Ioan Merches

Principles of Analytical Mechanics

As we have already mentioned in Chap. 1, a particle (or a system of particles) is subject to constraints if its motion is restricted by a constraint force on a certain surface, or on some curve, etc. The notion of constraint is essential in understanding the analytical mechanics formalism, and we shall begin this chapter with a thorough analysis of this basic concept. By definition, a constraint is a geometric or kinematic condition that limits the possibilities of motion of a mechanical system. For example, a body sliding on an inclined plane cannot leave the plane; or a pebble inside a soccer ball is compelled to move within a given volume, etc.

The Electromagnetic Field

The study of time-variable electric and magnetic fields showed the strong interdependence between them: a variable electric field produces a magnetic field and vice-versa.

Elements of Magnetofluid Dynamics

In the fourth decade of the 20th century appeared a new branch of physical sciences, as a border discipline between the electromagnetic field theory, on the one side, and the fluid mechanics, on the other, known as magnetofluid dynamics, or magnetohydrodynamics – MHD. By its object and applications, this discipline can be included in the larger framework of plasma physics.

Fields of Stationary Currents

If at the ends A and B (or at any two points) of a conductor is applied a potential difference or voltage, this will produce in the conductor an oriented displacement of electric charges.

General Theory of Relativity

Einstein developed his ideas about the relativistic approach to gravity over many years, culminating with his 1915 papers presented to the Prussian Academy of Science.

Relativistic Formulation of Electrodynamics in Minkowski Space

Consider a charged particle of mass (m_0) and charge e, moving in the electromagnetic field ({mathbf {E}},,{mathbf {B}}), defined in terms of the usual electromagnetic potentials (V,,{mathbf {A}}).

Analytical mechanics : solutions to problems in classical physics

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 Torricellis Principle Principles of Analytical Mechanics Dalemberts Principle Configuration Space Generalized Forces Hamiltons Principle The Simple Pendulum Problem Classical (Newtonian) Formalism Lagrange Equations of the first Kind Approach Lagrange Equations of the Second Kind Approach Hamiltons 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

Foundations of Newtonian Mechanics

In physics, by mechanical motion we mean the change in time of the position of a body with respect to another body, chosen as a reference. Generally speaking, the motion of a body does not reduce to its mechanical motion, since the body can be simultaneously animated by several types of motion (mechanical, chemical, biological, etc.) depending on its complexity. For the sake of simplicity we shall, nevertheless, call mechanical motion just motion.

Applications of the Lagrangian Formalism in the Study of Discrete Particle Systems

Consider a system of N bodies, supposed to be particles, and assume that they are subject only to internal forces F ij (gravitational, electrostatic, etc.). The problem of determining the motion of each body in the presence of the other N − 1 is known as the problem of theNbodies. The difficulty of solving such a problem is obviously dependent on the number of the bodies involved. The simplest case (N = 2) is found in classical systems, like Sun–Earth, or nucleus–electron, and is called the two-body problem.

Rigid Body Mechanics

By rigid or non-deformable body we mean a continuous or discrete system of particles, with the property that the distance between any two particles does not change during the motion. Under normal conditions, within certain pressure and temperature limits, bodies made out of metal, glass, stone, etc. can be considered rigid.