Conservative Force and Potential Energy

The relation

For any conservative force, there exists a potential energy function $U(x)$ such that:

$F = -\frac{dU}{dx}$

In three dimensions:

$\vec{F} = -\nabla U = -\left(\frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k}\right)$

Force is the negative gradient of potential energy.

Derivation

By definition, the work done by a conservative force equals the decrease in potential energy:

$dW = F \, dx = -dU$

$F = -\frac{dU}{dx}$

What this means physically

The force points in the direction of decreasing potential energy. A body naturally accelerates toward lower PE — like a ball rolling downhill.

• If $U$ increases with $x$: $\frac{dU}{dx} > 0$, so $F < 0$ (force in $-x$ direction, opposing motion)
• If $U$ decreases with $x$: $\frac{dU}{dx} < 0$, so $F > 0$ (force in $+x$ direction, aiding motion)
• If $U$ is constant: $F = 0$ (equilibrium)

Verification with known forces

Gravity: $U = mgy$ (upward $y$ positive)

$F_y = -\frac{d(mgy)}{dy} = -mg$

Force is $-mg$ (downward). ✓

Spring: $U = \frac{1}{2}kx^2$

$F = -\frac{d}{dx}\left(\frac{1}{2}kx^2\right) = -kx$

Hooke's Law recovered. ✓

Equilibrium points

At equilibrium, $F = 0$:

$-\frac{dU}{dx} = 0 \implies \frac{dU}{dx} = 0$

Equilibrium occurs at extrema of the potential energy curve.

Stable equilibrium: $\frac{d^2U}{dx^2} > 0$ (minimum of $U$) — body returns to equilibrium when displaced

Unstable equilibrium: $\frac{d^2U}{dx^2} < 0$ (maximum of $U$) — body moves further away when displaced

Neutral equilibrium: $\frac{d^2U}{dx^2} = 0$ (flat region) — body stays wherever placed

What makes a force conservative

A force is conservative if and only if:

1. Work done is path-independent (depends only on start and end points)
2. Work done over any closed path is zero
3. A potential energy function $U$ exists such that $\vec{F} = -\nabla U$

All three conditions are equivalent. Examples: gravity, spring force, electrostatic force.

Non-conservative forces (friction, air resistance) do not satisfy these conditions — no potential energy can be defined for them.