دانلود کتاب فیزیک نظری فشرده ۱ – مکانیک کلاسیک

This book provides a textbook on classical mechanics and is in particular suited for bachelor students in their irst year of studies in theoretical physics. The mathematical requirements include a knowledge of differentiation and integration and mathematical proofs are kept as simple as possible, however, still kept stringent. Elements of linear algebra are explained in detail in the text if needed in the context of coordinate transformations, rotations, Galilei or Lorentz transformations. After deining the physical quantities of interest in the kinematics of mass points in inertial systems, the transformations between different inertial systems are derived. After these preparatory chapters, the Newtonian dynamics is formulated and examples for the solution of the equations of motion are presented. Furthermore, the tight connection between Galilei invariance and the conservation laws of momentum and angular momentum are pointed out. In case of conservative forces a potential energy can be formulated that—together with the kinetic energy of mass points—gives the energy of the system. The conservation of the total energy for a closed system follows in a straight forward manner. Applications of Newtonian mechanics for 1/r2-forces lead to Kepler’s laws for the motion of planets and a gravitational ield for a static mass distribution can be deined. Another important application is the harmonic oscillator being damped or driven by an external periodic force. Since Maxwell’s equations for electrodynamics are not Galilei invariant a new transformation law is derived (Lorentz transformation), which keeps the velocity of light c invariant in all inertial systems moving with relative velocity υ < c. Some consequences are pointed out such as Lorentz contraction, time dilation, simultaneity, or causality of events. Mathematical aspects of Lorentz transformations are pointed out and the relativistic dynamics for mass points are derived accordingly. It is, furthermore, shown that the relativistic equations of motion merge with Newtonian dynamics for small velocities υ ≪ c. The formal structure of mechanics is addressed in the second part of this book that aims at an algebraic formulation of the dynamics, which is independent on the particular choice of coordinates of an observer. After introducing generalized coordinates, that account for constraints on the system of particles and avoid the introduction of coercive forces, we introduce the Lagrange function and a variational principle to derive the Lagrange equations of motion. A Legendre transformation of the Lagrange function to the Hamilton function will lead to a description of the dynamics in phase-space variables, i.e. generalized coordinates and momenta. The Lagrange equations of motion turn to Hamilton’s equations of motion, which can be expressed by Poisson brackets for the time evolution of an observable. The latter are shown to be invariant with respect to point transformations and extended canonical transformations of the phase-space variables such that a formal formulation of the classical mechanics is achieved that paves the way for the formulation of quantum mechanics, continuum mechanics, and statistical mechanics. In the appendices the relativistic Lagrange and Hamilton functions for characteristic problems are given as well as numerical algorithms for differentiation and integration. Furthermore, algorithms of different order for the solution of differential equations are presented.