Variational principles for second-order differential equations application of the Spencer theory to characterize variational sprays by J. Grifone

Cover of: Variational principles for second-order differential equations | J. Grifone

Published by World Scientific in Singapore, River Edge, NJ .

Written in English

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  • Differential equations, Partial.,
  • Variational principles.

Edition Notes

Includes bibliographical references (p. 213-215) and index.

Book details

StatementJoseph Grifone, Zoltán Muzsnay.
ContributionsMuzsnay, Zoltán.
LC ClassificationsQA377 .G748 2000
The Physical Object
Paginationx, 217 p. :
Number of Pages217
ID Numbers
Open LibraryOL3438820M
ISBN 109810237340
LC Control Number2005297863

Download Variational principles for second-order differential equations

The inverse problem of the calculus of variations was first studied by Helmholtz in and it is entirely solved for the differential operators, but only a few results are known in the more general case of differential equations.

This book looks at second-order differential equations and asks if they can be written as Euler-Lagrangian by: Variational Principles for Second-Order Differential Equations. This book looks at second-order differential equations and asks if they can be written as Euler–Lagrangian equations.

If the equations are quadratic, the problem reduces to the characterization of the connections which are Levi–Civita for some Riemann metric. This advanced graduate-level text examines variational methods in partial differential equations and illustrates their applications to a number of free-boundary problems.

Detailed statements of the standard theory of elliptic and parabolic operators make this treatment readable for engineers, students, and nonspecialists by: Variational Approaches to Kirchhoff-Type Second-Order Impulsive Differential Equations on the Half-Line.

Results in Mathematics, Vol. 73, Issue. 1, Results in Mathematics, Vol. 73, Issue. 1, CrossRefCited by: 1. The field of geometric variational problems is fast-moving and influential. These problems interact with many other areas of mathematics and have strong relevance to the study of integrable systems, mathematical physics and PDEs.

The workshop 'Variational Problems in Differential Geometry' held in at the University of Leeds brought together. ISBN: e-book (Adobe PDF color) ISBN: print (Paperback grayscale) Variational Principles in Classical Mechanics Contributors Author: Douglas Cline Illustrator: Meghan Sarkis Published by University of Rochester River Campus Libraries University of Rochester Rochester, NY Together with parabolic differential equations in general, heat-conduction equations occur with such regularity in important applications that variational principles leading to these equations have been an important topic for many years.

The variational technique is such a powerful one that many solutions have been proposed for the problem. The variational principle used in two-dimensional elasticity problems is the principle of virtual work, which is expressed by the following integral equation: () ɛ ɛ ∬ D σ x δ ɛ x + σ y δ ɛ y + τ x y δ γ x y tdxdy − ∬ D F x δ u + F y δ v tdxdy − ∫ S σ t ¯ x ∗ δ u + t ¯ y ∗ δ v t d s = 0.

This book explains the following topics: First Order Equations, Numerical Methods, Applications of First Order Equations1em, Linear Second Order Equations, Applcations of Linear Second Order Equations, Series Solutions of Linear Second Order Equations, Laplace Transforms, Linear Higher Order Equations, Linear Systems of Differential Equations, Boundary Value Problems and Fourier.

I want to point out two main guiding questions to keep in mind as you learn your way through this rich field of mathematics. Question 1: are you mostly interested in ordinary or partial differential equations. Both have some of the same (or very s. to the introductory book [5] and to [7–9] for numerical aspects on solving fractional Euler–Lagrange equations.

For applications of fractional-order models and variational principles in epidemics, biology, and medicine, see [10–14] and references therein. This book introduces the use of variational principles in classical mechanics.

A brief review of Newtonian mechanics compares and contrasts the relative merits of the intuitive Newtonian vectorial formulation, with the more powerful analytical variational formulations.

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The differential equations are transformed to integral equations. Then one starts with some initial conditions to make a first order estimate of the functions. From the Euler–Lagrange equations for the functional, one may calculate the auxiliary function g(u) at a first approximation by takingp(u) from the linear case (N50).

First of all, we generalize this transformation for Hamiltonian systems. We multiply the differential equation by png and apply the variational method.

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The left hand side of this equation is called the functional derivative of J[f] and is denoted δJ/δf(x). In general this gives a second-order ordinary differential equation which can be solved to obtain the extremal function f(x). The Euler–Lagrange equation is a. In this section we discuss the solution to homogeneous, linear, second order differential equations, ay'' + by' + c = 0, in which the roots of the characteristic polynomial, ar^2 + br + c = 0, are complex roots.

We will also derive from the complex roots the standard solution that is typically used in this case that will not involve complex numbers.

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We will concentrate mostly on constant coefficient second order differential equations. We will derive the solutions for homogeneous differential equations and we will use the methods of undetermined coefficients and variation of parameters to solve non homogeneous differential equations.

Dirichlet’s principle consists in constructing harmonic functions by minimizing the Dirichlet integral in an appropriate class of functions.

This idea is generalized, and minimizers of variational integrals are weak solutions of the associated differential equations of Euler and. Euler, Lagrange, Hamilton, and Jacobi, developed powerful alternative variational formulations based on the assumption that nature follows the principle of least action.

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A brief review of Newtonian mechanics compares and contrasts the relative merits of the intuitive Newtonian vectorial formulation, with the more powerful analytical variational formulations.

subject is currently under strong development [6]. We refer the interested reader to the introductory book [5] and to [7–9] for numerical aspects on solving fractional Euler–Lagrange equations. For applications of fractional-order models and variational principles in epidemics, biology, and medicine, see [10–14] and references therein.

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The. Abstract. The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion.

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Prerequisites: MA Analysis, a module on solving ordinary differential equations (and it is probably a good idea to revise at least separation of variables and linear constant coefficient ODEs) and MA Multivariable Calculus is also helpful.

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In this famous monograph, a distinguished mathematician presents an innovative approach to classical boundary value problems that employs the basic scheme first suggested by Hilbert and developed by Tonnelli. The treatment covers variational principles of the theory of conformal mapping, hydrodynamic applications and quasiconformal mappings, linear systems, and other subjects.


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