Construction of Lagrangians in continuum theories

Abstract

For physical systems the dynamics of which is formulated within the framework of Lagrange formalism, the dynamics is completely defined by only one function, namely the Lagrangian. For instance, the whole conservative Newtonian mechanics has been successfully embedded into this methodical concept. In continuum theories, however, the situation is different: no generally valid construction rule for the Lagrangian has been established in the past. In this paper general properties of Lagrangians in non–relativistic field theories are derived by considering universal symmetries, namely space– and time–translations, rigid rotations and Galilei boosts. These investigations discover the dual structure, i.e. the coexistence of two complementary representations of the Lagrangian. From the dual structure, relevant restrictions for the analytical form of the Lagrangian are derived which eventually result in a general scheme for Lagrangians. For two examples, namely Schrödinger's theory and the flow of an ideal fluid, the compatibility of the Lagrangian with the general scheme is demonstrated. The dual structure also has consequences for the balances which result from the respective symmetries by Noether's theorem: universally valid constitutive relations between the densities and the flux densities of energy, momentum, mass and centre of mass are derived. By an inverse treatment of these constitutive relations a Lagrangian for a given physical system can be constructed. This procedure is demonstrated for an elastically deforming body.