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For unit mass, incoming speed $v>0$, impact parameter $b>0$, and potential energy $\mp \kappa /r$, put $q=\kappa /v^2>0$. At closest approach, conservation of energy and conservation of angular momentum give the displayed radii and tangential speeds $u_{\mp }=bv/p_{\mp }$. Consequently $p_{-}{p}_{+}=b^2$ and $u_{-}{u}_{+}=v^2$. These relations compare equal force magnitudes. The attractive radial case $b=0$ is a singular collision, so the speed-product statement does not extend to that endpoint.

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.

Baroclinic instability converts potential energy of a stratified sheared basic flow into growing disturbances. In the Eady model, coupled Boundary Rossby waves provide a useful mechanism: their induced velocities displace boundary buoyancy anomalies, and suitable wavelengths permit phase locking. A single neutral boundary wave does not by itself imply exponential instability.

A bounded trajectory in the damped Binet equation with nonzero angular momentum asymptotically becomes circular. Energy loss is nonnegative in magnitude and vanishes only for zero radial velocity. The centrifugal effective potential energy excludes collision; bounded radius forces indefinitely increasing forward angular distance, so the reciprocal-radius oscillation decays. Escaping trajectories are excluded by the boundedness hypothesis.

For a quadratic potential energy, shifting a path to its classical solution leaves an exactly quadratic fluctuation action with homogeneous Dirichlet boundary conditions. The first variation vanishes by the Euler-Lagrange equation. The Gaussian path integral prefactor is an inverse square root of a regulated functional determinant and does not depend on the endpoints. At conjugate times the ordinary formula must instead be read by distributional continuation.

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 $V$.

For distinct point charges $q_i$ at positions $r_i$, the electrostatic potential energy isWhen some charges are fixed, the terms depending on a movable charge form its potential energy in their electric potential.

For a translating potential energy profile with equal values at the two ends and linear drag force $-\zeta \dot{x}$, a finite passing trajectory satisfies $v\Delta x=\int {\dot{x}}^{2}dt\ge 0$. Equivalently, since $\int Fdy=0$, $\Delta x=\int u^2/[v(v-u)]dy$. Compact support of the force alone is insufficient. At high speed the displacement is $v^{-2}\int (F/\zeta {)}^{2}dy+O(v^{-3})$.

A constant-depth layer initially at rest with a step in its free surface adjusts by radiating inertia-gravity waves. Conservation of linearized shallow-water potential vorticity leaves the bounded balanced profile ${\eta }_s={\eta }_0\mathrm{sgn}(x)(1-e^{-|x|/R_D})$, where $R_D=\sqrt{gH}/|f|$ is the Rossby deformation radius. The released potential energy per unit transverse length is $3\rho g{\eta }_0^2{R}_D/2$; the final kinetic energy is $\rho g{\eta }_0^2{R}_D/2$, leaving $\rho g{\eta }_0^2{R}_D$ in radiated waves.

Mechanical energy is the sum of kinetic energy and potential energy. It is conserved when all forces are conservative and the potential has no explicit time dependence.

Two point masses $m_1,m_2$ separated by distance $r$ have potential energyThe corresponding attractive force is $-\nabla V$.