An imbedding theorem in the calculus of variations for multiple integrals,addendum

It is shown how a set of canonical variables in the sense of Rund [5] can be associated with a given extremal of a multiple integral variational problem in a simple, direct manner. The definition of these variables in a previous paper [1], which is concerned with the problem of imbedding a given extremal in anr-geodesic field, is thereby clarified and abbreviated considerably. A theorem due essentially to Boerner, which is crucial to the imbedding theorem given in [1], is proved more easily and under less restrictive hypotheses than in [1]. Furthermore, it is shown how the present definition of the canonical variables allows one to eliminate from the geodesic field theory of Carathéodory the restriction that the Lagrangian be non-vanishing along the extremal.     

An existence theorem in the calculus of variations

In this paper, a nonparametric variational problem is considered in the setting of the theory of generalized curves. It is assumed that the integrand of the problem does not grow at infinity faster than the norm of the variable , for all values of the other variablest andx (which take their values in a compact product set). It is shown that a generalized curve exists such that the minimum of the functional over an appropriate set is achieved. This generalized curve does not in general have compact support.     

The Hessian and Jacobi morphisms for higher order calculus of variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler-Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.     

A formulation of Noether's theorem for fractional problems of the calculus of variations

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.     

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.     

The complete figure for second order multiple integral problems in the calculus of variations

Notation Throughout this paper Greek indices, , , and Latin indicesi, j, h, k, assume the values 1, ,m, and 1, ,n respectively. The summation convention is operative in respect of both sets of indices.This work was supported by the South African Council for Scientific and Industrial Research.At time of writing Professor Grässer was Visiting Scholar at the University of Arizona, Tucson, Arizona.     

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.     

Optimality conditions for discrete calculus of variations problems

Nonlinear discrete calculus of variations problems with variable endpoints and with equality type constraints on trajectories are considered. We derive new nontrivial first- and second-order necessary optimality conditions.