The tendency to minimum total potential energy is due to the second law of thermodynamics, which states that the entropy of a system will maximize at equilibrium. Given two possibilities - a low heat content and a high potential energy, or a high heat content and low potential energy, the latter will be the state with the highest entropy, and will therefore be the state towards which the system moves.
The principle of minimum total potential energy should not be confused with the related principle of minimum energy which states that for a system that changes without heat transfer, the total energy will be minimized.
Note that in most complex systems there is one global minimum and many local minima (smaller dips) in the potential energy. These are called metastable states. A system may reside in a local minimum for a long time — even an effectively infinite period of time.
The total potential energy, , is the sum of the elastic strain energy, U, stored in the deformed body and the potential energy, V, of the applied forces:
The equality between external and internal virtual work (due to virtual displacements) is:
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:
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:
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