William Thomson's Work On Mathematical Physics

William Thomson is a central figure in the Cambridge School of Mathematical Physics.His research was extremely extensive,and he has made remarkable achievements in geophysics,thermodynamics,electromagnetics,fluid mechanics,mathematics,engineering applications,etc.In particular,some general laws and mathematical methods for solving physical problems were innovatively proposed and developed by Thomson during the 1840s-1860 s period.For example,he innovatively proposed the identity between the theory of static electricity and the theory of heat conduction;established the method of electrical images by geometric rule;constructed a spherical harmonic function,etc.Thomson was a British physicist and engineer who has made important contributions and influence to the development of mathematical physics in the 19 th century.This article focuses on Thomson's work,and sorts out the work of important people in important links related to it,and traces the source of various ideas and methods.This thesis which is based on the original literature and related literature,and guided by "why mathematics",explores the work of Thomson in this historical process.Different from previous studies,the thesis plane to carefully analyzes the historical background,causes,development process and impact of Thomson's work.And systematically examine the internal relationship of Thomson's innovative work in these three aspects.The main research results obtained are as follows:1.Thorough analysis of Thomson's thought origins and background about analogy of electrical mathematical theory to heat conduction theory.Exploration of Thomson's simple method in order to determine the temperature at any point inside and outside the infinite solid,he introduced the potential function in Laplace's gravitational theory into the heat conduction theory to establish a temperature potential function,and he unifies the different physical phenomena with mathematical model,potential function.Briefly discusses Laplace and Poisson's work on gravitational theory and electrical theory.Specifically analyzed the reason of Laplace and Poisson failed to complete the unity,and Poisson's theory of electrical mathematics failed to solve the problem of determining equilibrium charge distribution condition.And explain the process of establishing the heat conduction equation by Fourier in determining the temperature at a certain point in an infinite solid.2.Through careful study of Thomson's use of mathematical models to unify the ideological methods of different physical phenomena,this thesis found that Thomson proposed the "replacement face theorem" under the guidance of Fourier's thermodynamic thoughts and mathematical techniques.And studied the similar physics between Thomson's establishment of thermal theory and electrical theory.Through this theorem Thomson puts the heat theory proposed by Fourier into the theory of electricity through the idea of linear propagation of the medium,and overcomingthe over-the-range effect of electrostatic force at that time,which is consistent with the force line thought proposed by Faraday.3.Excavate the source,background and purpose of Thomson's idea method of establishing an electric image method in 1847.It is revealed that Thomson established this law in order to determine the amount of induced charge on the surface of a certain shape conductor and the electrostatic field generated by it,revealing the "replacement surface" theorem,Reciprocal relationship of charge density in the paper,which is Green's "Application of Mathematical Analysis and Electromagnetic Theory",published in 1828,and his father Jameson Thomson's book,the Annotations and Illustrations of the First Six Volumes,Eleventh and Twelfth Volumes of Euclidean Geometry Proposition G,these results are the basis and methodological basis for Thomson's idea on establishing an electrical image in 1847.Analysed,on the basis of predecessors,why Thomson could successful breakthrough with success on previous problems,what difficulties were overcomed? Make it clear that the process of constructing progressively electrical image law,by interpreting Thomson's report in 1847 and the main article published in 1848.4.The purpose and innovative method of Thomson constructing spherical harmonic function are studied,and Make it clear that which concrete issues led Thomson to consider the spherical harmonic function,and reveals the effect of the properties of the electrophotographic method on the construction of the spherical harmonic function.Combing the early work of Legendre and Laplace on the Legendre coefficient and Laplace coefficient,which is the prime form of spherical harmonic function,and explained Thomson how to learn Laplace's work,and the inheritance relationship between Thomson's work and his predecessors' work.And investigated the process of constructing spherical harmonic function by Thomson,which types of physical problems can be solved in the way of Thomson's spherical harmonic function.