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## Über dieses Buch

This textbook takes a broad yet thorough approach to mechanics, aimed at bridging the gap between classical analytic and modern differential geometric approaches to the subject. Developed by the authors from over 30 years of teaching experience, the presentation is designed to give students an overview of the many different models used through the history of the field—from Newton to Hamilton—while also painting a clear picture of the most modern developments.

The text is organized into two parts. The first focuses on developing the mathematical framework of linear algebra and differential geometry necessary for the remainder of the book. Topics covered include tensor algebra, Euclidean and symplectic vector spaces, differential manifolds, and absolute differential calculus. The second part of the book applies these topics to kinematics, rigid body dynamics, Lagrangian and Hamiltonian dynamics, Hamilton–Jacobi theory, completely integrable systems, statistical mechanics of equilibrium, and impulsive dynamics, among others. This new edition has been completely revised and updated and now includes almost 200 exercises, as well as new chapters on celestial mechanics, one-dimensional continuous systems, and variational calculus with applications. Several Mathematica® notebooks are available to download that will further aid students in their understanding of some of the more difficult material.

Unique in its scope of coverage and method of approach, Classical Mechanics with Mathematica® will be useful resource for graduate students and advanced undergraduates in applied mathematics and physics who hope to gain a deeper understanding of mechanics.

## Inhaltsverzeichnis

### Chapter 1. Vector Space and Linear Maps

Abstract
In this chapter, the abstract definition of vector space, dimension of a vector space, bases, and contravariant components of a vector and their transformation properties are discussed. After giving some interesting examples of vector spaces, linear maps between vector spaces and their fundamental aspects are presented.

### Chapter 2. Tensor Algebra

Abstract
This chapter contains an introduction to tensor algebra. After defining covectors and dual bases, the space of covariant 2-tensor is introduced. Then, the results derived for this space are extended to the general space of the (r, s)-tensors. Finally, the operations of contraction and contracted multiplication are introduced.

### Chapter 3. Skew-Symmetric Tensors and Exterior Algebra

Abstract
This chapter is devoted to $$(0,r)-$$ skew-symmetric tensors. After determining their strict components, it is proved that they form a vector space $$\varLambda _r(E_n)$$. Then, the dimension of this space is determined together with the bases of this space and the transformation properties of strict components. Finally, the exterior algebra is formulated and oriented vector spaces are introduced.

### Chapter 4. Euclidean and Symplectic Vector Spaces

Abstract
In the preceding chapters we analyzed some properties of a vector space E$$_{\textit{n}}$$. In this chapter we introduce into E$$_{\textit{n}}$$ two other operations: the scalar product and the antiscalar product. A vector space equipped with the first operation is called a Euclidean vector space, whereas when it is equipped with the second operation, it is said to be a symplectic vector space. These operations allow us to introduce into E$$_{\textit{n}}$$ many other geometric and algebraic concepts such as length of a vector, orthogonality between two vectors, etc. Further, eigenvalues and eigenvectors of a linear map are analyzed together with orthogonal transformations of $$E_n$$. Finally, symplectic vector spaces are introduced and some their properties studied.

### Chapter 5. Duality and Euclidean Tensors

Abstract
In this chapter we show that when E$$_{\textit{n}}$$ is a Euclidean vector space, there is an isomorphism among the tensor spaces T$$_{\textit{s}}^{\textit{r}}$$(E$$_{\textit{n}}$$) for which r+s has a given value. In other words, we show the existence of an isomorphism between $$\textit{E}_{\textit{n}}$$ and $$\textit{E}_{\textit{n}}^{{*}}$$, of isomorphisms between T$$_{0}^{2}$$, T$$_{1}^{1}$$, and T$$_{2}^{0}$$, and so on. These isomorphisms lead to the definitions of Euclidean vectors, tensors, and to the covariant, contravariant, and mixed components of a tensor.

### Chapter 6. Differentiable Manifolds

Abstract
This chapter is an introduction to the wide subject of differentiable manifolds. A differentiable manifold, which generalizes the ordinary definitions of curves and surfaces, is presented here from an intrinsic point of view, that is, without referring to an ambient space in which it is embedded. Differentiable functions, tangent vectors, tensor fields, and r-forms are defined. Further, differential and codifferential of a map between manifolds are studied. In order to introduce metric evaluation on a manifold, the Riemann manifolds are introduced and the geodesics are defined. Many interesting exercises conclude the chapter.

### Chapter 7. One-Parameter Groups of Diffeomorphisms

Abstract
In this chapter the one-parameter transformation groups and Lie’s derivative are defined.

### Chapter 8. Exterior Derivative and Integration

Abstract
Two fundamental topics of differential geometry are presented in this chapter in introductory form: exterior derivative and integration of r-forms. The exterior derivative extends to r-forms the elementary definitions of gradient of a function, curl, and divergence of a vector field as well as the meaning of exact and closed 1-forms. The integration of r-forms allows to extend the definitions of surface and volume integrals as well as the Gauss and Stokes theorems.

### Chapter 9. Absolute Differential Calculus

Abstract
In this chapter, we address the fundamental problem of extending differential calculus to manifolds. This extension requires the introduction of a criterion to compare vectors belonging to different tangent spaces of the manifold. This criterion, except for some general rules that it must satisfy, is quite arbitrary. The choice of such a criterion corresponds to the introduction of an affine connection on the manifold. In this chapter we define the affine connections and study their properties. On a Riemannian manifold it is possible to define only one affine connection that is compatible with the metric. This means that the connection does not modify the scalar product between two vectors when they undergo a parallel transport along an arbitrary curve, provided that the parallelism is evaluated by the affine connection.

### Chapter 10. An Overview of Dynamical Systems

Abstract
This chapter is devoted to an overview of dynamical systems that play a fundamental role in building mathematical models of reality. After a brief introduction of modeling, we present some theorems of existence and uniqueness as well as the definitions of first integral and phase portrait. Then, we define Liapunov’s stability for autonomous systems together with some theorems of stability and instability of equilibrium. Poincare’s perturbation method is described with some applications. Finally, Weierstass’s qualitative analysis is presented.

### Chapter 11. Kinematics of a Point Particle

Abstract
Kinematics analyzes the trajectories, velocities, and accelerations of the points of a moving body, the deformations of its volume elements, and the dependence of all these quantities on the frame of reference. In many cases, when such an accurate description of motion is too complex, it is convenient to substitute the real body with an ideal body for which the analysis of the preceding characteristics is simpler, provided that the kinematic description of the ideal body is sufficiently close to the behavior of the real one. For instance, when the deformations undergone by a body under the influence of the acting forces can be neglected, we adopt the rigid body model, which is defined by the condition that the distances among its points do not change during the motion. More particularly, if the body is contained in a sphere whose radius is much smaller than the length of its position vectors relative to a frame of reference, then the whole body is sufficiently localized by the position of any one of its points. In this case, we adopt the model of a point particle. In this chapter we analyze the kinematics of a point particle defining velocity, acceleration, trajectory, and compound motion.

### Chapter 12. Kinematics of Rigid Bodies

Abstract
In this chapter a change of rigid frame of reference is considered and the general formula of velocity field in a rigid motion is given. Then, the translational, rotational, spherical, and planar motions are studied. Finally, the transformation formulae of velocity and acceleration from a rigid frame of reference to another one are determined. Exercises conclude the chapter.

### Chapter 13. Principles of Dynamics

Abstract
In this chapter the principle of inertia and inertial frames are introduced. In the inertial frames the axiomatic of classical forces is discussed. Then, the Newton laws are recalled together with some important consequences (momentum balance, angular momentum balance, kinetic energy, König’s theorem). Dynamics in noninertial frames both in the absence and in the presence of constraints is presented together with some exercises of point dynamics.

### Chapter 14. Dynamics of a Material Point

Abstract
In this chapter we analyze the motion of a material point acted upon by a Newtonian force and derive Kepler’s laws. The motion in the air of a material point subjected to its weight is described. Then, terrestrial dynamics is formulated and the motions of a simple pendulum, rotating simple pendulum, and Foucault’s pendulum are studied.

### Chapter 15. General Principles of Rigid Body Dynamics

Abstract
In this chapter the dynamical description of real bodies is improved by adopting the rigid body model. After defining the center of mass, linear momentum, angular momentum, and kinetic energy of a rigid body, the tensor of inertia is introduced and the fundamental properties of this tensor are discussed. Then, active and reactive forces are described and the balance equations of a rigid body are formulated.

### Chapter 16. Dynamics of a Rigid Body

Abstract
In this chapter the motion of a solid with a smooth fixed axis is analyzed. Then, some rigid motions of a solid with a (smooth) fixed point when the moment of forces vanishes are studied: Poinsot’s motions, uniform rotations, and inertial motions of a gyroscope. The different properties of motion of a heavy gyroscope and the gyroscopic effect are analyzed. Finally, the motions of Foucault’s gyroscope, gyroscopic compass, and free heavy solid are taken into account.

### Chapter 17. Lagrangian Dynamics

Abstract
In this chapter, Lagrangian dynamics is described. After introducing the configuration space for a system of constrained rigid bodies, the principle of virtual power and its equivalence to Lagrange’s equations is shown. Then, these equations are formulated in the case of conservative forces and forces deriving from a generalized potential energy. The fundamental relation between conservation laws (first integrals) and the symmetries of the Lagrangian function is proved (Noether’s theorem). Further, Lagrange’s equations for linear nonholonomic constraints are formulated. The chapter contains also the analysis of small oscillations about a stable equilibrium configuration (normal modes), and an introduction to variational calculus including Hamilton’s principle. Finally, two geometric formulations of Lagrangian’s dynamics are presented together with Legendre’s transformation that allows to transform Lagrange’s equations into Hamilton’s equations.

### Chapter 18. Hamiltonian Dynamics

Abstract
In this chapter we introduce the phase space and its symplectic structure. Then, canonical coordinates and generating functions are defined. The fundamental symplectic 2-form is introduced to define an isomorphism between vector fields and differential forms. Further, Hamiltonian vector fields and Poisson’s brackets are presented. The relation between first integrals and symmetries is analyzed together with Poincaré’s absolute and relative integral invariants. Liouville’s theorem and Poincaré’s theorem are proved and the volume form on invariant submanifolds is defined. Time-dependent Hamiltonian mechanics and contact manifolds are introduced together with contact coordinates and locally and globally Hamiltonian fields.

### Chapter 19. The Hamilton–Jacobi Theory

Abstract
We are able to solve Hamilton’s equations only in a few cases. Consequently, we must search for information about their solutions following other paths. In this chapter, we show that the Hamilton–Jacobi theory allows, at least in principle, to determine a set of canonical coordinates in which the Hamiltonian equations can be integrated at once. However, these coordinates can be determined solving the Hamilton–Jacobi nonlinear partial differential equation. Usually, to solve this equation is more difficult than to integrate Hamiltonian equations, but Jacobi proposed the method of separated variables that in some cases allows to solve the Hamilton–Jacobi equation.

### Chapter 20. Completely Integrable Systems

Abstract
This chapter contains advanced topics of  Hamiltonian mechanics. Arnold–Liouville’s theorem shows that the motion of completely integrable systems can be determined. These systems describe the behavior of many physical systems and represent the starting point for analyzing more complex mechanical systems. The study of completely integrable systems leads to the introduction of very convenient canonical coordinates: the angle–action variables that are fundamental in celestial mechanics. Further, they were used by Bohr–Sommerfeld to formulate the quantization rules that allowed a first interpretation of hydrogenoid atomic spectra. A sketch of the Hamiltonian perturbation theory and an overview of KAM theorem conclude the chapter.

### Chapter 21. Elements of Statistical Mechanics of Equilibrium

Abstract
Statistical mechanics has the purpose to show that, if the idea is accepted that bodies are formed by small particles with reasonable properties, then it is possible to derive the macroscopic behavior of the matter around us. This point of view was severely criticized by the academic world since at that time there were many doubts about the existence of elementary constituents of matter. This chapter is devoted to the study of statistical mechanics of equilibrium. We start with Maxwell’s kinetic theory that is based on velocity distribution. Then, we formulate Gibb’s distribution and analyze its consequences. Finally, the last sections of the chapter are devoted to an introduction to ergodic theory.

### Chapter 22. Impulsive Dynamics

Abstract
In this chapter, we formulate the equations of impulsive dynamics starting from the balance equations of momentum and angular momentum of rigid bodies. Then, the Lagrangian formulation of the impulsive dynamics is developed. Exercises conclude the chapter.

### Chapter 23. Introduction to Fluid Mechanics

Abstract
In this chapter we consider the simplest model of a deformable body: a compressible or incompressible perfect fluid. We start analyzing the fundamental principles underlying mechanical models in which the deformability of bodies is taken into account. After an introduction to kinematics of deformable bodies, we determine Euler’s equations of hydrodynamics and study their consequences at equilibrium and in dynamical conditions. In particular, we prove Archimedes’s principle, Thomson’s theorem, Lagrange’s theorem, and the fundamental Bernoulli theorem. Then, we formulate the boundary value problems for Euler’s equations and analyze some plane motions by using complex potentials, among which is the Joukovsky potential. After showing D’Alembert paradox, we prove Blausius formula. Finally, we analyze the propagation of small amplitude waves in a compressible fluid.

### Chapter 24. An Introduction to Celestial Mechanics

Abstract
Celestial mechanics is one of the most interesting applications of classical mechanics. This topic answers one of the oldest questions of facing humankind: what forces govern the motion of celestial bodies? How do celestial bodies move under the action of these forces? In this chapter we discuss the foundations of this subject. We first recall the two-body problem and introduce the orbital elements. Then, we analyze the restricted three-body problem in which the third body has a very small mass compared with the masses of the other two bodies. In particular, we describe the Lagrange equilibrium positions and their stability. Then, we consider the N–body problem showing that the Hamiltonian of this system is obtained by adding the Hamiltonian of the two-body problem, that describes a completely integrable system, to another term that can be regarded as a perturbation term. In the remaining part of the chapter, we show how the gravitation law for mass points can be extended to continuous mass distributions, and we evaluate the asymptotic behavior of the gravitational field produced by extended bodies. Then, we define the local inertial frames and tidal forces. Finally, we pose the problem of determining the form of self-gravitating bodies.

### Chapter 25. One-Dimensional Continuous Systems

Abstract
In this chapter, a brief introduction to one-dimensional continuous systems is presented. First, the balance equations relative to these systems are formulated: the mass conservation and the balance of momentum and angular momentum. Then, these equations are applied to describe the behavior of Euler’s beam. Further, the simplified model of wires is introduced to describe the suspension bridge and catenary. Finally, the equation of vibrating strings is derived and the corresponding Sturm–Liouville problem is analyzed. In particular, the homogeneous string is studied by D’Alembert solution.

### Chapter 26. An Introduction to Special Relativity

Abstract
It is well known that at the beginning of the 20th century, a deep fracture was detected between classical mechanics and electrodynamics. It was evident that if classical mechanics is accepted without modification, then electrodynamics is valid only in a frame of reference: the ether frame. All the attempts to localize this frame of reference failed. Einstein with the special theory of relativity overcame all the difficulties accepting electrodynamics and modifying mechanics. In this chapter, adopting Einstein’s original approach, we analyze the foundations of this theory together with some applications. Then, we discuss the four-dimensional formulation of the special relativity proposed by Minkowski and write the physical laws in tensor form.

### Chapter 27. Variational Calculus with Applications

Abstract
In Chapter 17, Hamilton’s principle was proved by using Euler’s simple approach according to which a one-parameter family of curves is introduced that reduces the search for a minimum of a functional to that of a minimum of a function depending on a single variable. In this chapter, we present a deeper approach to variational calculus that shows how to extend to functionals many concepts of differential calculus relative to functions. Further, we formulate the problems of constrained stationary points of functionals and show how to obtain the momentum equation of continuum mechanics by variational principle.