CSIR NET Physical Sciences Classical Mechanics Syllabus

Classify the fixed point of a 2D system when $\tau = 0$ and $\Delta > 0$.

The eigenvalues are purely imaginary, $\lambda = \pm i\sqrt{\Delta}$: the fixed point is a center. Trajectories are closed orbits around it (marginally stable, neutrally stable).

On the trace-determinant ($\tau$-$\Delta$) plane, what does the parabola $\tau^{2} = 4\Delta$ represent?

It is the boundary separating nodes (below, $\tau^{2}>4\Delta$, real eigenvalues) from spirals/foci (above, $\tau^{2}<4\Delta$, complex eigenvalues). On the parabola itself the eigenvalues are real and equal (degenerate/star or improper nodes).

How is the stability type of a fixed point related to the local potential energy $V(x)$ for a 1D conservative system?

A minimum of $V(x)$ ($V''>0$) is a stable equilibrium (a center in phase space); a maximum of $V(x)$ ($V''<0$) is an unstable equilibrium (a saddle/hyperbolic point in phase space).

For a 1D conservative system near a potential minimum, what is the frequency of small oscillations?

$$\omega = \sqrt{\frac{V''(x_0)}{m}}$$ where $x_0$ is the equilibrium position and $V''(x_0)>0$.

Define a Lyapunov function and state what it establishes.

A Lyapunov function $V(\vec{x})$ is a scalar function that is positive definite about a fixed point ($V(\vec{x}^{*})=0$, $V>0$ elsewhere nearby) with $\dot{V}\leq 0$ along trajectories. Its existence proves (Lyapunov) stability without solving the equations of motion; $\dot{V}<0$ proves asymptotic stability.

Distinguish Lyapunov (neutral) stability from asymptotic stability.

Lyapunov stable: trajectories starting near the fixed point stay near it for all time. Asymptotically stable: trajectories starting near it stay near AND converge to the fixed point as $t\to\infty$. Asymptotic stability is strictly stronger.

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.

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.

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.

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.

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.

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: $\lambda_{1,2} = -\gamma \pm \sqrt{\gamma^{2} - \omega_0^{2}}$.

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.

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.

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.

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).

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.

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 $r<0$.

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 $x=\pm\sqrt{r}$.

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.

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).

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).

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.

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 $\geq 3$.

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.

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.

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, $\dot{\phi}=\omega=\partial H/\partial J$.

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.

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).

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.