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PHYSICS FORMULA SHEET
Stability Analysis & Phase Space
1. Equilibrium Points & Stability Analysis
An equilibrium point \(x_0\) is defined by vanishing net conservative force:
\[ F(x_0) = -\left.\frac{dV}{dx}\right\vert_{x=x_0} = 0 \quad\Longrightarrow\quad \left.\frac{dV}{dx}\right\vert_{x=x_0}=0 \]
Classification via second derivative test
- Stable equilibrium: \(\left.\dfrac{d^2V}{dx^2}\right\vert_{x_0} > 0\) (local minimum of \(V\))
- Unstable equilibrium: \(\left.\dfrac{d^2V}{dx^2}\right\vert_{x_0} < 0\) (local maximum of \(V\))
- Neutral / higher-order equilibrium: \(\left.\dfrac{d^2V}{dx^2}\right\vert_{x_0} = 0\). Examine the first non-vanishing derivative \(V^{(n)}(x_0)\):
- \(n\) even, \(V^{(n)}(x_0) > 0 \Rightarrow\) stable
- \(n\) even, \(V^{(n)}(x_0) < 0 \Rightarrow\) unstable
- \(n\) odd \(\Rightarrow\) unstable (inflection-type, e.g. cubic term)
Exam Tip
GATE/JAM often give \(V(x)\) as a polynomial and ask you to locate all equilibria by solving \(V'(x)=0\), then classify each using \(V''(x)\) — always check every root, not just \(x=0\).
2. Classically Allowed and Forbidden Regions
For a particle of total energy \(E = T + V(x)\), momentum is
\[ p = \pm\sqrt{2m\big(E - V(x)\big)} \]
Region conditions
- Allowed region: \(E \ge V(x) \Rightarrow p \in \mathbb{R}\) (real momentum, motion possible)
- Forbidden region: \(E < V(x) \Rightarrow T<0\), \(p\) imaginary — classically inaccessible
- Turning point(s) \(x_t\): defined by \(E = V(x_t)\), i.e. \(p(x_t)=0\); the velocity reverses sign here
3. Bounded Motion & Small Oscillations About Stable Equilibrium
Expand \(V(x)\) in a Taylor series about a stable equilibrium \(x_0\), with \(\eta = x - x_0\):
\[ V(x) = V(x_0) + \underbrace{V'(x_0)}_{=0}\eta + \frac{1}{2}V''(x_0)\,\eta^2 + \mathcal{O}(\eta^3) \]
Effective SHM results
\[ k_{\text{eff}} = \left.\frac{d^2V}{dx^2}\right\vert_{x=x_0} \]
\[ m\ddot{\eta} + k_{\text{eff}}\,\eta = 0 \]
\[ \omega = \sqrt{\dfrac{k_{\text{eff}}}{m}} = \sqrt{\dfrac{V''(x_0)}{m}} \]
\[ T = \dfrac{2\pi}{\omega} = 2\pi\sqrt{\dfrac{m}{V''(x_0)}} \]
Multi-dimensional / coupled systems (secular equation):
\[ \det\big(\mathbf{V} - \omega^2\,\mathbf{T}\big) = 0, \qquad V_{ij} = \left.\frac{\partial^2 V}{\partial q_i\,\partial q_j}\right\vert_{eq}, \qquad T_{ij} = \text{kinetic-energy matrix element} \]
Exam Tip
For coupled oscillators (two masses/springs), always write the full \(\mathbf{T}\) and \(\mathbf{V}\) matrices first, then solve \(\det(\mathbf{V}-\omega^2\mathbf{T})=0\) for normal-mode frequencies — a very common CSIR‑NET question.
4. Phase Space Trajectories (\(p\) vs. \(x\))
From energy conservation \(E = \dfrac{p^2}{2m} + V(x)\), the phase curve is
\[ p(x) = \pm\sqrt{2m\big(E-V(x)\big)} \]
| System | Potential \(V(x)\) | Phase-curve shape | Key relation / features |
|---|---|---|---|
| (a) Free particle | \(V(x)=0\) | Horizontal straight lines | \(p=\pm\sqrt{2mE}=\text{const}\); unbounded motion |
| (b) Particle in gravitational field | \(V(x)=mgx\) | Parabolas | \(p=\pm\sqrt{2m(E-mgx)}\); opens toward \(x<0\) |
| (c) Harmonic oscillator | \(V(x)=\tfrac{1}{2}kx^2\) | Concentric ellipses | \(\dfrac{p^2}{2mE}+\dfrac{x^2}{2E/k}=1\); closed, bounded orbits |
| (d) Repulsive parabolic potential | \(V(x)=-\tfrac{1}{2}kx^2\) | Hyperbolas | \(\dfrac{p^2}{2mE}-\dfrac{x^2}{2E/k}=1\); unstable equilibrium at origin |
| (e) Cubic potential | \(V(x)=ax^2-bx^3\) | Closed loops + open branches | Local min → closed loop (oscillation); local max → separatrix |
| (f) Quartic potential | \(V(x)=\alpha x^4\) | Flattened closed ovals | \(p=\pm\sqrt{2m(E-\alpha x^4)}\); anharmonic, non-elliptical closed curves |
5. Key Exam Rules for Phase Portraits
Quick-check rules
- Direction of flow: In the upper half-plane (\(p>0\)), \(x\) increases \(\Rightarrow\) motion is clockwise; in the lower half-plane (\(p<0\)), \(x\) decreases \(\Rightarrow\) counter-clockwise.
- Non-intersection rule: Distinct phase trajectories never cross, except at equilibrium (fixed) points where \(p=0,\ V'(x)=0\) (uniqueness of solutions to the equations of motion).
- Separatrix: The special trajectory passing through an unstable equilibrium (\(V'(x_0)=0\), saddle point); it separates topologically distinct classes of motion (bounded oscillation vs. unbounded escape).
- Closed curve \(\Rightarrow\) periodic bounded motion; open curve \(\Rightarrow\) unbounded motion.