Energy is the conserved quantity associated to time-translation invariance by Noether's theorem.

It is usually split into (at least) two parts: Kinetic and potential energy. Kinetic energy is a measure of the "energy of motion" of particles. It can be classically defined as $\frac{1}{2}mv^2$. Potential energy is a relative concept, it is defined as the amount of work done against a conservative force to move the system from a reference state to the current state.

Conservation of energy can be derived classically from Newton's second law, $F=\frac{\mathrm{d}p}{\mathrm{d}t}$. Writing $p$ as $mv$, and writing $\frac{\mathrm{d}v}{\mathrm{d}t}$ as $v\frac{\mathrm{d}v}{\mathrm{d}x}$, and integrating, we get that $\frac{1}{2}mv^2+\int F\mathrm{d}x$ is a conserved quantity. The former term is kinetic energy, while the latter term is the potential energy, i.e. the work done by conservative forces.

By Einstein's mass-energy equivalence $E=mc^2$, all matter can be seen as a form of energy, too.

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