Principle of minimum energy

Under fixed constraints, a stable system occupies an accessible state of lowest energy. The principle predicts equilibrium when energy is the relevant quantity minimized.

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Potential energy: Stored energy associated with a system’s configuration or position in a force field. For conservative systems, equilibrium configurations occur at stationary points of potential energy.
Conservative force: A force whose work between two points is independent of the path taken. Conservative forces permit a potential-energy description of equilibrium.
Principle of maximum entropy: The principle that equilibrium maximizes entropy subject to specified constraints. For isolated systems, entropy maximization characterizes equilibrium where energy minimization alone does not.
Elastic energy: Energy stored when an elastic material is deformed from its relaxed configuration. Deformed structures settle toward configurations that reduce elastic energy while meeting constraints.
Energy landscape: A mapping of a system’s energy across its possible configurations. Landscape shape determines which minima exist and how difficult transitions between them are.
Constrained optimization: The mathematical search for extrema of a function subject to equality or inequality restrictions. Fixed constraints restrict which states can compete for the minimum.
Configuration space: The space of all possible positions or states of a system, represented by coordinates. The energy minimum is sought among the configurations allowed by the constraints.
Helmholtz free energy: A thermodynamic potential equal to internal energy minus temperature times entropy. At fixed temperature and volume, equilibrium minimizes this free energy rather than internal energy.
Buckling: A sudden change in a structure’s shape under compressive loading. Buckling occurs when a different deformation pattern becomes energetically favorable.
Ergodic hypothesis: The hypothesis that a system’s long-time trajectory explores its accessible state space in the relevant statistical sense. Failure to explore accessible states can prevent observed systems from reaching the predicted minimum.