2013/11/14 by Luca Vitagliano · 1 citation
Mathematics · Physics and Astronomy · #math.DG #math-ph #math.AP #math.MP #msc:35A10 #msc:35F61 #msc:35A30 #msc:35A18 #msc:35F21 #msc:58A20 #msc:53D10 #msc:53D05 #msc:70H20
paper · pdf · doi:10.1142/s0219887814600391
published as Int. J. Geom. Meth. Mod. Phys. 11 (2014) 1460039 · 26 pages, short elementary review submitted for publication on the Proceedings of XXII IFWGP
arxiv created 2013/11/14 · arxiv updated 2014/11/04
Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics and in Physics. I will review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e., those aspects which are invariant under general changes of coordinates. After a basically analytic introduction, I will pass to a modern, geometric point of view, presenting characteristics within the jet space approach to PDEs. In particular, I will discuss the relationship between characteristics and singularities of solutions and observe that: "wave-fronts are characteristic surfaces and propagate along bicharacteristics". This remark may be understood as a mathematical formulation of the wave/particle duality in optics and/or quantum mechanics. The content of the paper reflects the three hour minicourse that I gave at the XXII International Fall Workshop on Geometry and Physics, September 2-5, 2013, Evora, Portugal.