Energy is a conserved quantity; it comes from time-invariance in Noether's theorem.

It consists of 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 easily derived classically from Newton's second law, $F=\frac{dp}{dt}$. Writing $p$ as $mv$, and writing $\frac{dv}{dt}$ as $v\frac{dv}{dx}$, and integrating, we get that $\frac{1}{2}mv^2+\int Fdx$ is a conserved quantity. The former term is kinetic energy, while the latter term is potential energy, with the same expression as work.

Einstein showed, with his famous formula $E=mc^2$ that matter itself is a form of energy.

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