An Interaction of An Oscillator with An One-Dimensional Scalar Field. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. I: D'Alembert-Kirchhoff-like formulae

Abstract

This paper is an electronic application to my set of lectures, subject:`Formal methods in solving differential equations and constructing models of physical phenomena'. Addressed, mainly: postgraduates and related readers. Content: a very detailed discussion of the simple model of interaction based on the equation array: d^2 q(t)/dt^2 =-Ω^2(q(t)-Q(t))+f_0(t), d^2u(t,x)/t^2=c^{2}d^2 u(t,x)/dx ^2 -4γcδ(x-x_0)(Q(t)-q(t)) +f_1(t,x), Q(t) = u(t,x_0) . Besides, less detailed discussion of related models. Central mathematical points: d'Alembert-Kirchhoff-like formulae. Central physical points: phenomena of Radiation Reaction, Braking Radiation and Resonance.