Willy Sarlet

GENERALIZATIONS OF NOETHER'S THEOREM IN CLASSICAL MECHANICS*

In this paper, a review is presented of various approaches to the generalization of the version of Noether’s theorem, which is presented in most textbooks on classical mechanics. Its motivation is the controversy still persisting around the possible scope of a Noether-type theorem allowing for velocity-dependent transformations. Our analysis is centered around the one factor common to all known treatments, namely the structure of the related first integral. We first discuss the most general framework, in which a function of the above-mentioned structure constitutes a first integral of a given Lagrangian system, and show that one cannot really talk about an “interrelationship” between symmetries and first integrals there. We then compare different proposed generalizations of Noether’s theorem, by describing the nature of the restrictions which characterize them, when they are situated within the broadest framework. We prove a seemingly new equivalence-result between the two main approaches: that of invaria...

Derivations of differential forms along the tangent bundle projection II

We study derivations of the algebra of differential forms along the tangent bundle projection τ : TM → M and of the module of vector-valued forms along τ . It is shown that a satisfactory classification and characterization of such derivations requires the extra availability of a connection on TM . The present theory completely explains and generalizes the calculus of forms associated to a given second-order vector field, which was previously introduced by one of us.

Towards a geometrical understanding of Douglas's solution of the inverse problem of the calculus of variations

We describe a novel approach to the study of the inverse problem of the calculus of variations, which gives new insights into Douglass solution (1941) of the two degree of freedom case.

Lie algebroid structures and Lagrangian systems on affine bundles

Abstract As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.

A class of nonconservative Lagrangian systems on Riemannian manifolds

We generalize results of Rauch-Wojciechowski, Marciniak and Lundmark, concerning a class of nonconservative Lagrangian systems, from the Euclidean to the Riemannian case.

Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors

The theory of Dimakis and M?ller-Hoissen (Dimakis?A and M?ller-Hoissen?F 2000 J. Phys. A: Math. Gen. 33 957-74) concerning bi-differential calculi and completely integrable systems is related to bi-Hamiltonian systems of the Poisson-Nijenhuis type. In the special case where the ambient manifold is a cotangent bundle one is able to recover and elucidate the theory of Ibort et al (Ibort?A, Magri?F and Marmo?G 2000 J. Geom. Phys. 33 210-23), which is in turn a reworking in the bi-Hamitonian context of Benentis theory of Hamilton-Jacobi separable systems. In particular, it is shown that Benentis conformal Killing tensor, which is central to his theory, has an even more special form than has hitherto been realized and that when it is converted into a field of endomorphisms by raising an index with the ambient metric, it necessarily has vanishing Nijenhuis torsion.

Derivations of forms along a map: the framework for time-dependent second-order equations

Abstract A comprehensive theory is presented concerning derivations of scalar and vector-valued forms along the projection π : R × TM → R × M . It is the continuation of previous work on derivations of forms along the tangent bundle projection and is prompted by the need for a scheme which is adapted to the study of time-dependent second-order equations. The overall structure of the theory closely follows the pattern of this preceding work, but there are many features which are certainly not trivial transcripts of the time-independent situation. As before, a crucial ingredient in the classification of derivations is a non-linear connection on the bundle π. In the presence of a given second-order system, such a connection is canonically defined and gives rise to two important operations: the dynamical covariant derivative, which is a derivation of degree 0, and the Jacobi endomorphism, which is a type (1, 1) tensor field along π. The theory is developed in such a way that all results readily apply to the more general situation of a bundle π : J 1 E → E , where E is fibred over R, but need not be the trivial fibration R × M → R .

Addendum to: The Integrability Conditions in the Inverse Problem of the Calculus of Variations for Second-Order Ordinary Differential Equations

A novel approach to a coordinate-free analysis of the multiplier question in the inverseproblem of the calculus of variations, initiated in a previous publication, is completed in thefollowing sense: under quite general circumstances, the complete set of passivity or integrabilityconditions is computed for systems with arbitrary dimension n. The results are appliedto prove that the problem is always solvable in the case that the Jacobi endomorphism of thesystem is a multiple of the identity. This generalizes to arbitrary n a result derived byDouglas for n=2.

The inverse problem of the calculus of variations: The use of geometrical calculus in Douglas’s analysis

The main objective of this paper is to work out a full-scale application of the integrability analysis of the inverse problem of the calculus of variations, as developed in recent papers by Sarlet and Crampin. For this purpose, the celebrated work of Douglas on systems with two degrees of freedom is taken as the reference model. It is shown that the coordinate-free, geometrical calculus used in Sarlet and Crampins general theoretical developments provides effective tools also to do the practical calculations. The result is not only that all subcases distinguished by Douglas can be given a more intrinsic characterization, but also that in most of the cases, the calculations can be carried out in a more efficient way and often lead to sharper conclusions.

Adjoint symmetries for time-dependent second-order equations

The authors extend part of their previous work on autonomous second-order systems (Sarlet et al., 1987) to time-dependent differential equations. The main subject of the paper concerns the notion of adjoint symmetries: they are introduced as a particular type of 1-form, whose leading coefficients satisfy the adjoint equations of the equations determining symmetry vector fields. It is shown that all interesting properties of adjoint symmetries, known from the autonomous theory, have their counterparts in the present framework. Of particular interest is a result establishing that Lagrangian systems seem to be the only ones for which there is a natural duality between symmetries and adjoint symmetries. A number of examples illustrate how the construction of adjoint symmetries of a given system can be explored in a systematic way.