1st Edition

Analytical Mechanics Solutions to Problems in Classical Physics

By Ioan Merches, Daniel Radu Copyright 2015
    456 Pages 122 B/W Illustrations
    by CRC Press

    Giving students a thorough grounding in basic problems and their solutions, Analytical Mechanics: Solutions to Problems in Classical Physics presents a short theoretical description of the principles and methods of analytical mechanics, followed by solved problems. The authors thoroughly discuss solutions to the problems by taking a comprehensive approach to explore the methods of investigation. They carefully perform the calculations step by step, graphically displaying some solutions via Mathematica® 4.0.

    This collection of solved problems gives students experience in applying theory (Lagrangian and Hamiltonian formalisms for discrete and continuous systems, Hamilton-Jacobi method, variational calculus, theory of stability, and more) to problems in classical physics. The authors develop some theoretical subjects, so that students can follow solutions to the problems without appealing to other reference sources. This has been done for both discrete and continuous physical systems or, in analytical terms, systems with finite and infinite degrees of freedom. The authors also highlight the basics of vector algebra and vector analysis, in Appendix B. They thoroughly develop and discuss notions like gradient, divergence, curl, and tensor, together with their physical applications.

    There are many excellent textbooks dedicated to applied analytical mechanics for both students and their instructors, but this one takes an unusual approach, with a thorough analysis of solutions to the problems and an appropriate choice of applications in various branches of physics. It lays out the similarities and differences between various analytical approaches, and their specific efficiency.

    Fundamentals of Analytical Mechanics
    Classification Criteria for Constraints
    The Fundamental Dynamical Problem for a Constrained Particle
    System of Particles Subject to Constraints
    Lagrange Equations of the First Kind

    Elementary Displacements
    Real, Possible and Virtual Displacements

    Virtual Work and Connected Principles
    Principle of Virtual Work
    Principle of Virtual Velocities
    Torricelli’s Principle

    Principles of Analytical Mechanics
    D’alembert’s Principle
    Configuration Space
    Generalized Forces
    Hamilton’s Principle

    The Simple Pendulum Problem
    Classical (Newtonian) Formalism
    Lagrange Equations of the first Kind Approach
    Lagrange Equations of the Second Kind Approach
    Hamilton’s Canonical Equations Approach
    Hamilton-Jacobi Method
    Action-Angle Variables Formalism

    Problems Solved by Means of the Principle of Virtual Work

    Problems of Variational Calculus
    Elements of Variational Calculus
    Functionals. Functional Derivative
    Extrema of Functionals

    Problems whose solutions demand elements of variational calculus
    Brachistochrone problem
    Catenary problem
    Isoperimetric problem
    Surface of revolution of minimum area
    Geodesics of a Riemannian manifold

    Problems Solved by Means of the Lagrangian Formalism
    Atwood machine
    Double Atwood Machine
    Pendulum with Horizontally Oscillating Point of Suspension
    Problem of Two Identical Coupled Pendulums
    Problem of Two Different Coupled Pendulums
    Problem of Three Identical Coupled Pendulums
    Problem of Double Gravitational Pendulum

    Problems of Equilibrium and Small Oscillations

    Problems Solved By Means of the Hamiltonian Formalism

    Problems of Continuous Systems
    A. Problems of Classical Electrodynamics
    B. Problems of Fluid Mechanics
    C. Problems of Magnetofluid Dynamics and Quantum Mechanics



    Ioan Merches, Daniel Radu