Partial differential equations of first order and their applications to physics (Q2784337)

Linear, quasilinear and nonlinear partial differential equations of the first order play an important role in mathematics, mathematical physics and engineering science. These equations arise in various areas of analytical dynamics, geometrical optics, quantum physics, and fluid dynamics. One of the famous examples of such equations is the Hamilton-Jacobi equation that is used to describe dynamical systems. Another famous example of the first-order nonlinear equations is the eikonal equation which arises in nonlinear optics and also describes the propagation of wave fronts and discontinuities for acoustic wave equations.NEWLINENEWLINE This short book is essentially based on the collection of notes of the course on partial differential equations taught by Gustavo López at different Mexican and other universities. This book gives an easy manipulative treatment of the subject with special emphasis to physical applications rather than a rigorous and systematic mathematical theory.NEWLINENEWLINE It has six short chapters. Chapter I deals with geometrical concepts which are fundamental for an understanding of geometrical significance of the partial differential equations in \(\mathbb{R}^3\). The method of solution of the characteristic equations is described with examples and examples of applications flom analytical dynamics. Linear and quasilinear partial differential equations are discussed with examples in Chapter II. Chapter III is concerned with applications of linear partial differential equatiolls of the first-order to physics. Included are several examples dealing with angular momentum in quantum mechanics, heat propagation between two superconducting cables, renorinalization group equations, and particle multiplicity in high energy physics. Hamiltonian perturbation approach to accelerator physics, and constant of motion for a relativistic particle under perturbation force.NEWLINENEWLINE Chapter IV is devoted to nonlinear partial differential equations of the first order with examples. In Chapter V, author presents physical applications including the motion of a classical particle, the Lagrangian obtained directly from the Hamiltonian, relativistic particle moving in a Coulomb field, motion of a test particle in a Schwarzschild's space, Einstein's equations in vacuum and gravitational waves. The final chapter deals with characteristic surfaces of the second-order linear partial differential equations. This chapter is very short.NEWLINENEWLINE Several classical standard texts and treatises on the subject are listed at the end of each chapter. No exercises are provided which makes the book unsuitable for the adoption as a textbook for a course in partial differential equations. However, the book does not contain any wrong information or serious errors.NEWLINENEWLINE There are many modern books on linear and nonlinear partial equations such as, \textit{J. Smoller}, Shock waves and reaction-diffusion equations, Springer, New York (1983; Zbl 0508.35002), \textit{P. D. Lax}, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM (1973; Zbl 0268.35062), \textit{G. B. Whitlam}, Linear and Nonlinear Waves, John Wiley, New York (1974; Zbl 0373.76001)] \textit{L. Debnath}, Nonlinear partial differential equations for scientists and engineers, Birkhäuser, Boston (1997~; Zbl 0892.35001).NEWLINENEWLINE The author did not list any such modern books in his references. In the opinion of the reviewer, this short book is not really the top of its kind. However, it seems to be moderately successful as a reference book, especially for the audience of physical science.