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:
  • Fx = 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 V0+\frac{1}{2}k(x-x0)^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(x0+a)=f(x0)+a\left.\frac{df}{dx}\right|{x0}+\frac{a^2}{2!}\left.\frac{d^2f}{dx^2}\right|{x0}+\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.).