Differential algebra and mathematical physics

TL;DR

This paper applies modern differential algebra and geometry to analyze equations in mathematical physics, revealing new insights into their algebraic structure and connections to physical theories like general relativity and electromagnetism.

Contribution

It introduces a modern differential algebra approach to study nonlinear PDEs in physics, connecting differential duality with jet coordinates and homological algebra.

Findings

Linearized Einstein equations parametrize stress equations but are not themselves parametrizable.

Gravitation and electromagnetism depend only on second order jets of conformal Killing equations.

New methods are applied to contact transformations and Hamilton-Jacobi equations.

Abstract

Many equations of mathematical physics are described by differential polynomials, that is by polynomials in the derivatives of a certain number of functions. However, up to the knowledge of the author, differential algebra in a modern setting has never been applied to study the specific algebraic feature of such equations. The purpose of this short but difficult paper is to revisit a few domains like general relativity, conformal geometry and contact geometry in the light of a modern approach to nonlinear systems of partial differential equations, using new methods from Differential Geometry (D.C. Spencer, 1970), Differential Algebra (J.F. Ritt, 1950 and E. Kolchin, 1973) and Algebraic Analysis (M. Kashiwara, 1970). Identifying the differential indeterminates of Ritt and Kolchin with the jet coordinates of Spencer, the idea is to study Differential Duality by using only linear differential operators with coefficients in a differential field. In particular, the linearized second order Einstein equations are parametrizing the first order Cauchy stress equations but cannot themselves be parametrized. In the framework of Homological Algebra, this result is not coherent with the vanishing of certain first and second extension modules. As a byproduct, we shall prove that gravitation and electromagnetism must only depend on the second order jets (elations) of the system of conformal Killing equations. Finally, we shall use these new methods in order to study contact transformations in arbitrary odd dimension and apply these results to the study of Hamilton-Jacobi equations in mechanics.