minimumenergy

Minimum energy, or the minimum-energy principle, is a broad idea in physics and applied mathematics describing how systems settle into states that minimize energy under given constraints. In static problems, the equilibrium configuration corresponds to a minimum of the potential energy. For a particle in a conservative force field, equilibrium occurs where the gradient of the potential energy is zero; stability requires the second derivative (the Hessian) to be positive definite. When constraints are present, the method of Lagrange multipliers finds configurations that minimize energy subject to those constraints.

In quantum mechanics, a related idea is the variational principle: the ground state energy is the lowest possible expectation value of the Hamiltonian. Any trial wavefunction provides an upper bound to the true ground-state energy, and the energy is minimized by optimizing the wavefunction. The Rayleigh–Ritz method extends this approach to approximate eigenvalues using a finite basis set.

For continuous media and field theories, one often minimizes an energy functional with specified boundary conditions. Examples include elastic deformation energy, electrostatic energy, and magnetic energy. The stationary points of these energy functionals satisfy the Euler–Lagrange equations, which yield the governing equations for the system.

Thermodynamics notes that energy minimization applies most cleanly at zero temperature, where systems tend toward minimum energy states, whereas at finite temperature entropy effects compete with energy minimization. In practice, energy minimization problems are solved numerically via gradient-based methods, conjugate gradient techniques, or more advanced variational and finite-element approaches. The landscape can include multiple local minima, so global optimization strategies or physical reasoning are often required to identify the true minimum-energy configuration.