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2011 | Buch

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Classical Mechanics

Theory and Mathematical Modeling

verfasst von: Emmanuele DiBenedetto

Verlag: Birkhäuser Boston

Buchreihe : Cornerstones

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Classical mechanics is a chief example of the scientific method organizing a "complex" collection of information into theoretically rigorous, unifying principles; in this sense, mechanics represents one of the highest forms of mathematical modeling. This textbook covers standard topics of a mechanics course, namely, the mechanics of rigid bodies, Lagrangian and Hamiltonian formalism, stability and small oscillations, an introduction to celestial mechanics, and Hamilton–Jacobi theory, but at the same time features unique examples—such as the spinning top including friction and gyroscopic compass—seldom appearing in this context. In addition, variational principles like Lagrangian and Hamiltonian dynamics are treated in great detail. Using a pedagogical approach, the author covers many topics that are gradually developed and motivated by classical examples. Through `Problems and Complements' sections at the end of each chapter, the work presents various questions in an extended presentation that is extremely useful for an interdisciplinary audience trying to master the subject. Beautiful illustrations, unique examples, and useful remarks are key features throughout the text. Classical Mechanics: Theory and Mathematical Modeling may serve as a textbook for advanced graduate students in mathematics, physics, engineering, and the natural sciences, as well as an excellent reference or self-study guide for applied mathematicians and mathematical physicists. Prerequisites include a working knowledge of linear algebra, multivariate calculus, the basic theory of ordinary differential equations, and elementary physics.

Inhaltsverzeichnis

Frontmatter

1. Geometry of Motion

Abstract

${\mathbb{R}}^{3}$A triple of vectors {e ₁, e ₂, e ₃} is positively oriented if e _i ∧ e _j = e _k for every cyclic permutation {i, j, k} of the indices. A positive triadis a Cartesian reference system {O; e1, e2, e3} in ${\mathbb{R}}^{3}$with origin in O and positively oriented unit vectors {e ₁, e ₂, e ₃}.

Emmanuele DiBenedetto

2. Constraints and Lagrangian Coordinates

Abstract

Constraints are limitations imposed on thegeometrical or kinematic configuration of a mechanical system. For example, in a rigid motion any two points are required to beat constant mutual distance. This is a rigidity constraint. A system with one of its points constrained on a surface is an example of a constrained mechanical system. Assume that a point P moves, being constrained to a surface $\mathcal{S}\subset {\mathbb{R}}^{3}$.

Emmanuele DiBenedetto

3. Dynamics of a Point Mass

Abstract

A point mass {P; m} is in a uniform mechanical state if its velocity is constant. Departures from a uniform state occur only by variations of velocity caused by solicitations external to {P; m} and acting on it. Such external solicitations are called forces. The vector equation

Emmanuele DiBenedetto

4. Geometry of Masses

Abstract

A distribution of masses within a bounded set $E \subset {\mathbb{R}}^{3}$is described by a measure μ. The symbol dμ(P) is the elemental mass about P as measured by μ.

Emmanuele DiBenedetto

5. Systems Dynamics

Abstract

Let G(t) be the configuration at time t of a material system $\{\mathcal{M};d\mu \}$in motion from its initial configuration G _o. Every point P _o ∈ G _o follows its trajectory to arrive at the position P(t) ∈ G(t) at time t; vice versa, a point P ∈ G(t) may be regarded as originating from the motion of some P _o ∈ G _o.

Emmanuele DiBenedetto

6. The Lagrange Equations

Abstract

Let $\{\mathcal{M};d\mu \}$be a material system whose mechanical state is described by N Lagrangian coordinates $q = ({q}_{1},\ldots, {q}_{N})$. Since every point $P \in \{\mathcal{M};d\mu \}$is identified along its motion by the map (q, t) → P(q, t), the configuration of the system is determined, instant by instant, by the map $t \rightarrow q(t) : \mathbb{R} \rightarrow {\mathbb{R}}^{N}$. The latter can be regarded as the motion of some abstract point in some N-dimensional space, called configuration space. Since N is the least numberof parameters needed to identify uniquely the position of each point P of the system, each of the maps $\{\mathcal{M};d\mu \} \ni P \rightarrow \| \partial P/\partial {q}_{h}\|$, $h = 1,\ldots, N$, is not identically zero. Equivalently, we have the following lemma.

Emmanuele DiBenedetto

7. Precessions and Gyroscopes

Abstract

Let $\{\mathcal{M};d\mu \}$be a rigid system in precession abouta pole O. Introduce a fixed inertial triad Σ and a moving triad S, both with originat O, so that S is in rigid motion with respect to Σ with angular characteristic ω. The latter is the unknown of the motion.The system is acted upon by external forcesthat generate a resultant moment M ^(e)with respect to the pole O. The constraint that keeps O fixed and other possible constraintsgive rise to reactions of resultant moment $\mathcal{M}$with respect to O. It is assumed that the moment M ^(e) and $\mathcal{M}$are known functions of ω, or equivalently of the Euler angles.

Emmanuele DiBenedetto

8. Stability and small Oscillations

Abstract

Let $\{\mathcal{M};d\mu \}$be a mechanical system with N degrees of freedom, by the Lagrangian parameters $({q}_{1},\ldots, {q}_{N})$. We will assume that $\{\mathcal{M};d\mu \}$is subject to fixed holonomic constraints satisfying the principle of virtual work and is acted upon by conservative forces.

Emmanuele DiBenedetto

9. Variational Principles

Abstract

Given two points q _o and q ₁ in ${\mathbb{R}}^{N}$and an interval $[{t}_{o},{t}_{1}] \subset \mathbb{R}$, consider the convex set $\mathcal{K}$of all smooth curves parameterized with t ∈ [t _o, t ₁], and of extremities q _o and q ₁, i.e.,

$$\mathcal{K} = \left \{q \in {C}^{1}[{t}_{ o},{t}_{1}]\bigm |q({t}_{o}) = {q}_{o}, q({t}_{1}) = {q}_{1}\right \}.$$

Given $q \in \mathcal{K}$and a vector-valued function φ ∈ C∞o(to, t1), the curve {q + λφ} is still in $\mathcal{K}$for all $\lambda \in \mathbb{R}$.

Emmanuele DiBenedetto

10. Canonical Transformations

Abstract

Let $\mathcal{H}(p,q;t)$be the Hamiltonian of a mechanical system with N degrees of freedom. By the least action principle, an orbit t → (p(t), q(t)) in phase space ${\mathbb{R}}^{2N}$is a solution of the Hamilton canonical equations if and only if it is a stationary point of the action

Emmanuele DiBenedetto

11. Integrating Hamilton–Jacobi Equations and Canonical Systems

Abstract

A complete integral of the Hamilton–Jacobi equation

$${F}_{t} + \mathcal{H}(q,{\nabla }_{q}F;t) = 0$$

(1.1)

is a smooth function F of q and t that depends on N arbitrary parameters $Q = ({Q}_{1},,\ldots, {Q}_{N})$and that satisfies (1.1) as q and Q range within the domain of definition of F.

Emmanuele DiBenedetto

12. Introduction to Fluid Dynamics

Abstract

A bounded open connected set in ${E}_{o} \subset {\mathbb{R}}^{N}$deforms in time to E in the sense that points y ∈ E _o are in one-to-one correspondence with points x ∈ E through smooth, nonintersecting trajectories t → x(t) such that x(0) = y and x(t) = x.

Emmanuele DiBenedetto

Backmatter

Titel: Classical Mechanics
verfasst von: Emmanuele DiBenedetto
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-4648-6
Print ISBN: 978-0-8176-4526-7
DOI: https://doi.org/10.1007/978-0-8176-4648-6