Applied Calculus of Variations for Engineers - Louis Komzsik

Applied Calculus of Variations for Engineers

(Autor)

Buch | Hardcover
175 Seiten
2009
Crc Press Inc (Verlag)
978-1-4200-8662-1 (ISBN)
63,55 inkl. MwSt
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The subject of calculus of variations is to find optimal solutions to engineering problems where the optimum may be a certain quantity, a shape, or a function. This title addresses this very important mathematical area applicable to many engineering disciplines. It offers an application-oriented approach.
The subject of calculus of variations is to find optimal solutions to engineering problems where the optimum may be a certain quantity, a shape, or a function. Applied Calculus of Variations for Engineers addresses this very important mathematical area applicable to many engineering disciplines. Its unique, application-oriented approach sets it apart from the theoretical treatises of most texts. It is aimed at enhancing the engineer’s understanding of the topic as well as aiding in the application of the concepts in a variety of engineering disciplines.





The first part of the book presents the fundamental variational problem and its solution via the Euler–Lagrange equation. It also discusses variational problems subject to constraints, the inverse problem of variational calculus, and the direct solution techniques of variational problems, such as the Ritz, Galerkin, and Kantorovich methods. With an emphasis on applications, the second part details the geodesic concept of differential geometry and its extensions to higher order spaces. It covers the variational origin of natural splines and the variational formulation of B-splines under various constraints. This section also focuses on analytic and computational mechanics, explaining classical mechanical problems and Lagrange’s equations of motion.

Steady State, Transients, and Design with MATLAB

MATHEMATICAL FOUNDATION


The Foundations of Calculus of Variations


The fundamental problem and lemma of calculus of variations


The Legendre test


The Euler–Lagrange differential equation


Application: minimal path problems


Open boundary variational problems


Constrained Variational Problems


Algebraic boundary conditions


Lagrange’s solution


Application: iso-perimetric problems


Closed-loop integrals


Multivariate Functionals


Functionals with several functions


Variational problems in parametric form


Functionals with two independent variables


Application: minimal surfaces


Functionals with three independent variables


Higher Order Derivatives


The Euler–Poisson equation


The Euler–Poisson system of equations


Algebraic constraints on the derivative


Application: linearization of second order problems


The Inverse Problem of the Calculus of Variations


The variational form of Poisson’s equation


The variational form of eigenvalue problems


Application: Sturm–Liouville problems


Direct Methods of Calculus of Variations


Euler’s method


Ritz method


Galerkin’s method


Kantorovich’s method





ENGINEERING APPLICATIONS


Differential Geometry


The geodesic problem


A system of differential equations for geodesic curves


Geodesic curvature


Generalization of the geodesic concept


Computational Geometry


Natural splines


B-spline approximation


B-splines with point constraints


B-splines with tangent constraints


Generalization to higher dimensions


Analytic Mechanics


Hamilton’s principle for mechanical systems


Elastic string vibrations


The elastic membrane


Bending of a beam under its own weight


Computational Mechanics


Three-dimensional elasticity


Lagrange’s equations of motion


Heat conduction


Fluid mechanics


Computational techniques


Closing Remarks





References





Index

Erscheint lt. Verlag 1.6.2009
Zusatzinfo 25 Illustrations, black and white
Verlagsort Bosa Roca
Sprache englisch
Maße 156 x 235 mm
Gewicht 431 g
Themenwelt Technik Elektrotechnik / Energietechnik
Technik Umwelttechnik / Biotechnologie
ISBN-10 1-4200-8662-6 / 1420086626
ISBN-13 978-1-4200-8662-1 / 9781420086621
Zustand Neuware
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