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Applied Calculus of Variations for Engineers by Louis Komzsik



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Applied Calculus of Variations for Engineers by Louis Komzsik ,Chief numerical analyst of Architecture and Technology at Siemens PLM Software.
The topic of this book has a long history. Its fundamentals were laid down by icons of mathematics like Euler and Lagrange. It was once heralded as the panacea for all engineering optimization problems by suggesting that all one needs to do was to apply the Euler-Lagrange equation form and solve the resulting di erential equation.
This, as most all encompassing solutions, turned out to be not always true and the resulting di erential equations are not necessarily easy to solve. On the other hand, many of the di erential equations commonly used by engineers today are derived from a variational problem. Hence, it is important and useful for engineers to delve into this topic.
The book is organized into two parts: theoretical foundation and engineering applications. The rst part starts with the statement of the fundamental variational problem and its solution via the Euler-Lagrange equation. This is followed by the gradual extension to variational problems subject to constraints, containing functions of multiple variables and functionals with higher order derivatives. It continues with the inverse problem of variational calculus, when the origin is in the di erential equation form and the corresponding variational problem is sought. The rst part concludes with the direct solution techniques of variational problems, such as the Ritz, Galerkin and Kantorovich methods.
With the emphasis on applications, the second part starts with a detailed discussion of the geodesic concept of di erential geometry and its extensions to higher order spaces. The computational geometry chapter covers the variational origin of natural splines and the variational formulation of B-splines under various constraints.

Applied Calculus of Variations for Engineers by Louis Komzsik cover the following topics.


  • 1. The foundations of calculus of variations
    1.1 The fundamental problem and lemma of calculus of variations
    1.2 The Legendre test
    1.3 The Euler-Lagrange differential equation
    1.4 Application: Minimal path problems
    1.4.1 Shortest curve between two points
    1.4.2 The brachistochrone problem
    1.4.3 Fermat's principle
    1.4.4 Particle moving in the gravitational field
    1.5 Open boundary variational problems

  • 2. Constrained variational problems
    2.1 Algebraic boundary conditions
    2.2 Lagrange's solution
    2.3 Application: Iso-perimetric problems
    2.3.1 Maximal area under curve with given length
    2.3.2 Optimal shape of curve of given length under gravity
    2.4 Closed-loop integrals

  • 3. Multivariate functionals
    3.1 Functionals with several functions
    3.2 Variational problems in parametric form
    3.3 Functionals with two independent variables
    3.4 Application: Minimal surfaces
    3.4.1 Minimal surfaces of revolution
    3.5 Functionals with three independent variables

  • 4. Higher order derivatives
    4.1 The Euler-Poisson equation
    4.2 The Euler-Poisson system of equations
    4.3 Algebraic constraints on the derivative
    4.4 Application: Linearization of second order problems

  • 5. The inverse problem of the calculus of variations
    5.1 The variational form of Poisson's equation
    5.2 The variational form of eigenvalue problems
    5.2.1 Orthogonal eigensolutions
    5.3 Application: Sturm-Liouville problems
    5.3.1 Legendre's equation and polynomials

  • 6. Direct methods of calculus of variations
    6.1 Euler's method
    6.2 Ritz method
    6.2.1 Application: Solution of Poisson's equation
    6.3 Galerkin's method
    6.4 Kantorovich's method

  • 7. Di erential geometry
    7.1 The geodesic problem
    7.1.1 Geodesics of a sphere
    7.2 A system of differential equations for geodesic curves
    7.2.1 Geodesics of surfaces of revolution
    7.3 Geodesic curvature
    7.3.1 Geodesic curvature of helix
    7.4 Generalization of the geodesic concept

  • 8. Computational geometry
    8.1 Natural splines
    8.2 B-spline approximation
    8.3 B-splines with point constraints
    8.4 B-splines with tangent constraints
    8.5 Generalization to higher dimensions

  • 9. Analytic mechanics
    9.1 Hamilton's principle for mechanical systems
    9.2 Elastic string vibrations
    9.3 The elastic membrane
    9.3.1 Nonzero boundary conditions
    9.4 Bending of a beam under its own weight

  • 10. Computational mechanics
    10.1 Three-dimensional elasticity
    10.2 Lagrange's equations of motion
    10.2.1 Hamilton's canonical equations
    10.3 Heat conduction
    10.4 Fluid mechanics
    10.5 Computational techniques
    10.5.1 Discretization of continua
    10.5.2 Computation of basis functions
    Closing Remarks
    Notation
    List of Tables
    List of Figures
    References
    Index

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