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## Summary

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, ${\boldsymbol {\Pi }}$ , is the sum of the elastic strain energy, U, stored in the deformed body and the potential energy, V, associated to the applied forces:

${\boldsymbol {\Pi }}=\mathbf {U} +\mathbf {V}$ (1)

This energy is at a stationary position when an infinitesimal variation from such position involves no change in energy:

$\delta {\boldsymbol {\Pi }}=\delta (\mathbf {U} +\mathbf {V} )=0$ (2)

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:

$\int _{S_{t}}\delta \ \mathbf {u} ^{T}\mathbf {T} dS+\int _{V}\delta \ \mathbf {u} ^{T}\mathbf {f} dV=\int _{V}\delta {\boldsymbol {\epsilon }}^{T}{\boldsymbol {\sigma }}dV$ (3)

where

$\mathbf {u}$ = vector of displacements
$\mathbf {T}$ = vector of distributed forces acting on the part $S_{t}$ of the surface
$\mathbf {f}$ = vector of body forces

In the special case of elastic bodies, the right-hand-side of (3) can be taken to be the change, $\delta \mathbf {U}$ , 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:

$\mathbf {V} =-\int _{S_{t}}\mathbf {u} ^{T}\mathbf {T} dS-\int _{V}\mathbf {u} ^{T}\mathbf {f} dV$ where the minus sign implies a loss of potential energy as the force is displaced in its direction. With these two subsidiary conditions, (3) becomes:

$-\delta \ \mathbf {V} =\delta \ \mathbf {U}$ This leads to (2) as desired. The variational form of (2) is often used as the basis for developing the finite element method in structural mechanics.