Book Review

Authors: Andrzej Icha

Publish Date: 2015/07/25

Volume: 172, Issue: 10, Pages: 2951-2953

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Abstract

PAM Dirac one of the greatest physicists of all times said “A physical law must possess mathematical beauty” In this meaning the beauty of classical mechanics lies in fact that it can all be derived from the two postulates the relativity and the Hamilton’s principlesThis book is intended for an undergraduate course in classical theoretical mechanics taken by students majoring in physics applied mathematics or engineering The reader is not assumed to have any previous knowledge beyond that contained in standard courses in general physics mathematical analysis differential/integral calculus and differential equations basic differential geometry and linear algebra There is a set of exercises at the end of each chapter Harder sections and exercises are marked with a star and several chapters include the problems with hints which are rather longer than the standard exercises and which solutions involve studies in the library or on the InternetChapter 1 addresses basic issues of Calculus of Variations It has arisen from attempts to solve extreme problems that occur naturally in theoretical physics and mathematics Three classical minimisation problems taken from geometry and physics respectively are described and discussed These include the isoperimetric problem the brachistochrone problem and the Fermat’s principle of least time Later on the Frenet–Serret equations are derived for a light ray propagating in the nonuniform refractive mediumChapter 2 reviews the mathematical and physical aspects of Lagrange Mechanics including the four principles by which particle dynamics may be described It introduces variational principles through Maupertuis’ principle Jacobi’s principle Hamilton’s principle and d’Alembert’s constraint principle or principle of virtual work The fourstep Lagrangian method is elucidated as constructive approach to obtain the equations of motion in configuration space Then the several spectacular applications to the various equations of pendula to the bead on a rotation hoop problem and to the Atwood machine equations are considered together with the concise discussion of the Noether theorem for discrete Lagrangian systems The chapter ends with a presentation of the relationship between symmetries of the Lagrangian with conservation laws The role of centerofmass frame in mechanics is also clarifiedChapter 3 presents briefly the elements of Hamiltonian Mechanics and shows how the motion of a mechanical system with n degrees of freedom can be described in the 2ndimensional phase space The Hamilton’s canonical equations of motion are concisely derived from the Lagrangian equations of motion and the Hamilton–Jacobi equation for particle dynamics is sketched In addition the Hamiltonian optics and wave–particle duality and the motion of a charged particle in an electromagnetic field are discussed It illustrates the direct connection between Lagrangian and Hamiltonian methods and inspires further development of mathematical concepts Next a closedform formulation for onedimensional Hamiltonian dynamics is developed It first presents solutions for simple harmonic oscillator the motion of a particle in the Morse potential and the mathematical pendulum problem In conclusion analysis of the constrained motion of a point particle on a cone in the presence of gravity field is performedThe Motion in a CentralForce Field is one of the oldest and widely studied problems in theoretical mechanics Chapter 4 is focused on timeindependent centralforce planar problems in which the principles of conservation of energy and angular momentum are only needed to describe the geometry of the possible trajectories It first presents a general formulation of the planar dynamics and then extends the analysis to the case of homogeneous central potentials The discussion encompasses the determination of the full solutions of the Kepler problem the isotropic simple harmonic oscillator problem and the problem of internal reflection inside a potential well The importance of the Laplace–Runge–Lenz symmetry is also underlinedThe elastic scattering of two particles is examined within Collisions and Scattering Theory in Chapter 5 The kinematic and dynamic aspects of collisions processes are described and discussed Explicit expressions for scattering cross sections are obtained which utilize classical and modified Rutherford approach The softsphere problem and the elastic scattering by a hard surface are considered also

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