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Applicable Differential Geometry
The background: vector calculus 1. Affine spaces 2. Curves, functions and derivatives 3. Vector fields and flows 4. Volumes and subspaces: exterior algebra 5. Calculus of forms 6. Frobenius's theorem
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
Higher-order differential equations and higher-order lagrangian mechanics
On considere le role du champ d'equation differentielle d'ordre eleve Γ qui definit la dynamique. C'est un champ vectoriel sur T N M. On etudie la methode pour definir un lagrangien
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
The Hilbert-Caratheodory and Poincare-Cartan forms for higher-order multiple-integral variational problems
We consider higher-order homogeneous multiple-integral variational problems defined in the context of m-frame bundles, and construct an m-form (the Hilbert-Caratheodory form) with the same extremals
Cartan Geometries and their Symmetries: A Lie Algebroid Approach
In this book we first review the ideas of Lie groupoid and Lie algebroid, and the associated concepts of connection. We next consider Lie groupoids of fibre morphisms of a fibre bundle, and the
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