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Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators, quartic oscillators, minimal surfaces, and Schrödinger's, Einstein's and Newton's equations. Historically, problems in these areas were approached using the Fourier transform or path integrals, although in some cases (e.g., the case of quartic oscillators) these methods do not work. New geometric methods are introduced in the work that have the advantage of providing quantitative or at least qualitative descriptions of operators, many of which cannot be treated by other methods. And, conservation laws of the Euler–Lagrange equations are employed to solve the equations of motion qualitatively when quantitative analysis is not possible. Main topics include: Lagrangian formalism on Riemannian manifolds; energy momentum tensor and conservation laws; Hamiltonian formalism; Hamilton–Jacobi theory; harmonic functions, maps, and geodesics; fundamental solutions for heat operators with potential; and a variational approach to mechanical curves. The text is enriched with good examples and exercises at the end of every chapter. Geometric Mechanics on Riemannian Manifolds is a fine text for a course or seminar directed at graduate and advanced undergraduate students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics. It is also an ideal resource for pure and applied mathematicians and theoretical physicists working in these areas.

Riemannian manifolds. --- Global Riemannian geometry. --- Mechanics, Analytic. --- Differential equations, Partial. --- Partial differential equations --- Analytical mechanics --- Kinetics --- Riemannian geometry, Global --- Global differential geometry --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Fourier analysis. --- Global differential geometry. --- Differential equations, partial. --- Mathematical physics. --- Harmonic analysis. --- Mathematics. --- Fourier Analysis. --- Differential Geometry. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Abstract Harmonic Analysis. --- Applications of Mathematics. --- Math --- Science --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Physical mathematics --- Physics --- Analysis, Fourier --- Differential geometry. --- Partial differential equations. --- Physics. --- Applied mathematics. --- Engineering mathematics. --- Engineering --- Engineering analysis --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Differential geometry

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Differential geometry. Global analysis --- Partial differential equations --- Mathematics --- Mathematical physics --- differentiaalvergelijkingen --- toegepaste wiskunde --- differentiaal geometrie --- wiskunde --- fysica

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Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators, quartic oscillators, minimal surfaces, and Schrödinger's, Einstein's and Newton's equations. Historically, problems in these areas were approached using the Fourier transform or path integrals, although in some cases (e.g., the case of quartic oscillators) these methods do not work. New geometric methods are introduced in the work that have the advantage of providing quantitative or at least qualitative descriptions of operators, many of which cannot be treated by other methods. And, conservation laws of the Euler-Lagrange equations are employed to solve the equations of motion qualitatively when quantitative analysis is not possible. Main topics include: Lagrangian formalism on Riemannian manifolds; energy momentum tensor and conservation laws; Hamiltonian formalism; Hamilton-Jacobi theory; harmonic functions, maps, and geodesics; fundamental solutions for heat operators with potential; and a variational approach to mechanical curves. The text is enriched with good examples and exercises at the end of every chapter. Geometric Mechanics on Riemannian Manifolds is a fine text for a course or seminar directed at graduate and advanced undergraduate students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics. It is also an ideal resource for pure and applied mathematicians and theoretical physicists working in these areas.

Differential geometry. Global analysis --- Partial differential equations --- Mathematics --- Mathematical physics --- differentiaalvergelijkingen --- toegepaste wiskunde --- differentiaal geometrie --- wiskunde --- fysica

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This monograph is a unified presentation of several theories of finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. These kernels are important in the theory of parabolic operators because they describe the distribution of heat on a given manifold as well as evolution phenomena and diffusion processes. The work is divided into four main parts: Part I treats the heat kernel by traditional methods, such as the Fourier transform method, paths integrals, variational calculus, and eigenvalue expansion; Part II deals with the heat kernel on nilpotent Lie groups and nilmanifolds; Part III examines Laguerre calculus applications; Part IV uses the method of pseudo-differential operators to describe heat kernels. Topics and features: ¢comprehensive treatment from the point of view of distinct branches of mathematics, such as stochastic processes, differential geometry, special functions, quantum mechanics, and PDEs; ¢novelty of the work is in the diverse methods used to compute heat kernels for elliptic and sub-elliptic operators; ¢most of the heat kernels computable by means of elementary functions are covered in the work; ¢self-contained material on stochastic processes and variational methods is included. Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal reference for graduate students, researchers in pure and applied mathematics, and theoretical physicists interested in understanding different ways of approaching evolution operators.

Differential geometry. Global analysis --- Operator theory --- Partial differential equations --- Mathematical analysis --- Operational research. Game theory --- Mathematical physics --- differentiaalvergelijkingen --- differentiaal geometrie --- Fourierreeksen --- stochastische analyse --- mathematische modellen --- wiskunde --- kansrekening

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