Analytical Mechanics Solutions to Problems in Classical Physics

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 Constraints Classification Criteria for Constraints The Fundamental Dynamical Problem for a Constrained Particle System of Particles Subject to Constraints Lagrange Equations of the First KindElementary Displacements Generalities Real, Possible and Virtual Displacements Virtual Work and Connected Principles Principle of Virtual WorkPrinciple 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 ApproachHamilton’s Canonical Equations Approach Hamilton-Jacobi MethodAction-Angle Variables FormalismProblems Solved by Means of the Principle of Virtual WorkProblems of Variational Calculus Elements of Variational Calculus Functionals. Functional Derivative Extrema of Functionals Problems whose solutions demand elements of variational calculus Brachistochrone problem Catenary problemIsoperimetric problemSurface of revolution of minimum area Geodesics of a Riemannian manifold Problems Solved by Means of the Lagrangian Formalism Atwood machine Double Atwood MachinePendulum 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 PendulumProblems of Equilibrium and Small Oscillations Problems Solved By Means of the Hamiltonian FormalismProblems of Continuous Systems A. Problems of Classical ElectrodynamicsB. Problems of Fluid Mechanics C. Problems of Magnetofluid Dynamics and Quantum Mechanics APPENDICES REFERENCES