Some physical models with Minkowski spacetime structure and Lorentz group symmetry

Abstract Projecting the linear equation X = AX , A ∈ so (n, 1) , yields a primitive model. It is a prototype of several physical models, namely perfect elastoplasticity (on phase), spacetime of special relativity, special relativistic mechanics and so on, all of which may endow a Minkowskian spacetime and the Lorentz group left acts on it. The mathematical structure of the perfect elastoplastic equations is then compared with those of the spacetime of special relativity and of the special relativistic equations of motion of a massive charged particle; their similarities in group properties and subtle differences in phase spaces are discussed. It is remarkable that the evolutions from elastic constitutive equations to elastoplastic constitutive equations and from the Newtonian equations to the special relativistic equations are very similar in several facets, notably (a) from a linear theory to a non-linear theory, (b) the state space being enlarged from the usual Euclidean space to Minkowski spacetime, (c) from an n-space to a cone of (n+1)-space, (d) from a non-bounded state space to a bounded state space, (e) from a non-causal relation of states to a causal relation of augmented states.