Energy is the capacity to do work.
The mechanical energies are:
The energy is just a number but can be used to describe transformations in physical behaviors of systems. This tactic works very well, so well that some people believe that energy is something that does exist in the real world. It does not exist; there is no instrument that measures energy. Some instruments measure some other physics quantities and then calculate the energy. The concept of energy is just a mathematical method developed to facilitate the calculations.
These numbers (energies) are conserved quantities, that is, they do not increase or diminish for an isolated system.
It is important to know that:
The mechanical energy of an object subject to the action of conservative forces is constant and does not change. This implies that $$E_m(A) = E_m(B),$$ where \(E_m(A)\) and \(E_m(B)\) are the mechanical energy between two states \(A\) and \(B\) no matter what path the system took to go from \(A\) to \(B\).
Energy conservation can also be applied to mechanical systems that are not conservative, by including work \(w_{ext}\) performed by non - conservative forces. The change in system energy is equal to the work done on the system: $$\Delta E_k + \Delta E_p = w_{ext}$$ or $$ E_{k_i} + E_{p_i} = E_{k_f} + E_{p_f} + w_{ext}.$$
The sum of the works done by all the forces acting on a particle, between any two points, is equal to the change in kinetic energy of the particle between these points. Mathematically, it is: $$ \sum_i^n w_i = \Delta E_k $$