Lecture 13 – Potential Energy, Equilibrium & Small Oscillations

Potential Energy from Force

• Potential energy $V(x)$ defined by $F_x = - \frac{dV}{dx}$
• Integrate the negative of force:
  • $F<em>x = kx \;(x>0) \;\Rightarrow\; V(x)=\frac{1}{2}k x^2 + C$ • Fx = \frac{1}{x^2+a^2}\;(a>0) \;\Rightarrow\; V(x)= -\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C
  • $F_x = 5\sin 2\pi x \;\Rightarrow\; V(x)= -\frac{5}{2\pi}\cos 2\pi x + C$

Potential-Energy Curves & Equilibrium

• Equilibrium points: $\frac{dV}{dx}=0$
  • Stable: \frac{d^2V}{dx^2}>0 (minimum)
  • Unstable: \frac{d^2V}{dx^2}<0 (maximum)
• Near a stable equilibrium, motion is oscillatory; expansion $V(x)\approx V<em>0+\frac{1}{2}k(x-x</em>0)^2$

1-D Harmonic Oscillator

• Force: $F=-kx$ ⇒ $V(x)=\frac{1}{2}kx^2$
• Energy: $E=\frac{1}{2}kx^2+\frac{1}{2}mv^2$
• Velocity–position ellipse: $\frac{x^2}{2E/k}+\frac{v^2}{2E/m}=1$

Simple Pendulum

• Coordinates: angle $\theta$, length $l$
• Potential: $V(\theta)=mgl(1-\cos\theta)$
• Equilibria
  • $\theta=0$ stable (minimum, \frac{d^2V}{d\theta^2}=mgl>0)
  • $\theta=\pi$ unstable (maximum, $\frac{d^2V}{d\theta^2}=-mgl$)
• Energy: $E=\frac{1}{2}ml^2\dot\theta^2+mgl(1-\cos\theta)$
• Small-angle approximation $\cos\theta\approx1-\theta^2/2$ ⇒ simple harmonic motion
• Larger $E$ → larger oscillations; once $E\ge 2mgl$ motion becomes full circular (unbounded in $\theta$)

Taylor-Series Reminder (about equilibrium analysis)

• $f(x<em>0+a)=f(x</em>0)+a\left.\frac{df}{dx}\right|<em>{x</em>0}+\frac{a^2}{2!}\left.\frac{d^2f}{dx^2}\right|<em>{x</em>0}+\ldots$
• Used to approximate potentials near equilibrium for small displacements

Duffing Oscillator (Non-linear Example)

• Potential: $V(x)=\frac{1}{2}\beta x^2+\frac{1}{4}\alpha x^4$ with \alpha>0,\;\beta<0 ⇒ double-well shape
• Total energy: $E=\frac{1}{2}m\dot{x}^2+V(x)$
• Graphing strategy: find extrema (set $\frac{dV}{dx}=0$), zeros $V(x)=0$, then sketch behaviour for $|x|\to\infty$ (dominant $\alpha x^4/4$)

Key Takeaways

• Potential energy curves fully determine conservative 1-D motion.
• Equilibrium classification uses first & second derivatives of $V$.
• Small oscillations near a stable point are always approximately harmonic.
• Energy diagrams help distinguish bounded vs. unbounded motion (pendulum, Duffing, etc.).