Potential energy

A conservative force is the negative gradient of a potential energy: $F=-\nabla V$.

The portion of a fluid system's potential energy that can be released by admissible rearrangement relative to a stable reference state. In a uniformly stratified incompressible fluid, the quadratic density is $A={\rho }_0{N}^2{\xi }_z^2/2={\rho }_0{b}^2/(2N^2)$, where ${\xi }_z$ is vertical fluid displacement and $b$ the buoyancy perturbation. It is positive for stable stratification and measures the restoring energy of internal gravity waves.

A potential function $V$ assigns a scalar to each point of a configuration or spatial domain. In mechanics it represents potential energy, and its negative gradient gives the corresponding conservative force.

A central potential depends only on distance from a fixed centre, $V(\mathbf{x})=V(|\mathbf{x}|)$. Its force is radial, and rotational symmetry implies conservation of angular momentum.

A force is conservative when its work depends only on the endpoints. On a simply connected domain this is equivalent to $F=-\nabla V$ for a potential energy $V$.

A nonconservative force has path-dependent work. On a simply connected domain, a continuously differentiable force is nonconservative exactly when its curl does not vanish identically.

The potential $V(r)=-k/r$ generates the attractive inverse-square central force $F=-kr/r^3$.

The even potentialhas a cusp at the origin and approaches the finite value $V_{\infty }$. Motions below that value oscillate between symmetric turning points; motions above it escape.

Escape velocity is the least initial speed whose total energy permits motion to spatial infinity.

An equipotential surface is a level set of a scalar potential. Its gradient and the corresponding conservative force are normal to every smooth level surface.