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Applications of Lie Groups to Differential Equations

Chapter 4. Applications of Lie Groups to Differential Equations. Peter J. Olver. Email :: olver@umn.edu Home Page :: http://www.math.umn.edu/~olver. Presented by Mehdi Nadjafikhah Email :: m_nadjafikhah@iust.ac.ir Home Page :: webpages.iust.ac.ir/ m_nadjafikhah.

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Applications of Lie Groups to Differential Equations

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  1. Chapter 4 Applications of Lie Groups to Differential Equations Peter J. Olver Email :: olver@umn.eduHome Page ::http://www.math.umn.edu/~olver Presented by MehdiNadjafikhah Email :: m_nadjafikhah@iust.ac.ir Home Page :: webpages.iust.ac.ir/m_nadjafikhah Joseph-Louis Lagrange (1736 – 1813) http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange webpages.iust.ac.ir/m_nadjafikhah

  2. Introduction to Lie Groups 1.1. Manifolds Change of Coordinates, Maps Between , Manifolds, The Maximal Rank Condition, Submanifolds, Regular Submanifolds, Implicit Submanifolds, Curves and Connectedness 1.2. Lie Groups Lie Subgroups, Local Lie Groups, Local Transformation Groups, Orbits 1.3. Vector Fields Flows, Action on Functions, Differentials, Lie Brackets, Tangent Spaces and Vectors Fields on Submanifolds, Frobenius' Theorem 1.4. Lie Algebras One-Parameter Subgroups, Subalgebras, The Exponential Map, Lie Algebras of Local Lie Groups, Structure Constants, CommutatorTables, Infinitesimal Group Actions 1.5. Differential Forms Pull-Back and Change of Coordinates, Interior Products, The Differential, The de RhamComplex, Lie Derivatives HomotopyOperators, Integration and Stokes' Theorem CHAPTER 4 Symmetry Groups and Conservation Laws webpages.iust.ac.ir/m_nadjafikhah

  3. Calculus of variations is a field of mathematics, or more specifically calculus, that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. The calculus of variations may be said to begin with the brachistochrone curve problem raised by Johann Bernoulli (1696). The calculus of variations is nearly as old as the calculus, and the two subjects were developed somewhat in parallel. The enduring interest in the calculus of variations is in part due to its applications. Variational principles abound in physics and particularly in mechanics. The application of these principles usually entails finding functions that minimize definite integrals (e.g., energy integrals) and hence the calculus of variations comes naturally to the fore. Hamilton's Principle in classical mechanics is a prominent example. An earlier example is Fermat's Principle of Minimum Time in geometrical optics. webpages.iust.ac.ir/m_nadjafikhah

  4. Motivation Catrnary webpages.iust.ac.ir/m_nadjafikhah

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  6. Catenoid webpages.iust.ac.ir/m_nadjafikhah

  7. Brachystochrones webpages.iust.ac.ir/m_nadjafikhah

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  10. Dido's Problem webpages.iust.ac.ir/m_nadjafikhah

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  13. Geodesics webpages.iust.ac.ir/m_nadjafikhah

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  17. Example: Geodesics on a Sphere webpages.iust.ac.ir/m_nadjafikhah

  18. Minimal Surfaces webpages.iust.ac.ir/m_nadjafikhah

  19. Costa's Minimal Surface In mathematics, Costa's minimal surface is an embedded minimal surface and was discovered in 1982 by the Brazilian mathematician Celso Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact surface. Topologically, it is a thrice-punctured torus. webpages.iust.ac.ir/m_nadjafikhah

  20. Optimal Harvest Strategy webpages.iust.ac.ir/m_nadjafikhah

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  22. Hamilton's Principle webpages.iust.ac.ir/m_nadjafikhah

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  25. Example: Simple Pendulum webpages.iust.ac.ir/m_nadjafikhah

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  27. Example: Kepler problem webpages.iust.ac.ir/m_nadjafikhah

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  29. The First Variation webpages.iust.ac.ir/m_nadjafikhah

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  38. Example: Geodesics in the Plane webpages.iust.ac.ir/m_nadjafikhah

  39. Example: Catenary webpages.iust.ac.ir/m_nadjafikhah

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  41. Example: Brachystochrone webpages.iust.ac.ir/m_nadjafikhah

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  43. Some Generalizations Functionals Containing Higher-Order Derivatives webpages.iust.ac.ir/m_nadjafikhah

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  48. Several Dependent Variables webpages.iust.ac.ir/m_nadjafikhah

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