A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

https://core.ac.uk/download/pdf/11469435.pdf

Hamiltonian formulation of distributed-parameter systems with boundary energy flow

Preface. 1. The geometry of tangent bundle. 2. Finsler spaces. 3. Lagrange spaces. 4. The geometry of cotangent bundle. 5. Hamilton spaces. 6. Cartan spaces. 7. The duality between Lagrange and Hamilton spaces. 8. Symplectic transformations of the differential geometry of T* M. 9. The dual bundle of a k-osculator bundle. 10. Linear connections on the manifold T*2M. 11. Generalized Hamilton spaces of order 2. 12. Hamilton spaces of order 2. 13. Cartan spaces of order 2. Bibliography. Index.

The Geometry of Hamilton and Lagrange Spaces

Two types of nonholonomic systems with symmetry are treated: (i) the configuration space is a total space of a G-principal bundle and the constraints are given by a connection; (ii) the configuration space is G itself and the constraints are given by left-invariant forms. The proofs are based on the method of quasicoordinates. In passing, a derivation of the Maurer-Cartan equations for Lie groups is obtained. Simple examples are given to illustrate the algorithmical character of the main results.

Reduction of some classical non-holonomic systems with symmetry

The inverse problem in the calculus of variations and the geometry of the tangent bundle

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 Noether's 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.

On the multisymplectic formalism for first order field theories

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

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

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

We consider multiple-integral variational problems where the Lagrangian function, defined on a frame bundle, is homogeneous. We construct, on the corresponding sphere bundle, a canonical Lagrangian form with the property that it is closed exactly when the Lagrangian is null. We also provide a straightforward characterization of null Lagrangians as sums of determinants of total derivatives. We describe the correspondence between Lagrangians on frame bundles and those on jet bundles: under this correspondence, the canonical Lagrangian form becomes the fundamental Lepage equivalent. We also use this correspondence to show that, for a single-determinant null Lagrangian, the fundamental Lepage equivalent and the Caratheodory form are identical.

Michael Crampin

Papers

On the multisymplectic formalism for first order field theories

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

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

On null Lagrangians

Reduction of degenerate Lagrangian systems