Local Variational Principle Research Articles

Variational principles for locally variational forms

We present the theory of higher order local variational principles in fibered manifolds, in which the fundamental global concept is a locally variational dynamical form. Any two Lepage forms, defining a local variational principle for this form, differ on intersection of their domains, by a variationally trivial form. In this sense, but in a different geometric setting, the local variational principles satisfy analogous properties as the variational functionals of the Chern–Simons type. The resulting theory of extremals and symmetries extends the first order theories of the Lagrange–Souriau form, presented by Grigore and Popp, and closed equivalents of the first order Euler–Lagrange forms of Haková and Krupková. Conceptually, our approach differs from Prieto, who uses the Poincaré–Cartan forms, which do not have higher order global analogues.

A local variational principle for the topological entropy

In this paper we introduce two notions of measure-theoretical entropy for covers and one notion of topological entropy for open covers. These notions extend the classical concept of measure-theoretical entropy of measurable partitions to measurable covers. Using these notions we prove a local variational principle for any given open cover.

Local variational principle

A generalization of the Gibbs-Bogoliubov-Feynman inequality for spinless particles is proven and then illustrated for the simple model of a symmetric double-well quartic potential. The method gives a pointwise lower bound for the finite-temperature density matrix and it can be systematically improved by the Trotter composition rule. It is also shown to produce ground state energies better than the ones given by the Rayleigh-Ritz principle as applied to the ground state eigenfunctions of the reference potentials. Based on this observation, it is argued that the local variational principle performs better than the equivalent methods based on the centroid path idea and on the Gibbs-Bogoliubov-Feynman variational principle, especially in the range of low temperatures.

A local variational principle and its application to an infinite strip containing a central transverse crack

A local variational principle and its application to an infinite strip containing a central transverse crack

Analogy between dynamics and statics related to variational mechanics

Using modern differential geometry we complete the classical analogy between dynamics and statics in mechanics. For every mechanical system satisfying Maxwell’s Principle a local variational principle is valid which in some cases can be extended to a global one. If a further regularity condition holds then we can get a Lagrangian function. There exist mechanical systems in Nature which satisfy Maxwell’s Principle but do not have a global variational formalism: such an example is the case of a polarized particle or a particle with spin.