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## Minimum total potential energy principleThe ## Additional recommended knowledgeThe 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. ## Some examples- A free proton and free electron will tend to combine to form the lowest energy state (the ground state) of a hydrogen atom, the most stable configuration. This is because that state's energy is 13.6 electron volts (eV) lower than when the two particles separated by an infinite distance. The dissipation in this system takes the form of spontaneous emission of electromagnetic radiation, which increases the entropy of the surroundings.
- A rolling ball will end up stationary at the bottom of a hill, the point of minimum potential energy. The reason is that as it rolls downward under the influence of gravity, friction produced by its motion adds to the heat of the surroundings with an attendant increase in entropy.
- A protein folds into the state of lowest energy. In this case, the dissipation takes the form of vibration of atoms within or adjacent to the protein.
## Structural MechanicsThe total potential energy, , is the sum of the elastic strain energy, 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 - = vector of displacements
- = vector of distributed forces acting on the part
*S*_{t}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 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. Categories: Thermodynamics | Solid mechanics |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Minimum_total_potential_energy_principle". A list of authors is available in Wikipedia. |