

17. What is a Lyapunov exponent and what does its sign indicate?

    A Lyapunov exponent $\lambda$ measures the average exponential rate of separation of initially nearby trajectories: $|\delta\vec{x}(t)| \approx |\delta\vec{x}(0)|e^{\lambda t}$. A positive largest exponent ($\lambda>0$) signals sensitive dependence on initial conditions, i.e. chaos; $\lambda<0$ indicates convergence/stability.

18. What characterizes a strange attractor?

    A strange attractor is an attractor with fractal (non-integer) dimension on which the motion is chaotic — exhibiting sensitive dependence on initial conditions (positive Lyapunov exponent) while remaining bounded. It arises only in dissipative nonlinear systems of dimension $\geq 3$.

19. State the structural distinction between an integrable and a chaotic Hamiltonian system in phase space.

    An integrable system with $N$ degrees of freedom has $N$ independent conserved quantities in involution; its motion lies on $N$-dimensional invariant tori (quasiperiodic). A non-integrable/chaotic system lacks enough conserved quantities, the tori break up (KAM scenario), and trajectories explore higher-dimensional regions of the energy surface.

20. What does the KAM (Kolmogorov–Arnold–Moser) theorem state qualitatively?

    Under a sufficiently small perturbation of an integrable Hamiltonian, most invariant tori (those with sufficiently irrational/non-resonant frequency ratios) survive and are only slightly deformed; resonant tori break up, seeding regions of chaos. Integrable motion is thus robust against weak perturbations.

21. What is the action variable $J$ for a periodic 1D Hamiltonian system, and why is it useful?

    $$J = \frac{1}{2\pi}\oint p\,dq$$ It is the phase-space area enclosed by the orbit divided by $2\pi$. It is an adiabatic invariant and a conserved action in action-angle variables, making the dynamics trivially solvable: the conjugate angle advances uniformly, $\dot{\phi}=\omega=\partial H/\partial J$.

22. What is an adiabatic invariant?

    An adiabatic invariant is a quantity (such as the action $J=\oint p\,dq$) that remains approximately constant when a parameter of the system is varied slowly compared to the system's characteristic oscillation period.

23. Compare a conservative and a dissipative system in terms of phase-space volume and long-term behaviour.

    Conservative (Hamiltonian): $\nabla\cdot\vec{F}=0$, phase volume preserved (Liouville), no attractors, fixed points are centers or saddles, motion lies on tori. Dissipative: $\nabla\cdot\vec{F}<0$, phase volume contracts, attractors exist (stable nodes/foci, limit cycles, strange attractors).

24. Why is a center classified as a 'marginal' or 'borderline' case in linear stability analysis?

    For a center the Jacobian has purely imaginary eigenvalues ($\mathrm{Re}\,\lambda=0$), so the linearization predicts neutral oscillation. Because the real part is exactly zero, nonlinear (higher-order) terms can tip it either to a stable or unstable spiral — linear analysis alone is inconclusive for the full nonlinear system.

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