Energy

Energy is a conserved scalar quantity associated with time-translation symmetry.

For a Lagrangian $L(q,\dot{q},t)$, its energy function isIt is conserved along Euler-Lagrange trajectories when $L$ has no explicit time dependence.

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.

The work done by a force along a path is $W=\int F\cdot dr$.

The net work on a particle equals its change in kinetic energy:

Power is the rate of energy transfer, $P=dW/dt$.

The nonrelativistic kinetic energy of a particle is $T=m{v}^2/2$.

Writing particle positions relative to the center of mass as ${\rho }_i=r_i-R$ makes ${\sum }_i{m}_i{\rho }_i=0$, so the cross term in the squared velocities vanishes. The total kinetic energy splits into center-of-mass translation and relative motion as displayed. For a rigid body, the latter is $\frac{1}{2}I{\omega }^2$, where $I$ is the moment of inertia about the instantaneous rotation axis through the center of mass.

For a twice differentiable position away from the origin, write $\mathbf{r}=r\mathbf{u}$ with radial unit vector $\mathbf{u}$. Its unit-length constraint gives $\mathbf{u}\cdot \dot{\mathbf{u}}=0$ and $\mathbf{u}\cdot \ddot{\mathbf{u}}=-|\dot{\mathbf{u}}{|}^2$, separating radial and directional contributions to speed and kinetic energy. The identity is kinematic and does not require a particular force law.

The kinetic energy density of a continuum with mass density $\rho$ and velocity $\mathbf{v}$ is $\rho |\mathbf{v}{|}^2/2$. For a displacement field, the material velocity is its time derivative.

A rigid body rotating about a fixed axis with moment of inertia $I$ and angular speed $\omega$ has kinetic energy $K=I{\omega }^2/2$.

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.

A turning point of one-dimensional motion is a position where the kinetic energy and instantaneous velocity vanish before the direction of motion reverses.