History of variational principles in physics involves integrals over an interval of time and (for fields) extended region of space. Moreover, in the usual formulation of classical action principles, the initial and final states of the system are fixed, e.g., In particular, the fixing of the "final" state appears to give the action principle a teleological character which has been controversial historically. This apparent teleology is eliminated in the quantum mechanical version of the action principle. History of variational principles in physics A variational principle in physics is an alternative method for determining the state or dynamics of a physical system, by identifying it as an
Yukawa interaction explained by the fact that the Yukawa particle has spin zero and even spin always results in an attractive potential. The negative sign of the exponential gives the interaction an effectively finite range, so that particles at great distances will hardly interact any longer (interaction forces drop off exponentially with increasing separation). Now suppose that the potential formula_7 has a minimum not at formula_18 but at some non-zero value formula_19. This can happen, for example, with a potential form such as formula_20 with formula_9 set to an imaginary value. In this case, the Lagrangian exhibits spontaneous symmetry breaking. This is