Minimum total potential energy principle

The minimum total potential energy principle is a fundamental concept used in physics, chemistry, biology, 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).

The total potential energy, ${\displaystyle {\boldsymbol {\Pi }}}$, 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 (3) can be taken to be the change, ${\displaystyle \delta \mathbf {U} }$, 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 (3) can be seen as the change in the potential energy function V of the forces. The function V is defined as:

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