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We show that given an ordinary differential equation of order four, it may be possible to determine a Lagrangian if the third derivative is absent (or eliminated) from the equation. This represents a subcase of Fels'conditions [M. E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348 (1996) 5007-5029] which ensure the existence and uniqueness of the Lagrangian in the case of a fourth-order equation. The key is the Jacobi last multiplier as in the case of a second-order equation. Two equations from a Number Theory paper by Hall, one of second and one of fourth order, will be used to exemplify the method. The known link between Jacobi last multiplier and Lie symmetries is also exploited. Finally the Lagrangian of two fourth-order equations drawn from Physics are determined with the same method.

This paper sheds light on the essential characteristics of geodesics, which frequently occur in considerations from motion in Euclidean space. Focus is mainly on a method of obtaining them from the calculus of variations, and an explicit geodesic computation for a Riemannian hypersurface.

In a first part we propose an introduction to multisymplectic formalisms, which are generalisations of Hamilton's formulation of Mechanics to the calculus of variations with several variables: we give some physical motivations, related to the quantum field theory, and expound the simplest example, based on a theory due to T. de Donder and H. Weyl. In a second part we explain quickly a work in collaboration with J. Kouneiher (math-ph/0211046) on generalizations of the de Donder--Weyl theory (known as Lepage theories). Lastly we show that in this framework a perturbative classical field theory (analog of the perturbative quantum field theory) can be constructed.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

"On the Inverse Problem of the Calculus of Variations" is an article from The Annals of Mathematics, Volume 9 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967456

This text is meant for students of higher schools and deals with the most important sections of mathematics-differential equations and the calculus of variations. The book contains a large number of examples and problems with solutions involving applications of mathematics to physics and mechanics. The book was translated from the Russian by George Yankovsky and was first published by Mir Publishers in 1970. There were two reprints one in 1973 and one in 1977. The book here is to the third reprint. Contents Chapter 1. First-Order Differential Equations 19 1. First-Order Differential Equations Solved for the Derivative 19 2. Separable Equations 23 3. Equations That Lead to Separable Equations 29 4. Linear Equations of the First Order 32 5. Exact Differential Equations 37 6. Theorems of the Existence and Uniqueness of Solution of the equation dy/dx = f(x,y) 44 7. Approximate Me~hods of Integrating First-Order Equations 66 8. Elementary Types of Equations Not Solved for the Derivative 73 9. The Existence and Uniqueness Theorem for Differential Equations Not Solved for the Derivative. Singular Solutions 81 Problems 88 Chapter 2. Differential Equations of the Second Order and Higher 91 1. The Existence and Uniqueness Theorem for an nth Order Differential Equation 91 2. The Most Elementary Cases of Reducing the Order 93 3. Linear Differential Equations of the nth Order 98 4. Homogeneous Linear Equations with Constant Coefficients and Euler's Equations 112 5. Nonhomogeneous Linear Equations 119 6. Nonhomogeneous Linear Equations with Constant Coefficients and Euler's Equations 130 7. Integration of Differential Equations by Means of Series 143 8. The Small Parameter Method and Its Application in the Theory of Quasilinear Oscillations 153 9. Boundary-Value Problems. Essentials Problems Chapter 3. Systems of Differential Equations 1. Fundamentals 2. Integrating a System of Differential Equations by Reducing It to a Single Equation of Higher Order 179 3.Finding Integrable Combinations 186 4.Systems of Linear Differential Equations 189 5. Systems of Linear Differential Equations with Constant Coefficients 200 6. Approximate Methods of Integrating Systems of Differential Equations and Equations of Order n 206 Problems 209 Chapter 4. Theory of Stability 211 1. Fundamentals 211 2. Elementary Types of Rest Points 214 3. Lyapunov's Second Method 223 4. Test for Stability Based on First Approximation 229 5. Criteria of Negativity of the Real Parts of All Roots of a Polynomial 236 6. The Case of a Small Coefficient of a Higher-Order Derivative 238 7. Stability Under Constantly Operating Perturbations 244 Problems 247 Chapter 5. First-Order Partial Differential Equations 251 1. Fundamentals 2. Linear and Quasilinear First-Order Partial Differential Equations 253 3. Pfaffian Equations 265 4. First-Order Nonlinear Equations 271 Problems 288 PART TWO THE CALCULUS OF VARIATIONS 293 Chapter 6. The Method of Variations in Problems with Fixed Boundaries 297 1. Variation and Its Properties 297 2. Euler's Equation 304 3. Functionals of the Form \int^{x_1} _{x_n} F (x,y_1, y_2,...y_n, y'_1, y'_2,...y'_n)dx 318 4. Functionals Dependent on Higher-Order Derivatives 321 5. Functionals Dependent on the Functions of Several Independent Variables 325 6. Variational Problems in Parametric Form 330 7. Some Applications 333 Problems 338 Chapter 7: Variational Problems with Moving Boundaries and Certain Other Problems 341 1. An Elementary Problem with Moving Boundaries 341 2. The Moving-Boundary Problem for a Functional.r,of the Form \int_{x_0}^{x_1}F(x, y, z, y', z') dx 347 3. Extremals with Corners 352 4. One-Sided Variations 360 Problems 363 Chapter 8. Sufficient Conditions for an Extremum 365 1. Field of Extremals 365 2. The Function E (x, y, x', y') 371 3. Transforming the Euler Equations to the Canonical Form 383 Problems 387 Chapter 9. Variational Problems Involving a Conditional Extremum 389 1. Constraints of the Form \phi (x, y_1, y_2, ., y_n) = 0 389 2. Constraints of the Form \phi (x, y_1, y_2, ., y_n, y'_1, y'_2,., y'_n) = 0 396 3. Isoperimetric Problems 399 Problems 407 Chapter 10. Direct Methods In Variational Problems 408 1. Direct Methods 408 2. Euler's Finite-Difference Method 409 3. The Ritz Method 411 4. Kantorovich's Method 420 Problems 427 Answers to Problems 429 Recommended literature 436 Index 437

Indeks;Reprinted.;XV, [1], 980 p. : il. ; 21 cm

In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of $Y$, the manifold of dependent and independent variables underlying a given problem, as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over $Y$. Explicit examples of natural boundary conditions are obtained when $Y$ is an $(n+1)$--dimensional domain in $\R^{n+1}$, and the Lagrangian is first-order (in particular, the hypersurface area).

We propose a unified approach to the theory of connections in the geometry of sprays and Finsler metrics which, in particular, gives a simple explanation of the well-known fact that all the classical Finslerian connections provide exactly the same formulas appearing in the calculus of variations.

We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a particular new class of forms. We establish relation of the complex of variational derivatives and the variational complex. Certain calculus of Lagrangians of multidimensional paths is developed. It is shown how covariant Lagrangians of higher order can be used to represent characteristic classes.

We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are connected to: i) the Caratheodory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the theorems of Noether; iii) regularity conditions for the minimizers and the work pioneered by Tonelli.

Lectures from the National Programme on Technology Enhanced Learning - Mathematics - Calculus of Variations and Integral Equations

Lectures from the National Programme on Technology Enhanced Learning - Mathematics - Calculus of Variations and Integral Equations

The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of the first--order variational integrals related to the second--order Euler--Lagrange systems.

We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized Noether type theorem.

"Jacobi's Condition for the Problem of Lagrange in the Calculus of Variations" is an article from Transactions of the American Mathematical Society, Volume 17 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1988832

We study in this paper the continuous and discrete Euler-Lagrange equations arising from a quadratic lagrangian. Those equations may be thought as numerical schemes and may be solved through a matrix based framework. When the lagrangian is time-independent, we can solve both continuous and discrete Euler-Lagrange equations under convenient oscillatory and non-resonance properties. The convergence of the solutions is also investigated. In the simplest case of the harmonic oscillator, unconditional convergence does not hold, we give results and experiments in this direction.

We study in this paper the continuous and discrete Euler-Lagrange equations arising from a quadratic lagrangian. Those equations may be thought as numerical schemes and may be solved through a matrix based framework. When the lagrangian is time-independent, we can solve both continuous and discrete Euler-Lagrange equations under convenient oscillatory and non-resonance properties. The convergence of the solutions is also investigated. In the simplest case of the harmonic oscillator, unconditional convergence does not hold, we give results and experiments in this direction.

We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.

We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.

We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.

We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.

65 p. 28 cm

65 p. 28 cm

65 p. 28 cm

65 p. 28 cm

"The Calculus of Variations" is an article from The Annals of Mathematics, Volume 9 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967518

First Reprinting, 1963, Johnson Reprint Corporation

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

The biggest step from derivatives with one variable to derivatives with many variables is from one to two. After that, going from two to three was just more algebra and more complicated pictures. Now the step will be from a �nite number of variables to an in�nite number. That will require a new set of tools, yet in many ways the techniques are not very di_erent from those you know. If you�ve never read chapter 19 of volume II of the Feynman Lectures in Physics, now would be a good time. It�s a classic introduction to the area. For a deeper look at the subject, pick up MacCluer�s book referred to in the Bibliography at the beginning of this book.

Contents: The Lagrangian function and the parametric integrand Extremal curves; The Euler-Lagrange equation Lagrangian functions which are linear in x sub t The Legendre condition for a minimal curve Proof of the Legendre condition Constrained problems; The Hamilton canonical equations The reciprocity between L and H; The transversality conditions Extremal fields; The Hilbert invariant integral The Weierstrass E-function; Positively regular problems A simple example of the construction of an extremal field; Rayleigh quotients and the method of Rayleigh-Ritz The principle of maupertuis; The propagation of waves Problems whose Lagrangian functions involve derivatives of higher order than the first Multiple-Integral problems of the calculus of variations Constrained problems; Characteristic numbers Multiple-integral problems whose Lagrangian functions involve derivatives of higher order than the first The Courant maximum-minimum principle

Contents: The Lagrangian function and the parametric integrand Extremal curves; The Euler-Lagrange equation Lagrangian functions which are linear in x sub t The Legendre condition for a minimal curve Proof of the Legendre condition Constrained problems; The Hamilton canonical equations The reciprocity between L and H; The transversality conditions Extremal fields; The Hilbert invariant integral The Weierstrass E-function; Positively regular problems A simple example of the construction of an extremal field; Rayleigh quotients and the method of Rayleigh-Ritz The principle of maupertuis; The propagation of waves Problems whose Lagrangian functions involve derivatives of higher order than the first Multiple-Integral problems of the calculus of variations Constrained problems; Characteristic numbers Multiple-integral problems whose Lagrangian functions involve derivatives of higher order than the first The Courant maximum-minimum principle

Contents: The Lagrangian function and the parametric integrand Extremal curves; The Euler-Lagrange equation Lagrangian functions which are linear in x sub t The Legendre condition for a minimal curve Proof of the Legendre condition Constrained problems; The Hamilton canonical equations The reciprocity between L and H; The transversality conditions Extremal fields; The Hilbert invariant integral The Weierstrass E-function; Positively regular problems A simple example of the construction of an extremal field; Rayleigh quotients and the method of Rayleigh-Ritz The principle of maupertuis; The propagation of waves Problems whose Lagrangian functions involve derivatives of higher order than the first Multiple-Integral problems of the calculus of variations Constrained problems; Characteristic numbers Multiple-integral problems whose Lagrangian functions involve derivatives of higher order than the first The Courant maximum-minimum principle

xvi, 292 p. 27 cm

"Some Inverse Problems in the Calculus of Variations" is an article from The American Mathematical Monthly, Volume 20 . View more articles from The American Mathematical Monthly . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2972961

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

For a Hamiltonian $H \in C^2(\R^{N \by n})$ and a map $u:\Om \sub \R^n !\larrow \R^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,\Om) \ :=\ \big\|H(Du)\big\|_{L^\infty(\Om)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system [ \label{2} A_\infty u := \Big(H_P \ot H_P + H[H_P]^\bot ! H_{PP}\Big)(Du):D^2 u = 0. \tag{2}] \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author \cite{K1}-\cite{K6}. Herein we show that the Dirichlet problem for \eqref{2} admits for all $n=N\geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top ! P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ uniqueness theory of the scalar case has no counterpart when $N\geq 2$. The key idea in the proofs is to recast \eqref{2} as a first order differential inclusion $Du(x) \in \mK \sub \R^{n\by n}$, $x\in \Om$.

For a Hamiltonian $H \in C^2(\R^{N \by n})$ and a map $u:\Om \sub \R^n !\larrow \R^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,\Om) \ :=\ \big\|H(Du)\big\|_{L^\infty(\Om)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system [ \label{2} A_\infty u := \Big(H_P \ot H_P + H[H_P]^\bot ! H_{PP}\Big)(Du):D^2 u = 0. \tag{2}] \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author \cite{K1}-\cite{K6}. Herein we show that the Dirichlet problem for \eqref{2} admits for all $n=N\geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top ! P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ uniqueness theory of the scalar case has no counterpart when $N\geq 2$. The key idea in the proofs is to recast \eqref{2} as a first order differential inclusion $Du(x) \in \mK \sub \R^{n\by n}$, $x\in \Om$.

For a Hamiltonian $H \in C^2(\R^{N \by n})$ and a map $u:\Om \sub \R^n !\larrow \R^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,\Om) \ :=\ \big\|H(Du)\big\|_{L^\infty(\Om)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system [ \label{2} A_\infty u := \Big(H_P \ot H_P + H[H_P]^\bot ! H_{PP}\Big)(Du):D^2 u = 0. \tag{2}] \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author \cite{K1}-\cite{K6}. Herein we show that the Dirichlet problem for \eqref{2} admits for all $n=N\geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top ! P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ uniqueness theory of the scalar case has no counterpart when $N\geq 2$. The key idea in the proofs is to recast \eqref{2} as a first order differential inclusion $Du(x) \in \mK \sub \R^{n\by n}$, $x\in \Om$.

Translation of [Lekt︠s︡ii po variat︠s︡ionnomu ischislenii︠u︡]

In this paper, I'll derive the Hamilton-Jacobi (HJ) equation for Merton's problem in Utility Optimization Theory using a Calculus of Variations (CoV) Approach. For stochastic control problems, Dynamic Programming (DP) has been used as a standard method. To the best of my knowledge, no one has used CoV for this problem. In addition, while the DP approach cannot guarantee that the optimum satisfies the HJ equation, the CoV approach does. Be aware that this is the first draft of this paper and many flaws might be introduced.

We suggest a method of derivation of the long-wave action of the model spin Hamiltonians using the non-linear partial differential equations of motions of the individual spins. According to the Vainberg's theorem the set of these equations are (formal) potential if the symmetry analysis for the Frechet derivatives of the system is true. The case of Heisenberg (anti)ferromagnets is considered. It is shown the functional whose stationary points are described by the equations coincides with the long-wave action and includes the non-trivial topological term (Berry phase).

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann-Liouville fractional derivative. Using the Grunwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on mesh points. The problem is then transformed to an ordinary static optimization problem. The solution to the latter problem gives an approximation to the original fractional problem on mesh points.