1st Edition
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 Kind
Elementary Displacements
Generalities
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
APPENDICES
REFERENCES
Biography
Ioan Merches, Daniel Radu