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).

Some examples

Structural mechanics

The total potential energy, , is the sum of the elastic strain energy, U, stored in the deformed body and the potential energy, V, associated to the applied forces:[1]






This energy is at a stationary position when an infinitesimal variation from such position involves no change in energy:[1]






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:







= vector of displacements
= vector of distributed forces acting on the part of the surface
= vector of body forces

In the special case of elastic bodies, the right-hand-side of (3) can be taken to be the change, , 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:[2]

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:

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.


  1. ^ a b Reddy, J. N. (2006). Theory and Analysis of Elastic Plates and Shells (2nd illustrated revised ed.). CRC Press. p. 59. ISBN 978-0-8493-8415-8. Extract of page 59
  2. ^ Reddy, J. N. (2007). An Introduction to Continuum Mechanics. Cambridge University Press. p. 244. ISBN 978-1-139-46640-0. Extract of page 244