The Hamilton formalism with fractional derivatives

Recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton's equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, the results are obtained to be in exact agreement with Agrawal's formalism.     

Canonical formalism for parameter-invariant integrals in the Calculus of Variations whose Lagrange functions involve second order derivatives

Summary In a recent paper [4] a general theory of parameter-invariant integrals in the Calculus of Variations whose Lagrangians involve higher derivatives was developed, and in particular a certain canonical formalism for such problems was discussed. From the point of view of applications it was found that this formalism proved inadequate inas-much as the suggested Hamiltonian function did not depend explicitly on the first derivatives of the positional coordinates. In the present note an alternative Hamiltonian function is defined, which gives rise to a new canonical formalism. The latter is less complicated than the formalism suggested in [4] and is more readily applicable to special problems. A brief discussion of the resulting Hamilton-Jacobi theory is given, and in conclusion the method is illustrated explicitly by means of an example of fairly general character.     

On variational principles in which the Lagrangian function involves third order derivatives

Summary Recently variational principles whose Lagrangian functions involve third order derivatives of the position vector have been considered with a view to applying them to certain aspects of elementary particle theory. It is known that definite consistency conditions arise when parameter invariance of the associated action integral is required. By invoking two additional assumptions, which have been adopted in the past, a general parameter invariant Lagrangian is deduced. The structure of the corresponding momentum vector is investigated. It is shown that a large class of the resulting equations of motion are derivable from a different approach due to Rund. Borelowski's equations of motion are also derived. Although no radiation effects are considered the parameter invariant counterpart of the Abraham vector plays an important role in the theory.     

ADJOINT PROBLEMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

Abstract

A general theory of adjoint variational problems is formulated for essentially arbitrary Lagrangians involving m independent and n dependent variables, together with the first derivatives of the latter, This approach contains as a special case the theory of Haar [4], in which the Lagrangian may depend solely on the derivatives of a single dependent function of two arguments. Because of the eventual occurrence of possibly incompatible sets of integrability conditions, the basic theory is developed against the background of non-integrable m-dimensional subspaces, which is in sharp contrast to the traditional approach to the calculus of variations. Relatively self-adjoint Lagrangians are defined and completely characterized in terms of an arbitrary Riemannian metric. In the course of the general theory certain geometric object fields are encountered in a very natural manner, some of which had arisen previously in the canonical formalism proposed by Caratheodory [2]. Accordingly the analysis of the present paper may serve to shed some light on this conceptually extremely difficult formalism.     

Fractional Noether’s theorem in the Riesz-Caputo sense

We prove a Noether’s theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.     

Hamiltonian stationary tori in K?hler manifolds

A Hamiltonian stationary Lagrangian submanifold of a K?hler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a K?hler manifold of real dimension four to guarantee the existence of a family of small Hamiltonian stationary Lagrangian tori.     

Non-differentiable embedding of Lagrangian systems and partial differential equations

We develop the non-differentiable embedding theory of differential operators and Lagrangian systems using a new operator on non-differentiable functions. We then construct the corresponding calculus of variations and we derive the associated non-differentiable Euler-Lagrange equation, and apply this formalism to the study of PDEs. First, we extend the characteristics method to the non-differentiable case. We prove that non-differentiable characteristics for the Navier-Stokes equation correspond to extremals of an explicit non-differentiable Lagrangian system. Second, we prove that the solutions of the Schrödinger equation are non-differentiable extremals of the Newton?s Lagrangian.     

Hamiltonians and conjugate Hamiltonians of some fourth-order nonlinear ODEs

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels [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], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi-Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painlevé test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.     

A new sufficiency theorem for strong minima in the calculus of variations

Sufficiency for strong local optimality in the calculus of variations involves, in the classical theory, the strengthened condition of Weierstrass. A proof of sufficiency for strong minima, modifying this condition under certain uniform continuity assumptions on the functions delimiting the problem, is presented. The proof is direct in nature as it makes no use of fields, Hamilton–Jacobi theory, Riccati equations or conjugate points. Some examples illustrate clear advantages of the new sufficient condition over the classical one.     

Fractional variational principles and their applications

Variational calculus and fractional calculus have played a significant role in various areas of applied sciences such as, among others, Physics, Engineering and Economics. This topic is deeply connected to the very recent developments in theoretical aspects and especially in the numerical schemes of fractional differential equations. Based on 1+1 field formalism, a new fractional Lagrangian and Hamiltonian formalisms are presented within the Riemann-Liouville fractional derivatives and the an-harmonic oscillator is analyzed. This formalism can be applied to analyze the control problems as well as for the fractional quantization procedure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An iterated graph construction and periodic orbits of Hamiltonian delay equations

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.     

On the nature of the de Branges Hamiltonian

We prove the theorem announced by the author in 1995 in the paper “A criterion for the discreteness of the spectrum of a singular canonical system” (Funkts. Anal. Prilozhen., 29, No. 3).In developing the theory of Hilbert spaces of entire functions (we call them Krein-de Branges spaces), de Branges arrived at a certain class of canonical equations of phase dimension 2. He showed that, for any given Krein-de Branges space, there exists a canonical equation of the class indicated that restores a chain of Krein-de Branges spaces imbedded one into another. The Hamiltonians of such canonical equations are called de Branges Hamiltonians. The following question arises: Under what conditions will the Hamiltonian of a certain canonical equation be a de Branges Hamiltonian? The main theorem of the present work, together with Theorem 1 of the paper cited above, gives an answer to this question.     

THE CONSTRUCTION OF THE COMPLETE FIGURE IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS: A DIRECT APPROACH

Abstract

The field-theoretic approach to the calculus of variations of multiple integrals, as initiated by Carathéodory, is based on the systematic use of certain field components that exist if and only if a determinant whose entries are defined uniquely in terms of the Lagrangian and its derivatives does not vanish. If one does not introduce an assumption of this kind ab initio, entirely different techniques must be adopted. It is shown that these give rise to a surprisingly simple and direct construction of the complete figure by means of which the solutions of the standard variational problem may be realized.     

Canonical transformations associated with second order problems in the calculus of variations

Summary Our object is a systematic investigation of some of the properties of canonical transformations associated with second order problems in the calculus of variations. After the introduction of such transformations, together with the related concepts of Lagrange and Poisson brackets, the bracket relationships are found which characterize canonical transformations. This characterization is also achieved by means of so-called reciprocity relations between the original transformation and its inverse (which always exists). The effect of the canonical transformation on the underlying variational problem is discussed. It is also shown that the Jacobian of such a transformation always has the value unity. The special case when the canonical transformation is independent of the parameter (a generalization of the so-called time-independent canonical transformation of mechanics) is treated in some detail. Finally it is indicated how the present theory can be extended to problems of higher order. Some of the results of this paper are contained in a doctoral thesis ([2]) which was presented to the University of South Africa. The writer wishes to express his gratitude to his supervisor, ProfessorH. Rund, for his interest, encouragement and advice concerning this work.     

Investigation of variational problems by direct methods

A direct method is proposed for solving variational problems in which an extremal is represented by an infinite series in terms of a complete system of basis functions. Taking into account the boundary conditions gives all the necessary conditions of the classical calculus of variations, that is, the Euler-Lagrange equations, transversality conditions, Erdmann-Weierstrass conditions, etc. The penalty function method reduces conditional extremum problems to variational ones in which the isoperimetric conditions described by constraint equations are taken into account by Lagrangian multipliers. The direct method proposed is applied to functionals depending on functions of one or two variables.     

Beyond polyconvexity: an existence result for a class of quasiconvex hyperelastic materials

This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non‐interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split‐quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.     

Stochastic differential equations

When a system is acted upon by exterior disturbances, its time-development can often be described by a system of ordinary differential equations, provided that the disturbances are smooth functions. But for sound reasons physicists and engineers usually want the theory to apply when the noises belong to a larger class, including for example “white noise.” If the integrals in the system derived for smooth noises are reinterpreted as Itô integrals, the equations make sense; but in nonlinear cases they often fail to describe the time-development of the system. In this paper (extending previous work of the author) a calculus is set up for stochastic systems that extends to a theory of differential equations. When the equations are known that describe the development of the system when noises are smooth, an extension to the larger class of noises is proposed that in many cases gives results consistent with the smooth-noise case and also has “robust” solutions, that change by small amounts when the noises undergo small changes. This is called the “canonical” extension.Nevertheless, there are certain systems in which the canonical equations are inappropriate. A criterion is suggested that may allow us to distinguish when the canonical equations are the right choice and when they are not.     

Differential properties of a generalized solution to a hyperbolic system of first-order differential equations

We study some questions of the qualitative theory of differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations whose right-hand sides contain some discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. In particular, its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts the common approach which uses the reduction of a system of two first-order equations to a single second-order equation.