Michael Crampin

On the multisymplectic formalism for first order field theories

Abstract The general purpose of this paper is to attempt to clarify the geometrical foundations of first order Lagrangian and Hamiltonian field theories by introducing in a systematic way multisymplectic manifolds, the field theoretical analogues of the symplectic structures used in geometrical mechanics. Much of the confusion surrounding such terms as gauge transformation and symmetry transformation as they are used in the context of Lagrangian theory is thereby eliminated, as we show. We discuss Noethers theorem for general symmetries of Lagrangian and Hamiltonian field theories. The cohomology associated to a group of symmetries of Hamiltonian or Lagrangian field theories is constructed and its relation with the structure of the current algebra is made apparent.

Tangent bundle geometry Lagrangian dynamics

Various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined, and the results applied to time-independent Lagrangian dynamics. It is shown that a certain type (1, 1) tensor field which is part of the intrinsic geometry of a tangent bundle, being a tensorial equivalent of the projection map of tangent vectors, plays a role in Lagrangian theory scarcely less important than that of the canonical one-form on a cotangent bundle in Hamiltonian theory. Recent results in Lagrangian theory are interpreted from this new viewpoint.

On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics

The conditions for a system of second-order differential equations to be derivable from a Lagrangian-the conditions of self-adjointness, in the terminology of Santilli (1978) and others-are related, in the time-independent case, to the differential geometry of the tangent bundle of configuration space. These conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second-order differential equations. Necessary and sufficient conditions for such a vector field to be derivable from a Lagrangian may be stated as the existence of a two-form with certain properties: it is interesting that it is a deduction, not an assumption, that this two-form is closed and thus defines a symplectic structure. Some other differential geometric properties of Euler-Lagrange second-order differential equations are described.

A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics

Appropriate geometrical machinery for the study of time-dependent Lagrangian dynamics is developed. It is applied to the inverse problem of the calculus of variations, and a set of necessary and sufficient conditions for the existence of a Lagrangian are given, in terms of the existence of a 2-form with suitable properties, which are exactly equivalent to the Helmholtz conditions.

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.

Reduction of degenerate Lagrangian systems

Abstract The geometrical structure of (finite dimensional) degenerate Lagrangian systems is studied and a reduction scheme, leading to a regular Lagrangian description of these systems on a reduced velocity phase space, is developed. The connection with the canonical approach to the regularization problem of degenerate systems (Diracs theory) and the reduction of systems with symmetry (Marsden-Weinstein theory) is investigated. Some examples and applications are discussed.

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.

On the Legendre map in higher-order field theories

The authors show how the construction of a Cartan form in higher-order field theories defines a Legendre map, and how the regularity of this map may be described in terms of a sequence of maps between intermediate phase spaces. They also show how semi-holonomic jets are important in this context: they allow the definition of regularity to be applied without ambiguity to the Lagrangian itself, and they permit the specification of a unique Cartan form, Legendre map and covariant phase space for second-order Lagrangians.

Solitons and SL(2, R)

Abstract The known relationship between non-linear partial differential equations which have soliton solutions, and SL (2, R), is developed to the point where it provides a framework for discussing Backlund transformations, and equations for the inverse scattering method.

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.