Law of Conservation of Energy

The law states that, if a body or system of bodies is in motion under a conservative system of forces, the sum of its kinetic energy and potential energy is constant.

Explanation

From the principle of work and energy, Work done = change in the kinetic energy

( i.e) W_1-2 = E_k2 - E_k1 ...(1)

If a body moves under the action of a conservative force, work done is stored as potential energy.

W_1-2 = - (E_P2 - E_P1) ...(2)

Work done is equal to negative change of potential energy. Combining the equation (1) and (2),

Ek₂ - E_k1 = - (E_P2 - E_P1) (or) E_p1 + E_k1 = E_P2 + E_k2 ...(3)

which means that the sum of the potential energy and kinetic energy of a system of particles remains constant during the motion under the action of the conservative forces.

The conservation of energy is a common feature in many physical theories. From a mathematical point of view it is understood as a consequence of Noether's theorem, which states every symmetry of a physical theory has an associated conserved quantity; if the theory's symmetry is time invariance then the conserved quantity is called "energy". The energy conservation law is a consequence of the shift symmetry of time; energy conservation is implied by the empirical fact that the laws of physics do not change with time itself. Philosophically this can be stated as "nothing depends on time per se". In other words, if the theory is invariant under the continuous symmetry of time translation then its energy (which is canonical conjugate quantity to time) is conserved. Conversely, theories which are not invariant under shifts in time (for example, systems with time dependent potential energy) do not exhibit conservation of energy unless we consider them to exchange energy with another, external system so that the theory of the enlarged system becomes time invariant again. Since any time-varying theory can be embedded within a time-invariant meta-theory energy conservation can always be recovered by a suitable re-definition of what energy is. Thus conservation of energy for finite systems is valid in such modern physical theories as special relativity and quantum theory (including QED) in the flat space-time.