Energy
Consider a physical system, such as a moving ball or a stretched spring. The energy of the system is a number that measures the system’s capacity to do work, that is, to push something through a distance. Energy is measured in joules (J).
Consider a particle of mass $m$ at position $x$, moving with velocity $v$ in a conservative force field with potential energy $V(x)$. The (mechanical) energy of the particle is the sum of its kinetic energy and its potential energy:
\[E = \frac{1}{2} m \norm{v}^2 + V(x).\]Properties
- Energy is a scalar, which has a size but no direction.
- The total energy of an isolated system is constant, but energy can change form. For example, in an inelastic collision, some kinetic energy is lost to other forms of energy, such as heat or sound.
- The kinetic energy and the momentum $p = m v$ of a particle are related by $K = \norm{p}^2 / (2 m)$.
- Energy is the conserved quantity that comes from symmetry under shifts in time (Noether’s theorem).
Notes
One joule is $1 \ \mathrm{kg} \cdot \mathrm{m}^2 / \mathrm{s}^2$, or one newton meter.
For a conservative system, the Hamiltonian is the total energy.
In signal processing, the “energy” of a signal $y$ is $\int \abs{y(t)}^2 \, \mathrm{d}t$, which is a different, though related, quantity.