Minimum total potential energy principle

The minimum total potential energy principle is a fundamental concept used in physics and engineering. It dictates that at low temperatures a structure or body shall deform or displace to a position that (locally) minimizes the total potential energy, with the lost potential energy being converted into kinetic energy (specifically heat).
Structural mechanics.
The total potential energy, formula_1, is the sum of the elastic strain energy, U, stored in the deformed body and the potential energy, V, associated to the applied forces:
This energy is at a stationary position when an infinitesimal variation from such position involves no change in energy:
The principle of minimum total potential energy may be derived as a special case of the virtual work principle for elastic systems subject to conservative forces.
The equality between external and internal virtual work (due to virtual displacements) is:
where
In the special case of elastic bodies, the right-hand-side of () can be taken to be the change, formula_6, of elastic strain energy U due to infinitesimal variations of real displacements.
In addition, when the external forces are conservative forces, the left-hand-side of () can be seen as the change in the potential energy function V of the forces. The function V is defined as:
where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, () becomes:
This leads to () as desired. The variational form of () is often used as the basis for developing the finite element method in structural mechanics.