Symmetries and geometrically implied nonlinearities in mechanics and field theory

TL;DR

This paper explores the fundamental role of symmetry in nonlinear dynamical models, emphasizing non-perturbative nonlinearities and affine invariance, with applications in mechanics, field theory, and gravity.

Contribution

It develops models with affine symmetry that inherently exhibit Born-Infeld type nonlinearities, offering new insights into nonlinear elasticity, relativistic continua, and gravity theories.

Findings

Models with affine invariance naturally have Born-Infeld structure

Non-perturbative nonlinearities are fundamentally different from linear corrections

Applications suggested in elasticity, relativistic mechanics, and gravitation

Abstract

Discussed is relationship between nonlinearity and symmetry of dynamical models. The special stress is laid on essential, non-perturbative nonlinearity, when none linear background does exist. This is nonlinearity essentially different from ones given by nonlinear corrections imposed onto some linear background. In a sense our ideas follow and develop those underlying Born-Infeld electrodynamics and general relativity. We are particularly interested in affine symmetry of degrees of freedom and dynamical models. Discussed are mechanical geodetic models where the elastic dynamics of the body is not encoded in potential energy but rather in affinely-invariant kinetic energy, i.e., in affinely-invariant metric tensors on the configuration space. In a sense this resembles the idea of Maupertuis variational principle. We discuss also the dynamics of the field of linear frames, invariant under the action of linear group of internal symmetries. It turns out that such models have automatically the generalized Born-Infeld structure. This is some new justification of Born-Infeld ideas. The suggested models may be applied in nonlinear elasticity and in mechanics of relativistic continua with microstructure. They provide also some alternative models of gravitation theory. There exists also some interesting relationship with the theory of nonlinear integrable lattices.