Cambridge International Science Publishing
MECHANICS, TENSORS & VIRTUAL WORKS
Yves Talpaert
The various chapters connect the notions of mechanics of first and second year with the ones which are developed in more specialized subjects as continuum mechanics at first, and fluid-dynamics, quantum mechanics, special relativity, general relativity, electromagnetism, stellar dynamics, celestial mechanics, meteorology, applied differential geometry, and so on.
This book is the ideal mathematical and mechanical preparation for the above-mentioned specialized disciplines. This is a course of Analytical Mechanics which synthesizes the notions of first level mechanics and leads to the various mentioned disciplines by introducing mathematical concepts as tensor and virtual work methods. Analytical mechanics is not only viewed as a self-sufficient mathematical discipline, but as a subject of mechanics preparing for theories of physics and engineering too. One of the author's goals has been to reduce the gap between first and second academic cycles. The intensive use of the tensor calculus contributes to this reduction. The main subjects of mechanics of the first academic cycle are set in this context and let "handle" tensors. Many books relating to the developments of tensor theory are either too abstract since aimed at algebraists only, or too quickly applied to physicists and engineers. The author has found a right compromise which allows to bring closer these points of view; the book chapters are intended for mathematicians, which will find an illustrated presentation of mathematical concepts and solved problems in mechanics, and for physics and engineering students too, since the mathematical foundations are introduced in a practical way. It is the first time that a mechanics course so much develops the tensor calculus by taking into account the two previous sides! Besides the tensors, another mathematical concept is systematically used and largely developed: The virtual work. This notion is clearly introduced from the one of virtual displacements and virtual velocities are also considered.
The introduction of mathematical "tools", namely tensors and virtual works, gives the mechanics treated subjects a great unity; these mathematical notions easily lead to geometrical ways in the frame of differential geometry, as well as to other methods in continuum mechanics for example. In Europe in particular, given its style, this book is aimed to contribute to teaching mechanics European programs, the scientific English language is simple and should be used as a common language for concerned students of all European countries. All definitions and propositions are written with the peculiar strictness proper to mathematicians, but they are always illustrated with numerous examples, remarks and exercises. The concise style of differential geometry books of the author is also found in this mechanics book. When writing this new book, the author had the following assertion fresh in his mind: Pedagogy contributes to Rigor.
First, the notion of dynam is introduced in Chapter 0. So, the representation of dynams, properties and operation on dynams are recalled as well as the particular dynam of velocities.
In Chapter 1, the study of Statics is divided into two parts. First the classic method is considered with mechanical actions, classification of forces, equilibrium conditions, equilibrium of rigid bodies and structures, stress and contact dynam, types of constraints, free-body diagram and so on. Next, the method of virtual work is largely expanded with the notions of generalized coordinates, virtual displacements, virtual velocities, two formulations of the virtual work and numerous exercises of classical mechanics.
The tensor theory is very developed and illustrated in Chapter 2. Tensors are intrinsic mathematical entities and are suitable for the expression of laws of mechanics (and physics) regardless of the choice of coordinate system. This property by oneself justifies the extent of this study of tensors. This chapter deals with multilinear forms, dual space, vectors and covectors, tensors of various types, tensor algebra, contraction and tensor criteria, pre-Euclidean vector spaces, canonical isomorphism and conjugate tensor, Euclidean vector spaces, exterior algebra with p-forms and q-vectors, point space and natural frame, tensor fields and metric element, Christoffel symbols, absolute differential, covariant derivative, geodesic, volume form and adjoint, differential operators (gradient, divergence, curl, Laplacian) and so on. Exercises (39 in number) are completely solved. © So, the reader will better view the notion of inertia tensor within the widen context of tensors.
Devoted to this particular tensor, Chapter 3 especially prepares for the study of continuum mechanics. Thus mass distribution and integrals, center of mass, inertia tensors, inertia ellipsoid, principal axes, theorem of Steiner, etc. are reviewed in this new context.
Chapter 4 presents Newton's postulates, Galilean relativity, kinetic energy, linear momentum, angular momentum, kinetic energy and other notions with the help of dynams. But the tensor calculus is also very helpful to write theorems deduced from postulates.
Lagrangian dynamics and variational principles are at the root of analytical mechanics introduced in Chapter 5. Lagrangian dynamics, with the d'Alembert-Lagrange principle, shows another formulation of motion equations from the notion of virtual displacements; so, the profitable use of scalar functions leads to the Lagrange equations. In particular, the Lagrange equations with multipliers are greatly studied. Euler's equations and Hamilton's principle are considered in the context of the variational calculus. One-parameter groups of diffeomorphisms and the Euler-Noether theorem conclude this essential chapter.
Hamiltonian mechanics constitutes the last chapter. It is dealt with the N-body problem, Legendre transformation, canonical equations, Liouville's theorem in statistical mechanics, symplectic matrix, Lagrange and Poisson brackets, Hamilton-Jacobi equation, Jacobi's theorem, separability and so on.
