Definition: A force is conservative if the work done by it depends only on the initial and final positions, NOT on the path taken.

Equivalent conditions (any one implies the others):

1. Work done in a closed loop = 0
2. Work is path-independent
3. Force can be written as F = -dU/dx (negative gradient of PE)
4. curl(F) = 0

Examples:

• Conservative: gravity, spring force, electrostatic force
• Non-conservative: friction, air resistance, viscous force, tension in a real string

Test for conservativeness (2D): For F = $F_x$*i + $F_y$*j: conservative if $dF_x$/dy = $\frac{dF_y}{dx}$

Non-conservative forces:

• Work depends on the path (longer path = more work by friction)
• No PE can be defined
• Energy is dissipated (converted to heat, sound, deformation)
• Work by friction in a round trip = -2*$mu_k$mgd (NOT zero)