Homogenous symplectic formulation of field dynamics and the poincaré-cartan form

We consider Noether symmetries within Hamiltonian setting as transformations that preserve Poincar?-Cartan form, i.e., as symmetries of characteristic line bundles of nondegenerate 1-forms. In the case when the Poincar?-Cartan form is contact, the explicit expression for the symmetries in the inverse Noether theorem is given. As examples, we consider natural mechanical systems, in particular the Kepler problem. Finally, we prove a variant of the theorem on complete (non-commutative) integrability in terms of Noether symmetries of time-dependent Hamiltonian systems.

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Hypercomplex Algebras and Calculi Derived from Generalized Kinematics

10.22541/au.162376453.35529708/v1 ◽

2021 ◽

Author(s):

Danail Brezov

Keyword(s):

Parallel Transport ◽

Holomorphic Functions ◽

Smooth Curves ◽

Commutative Algebras ◽

Cartan Form ◽

Physical Problems ◽

Analytical Properties ◽

Dual Extension ◽

Real Forms ◽

Complex Cases

The paper provides a study of the commutative algebras generated by iteration of the cross products in $\mathbb{C}^3$. Focusing on particular real forms we also consider the analytical properties of the corresponding rings of functions and relate them to different physical problems. Familiar results from the theory of holomorphic and bi-holomorphic functions appear naturally in this context, but new types of hypercomplex calculi emerge as well. The parallel transport along smooth curves in $\mathbb{E}^3$ and the associated Maurer-Cartan form are also studied with examples from kinematics and electrodynamics. Finally, the dual extension is discussed in the context of screw calculus and Galilean mechanics; a similar construction is studied also in the multi-dimensional real and complex cases.

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SUB-WEYL GEOMETRY AND ITS LINEAR CONNECTIONS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812500478 ◽

2012 ◽

Vol 09 (05) ◽

pp. 1250047 ◽

Cited By ~ 3

Author(s):

OANA CONSTANTINESCU ◽

MIRCEA CRASMAREANU

Keyword(s):

Tangent Bundle ◽

Natural Extension ◽

Conformal Structure ◽

Point Of View ◽

Dimensional Manifold ◽

Finsler Spaces ◽

Weyl Geometry ◽

Cartan Form ◽

Linear Connections ◽

Local Expression

The aim of this paper is to study from the point of view of linear connections the data [Formula: see text] with M a smooth (n+p)-dimensional real manifold, [Formula: see text] an n-dimensional manifold semi-Riemannian distribution on M, [Formula: see text] the conformal structure generated by g and W a Weyl substructure: a map [Formula: see text] such that W(ḡ) = W(g) - du, ḡ = eug;u ∈ C∞(M). Compatible linear connections are introduced as a natural extension of similar notions from Weyl geometry and such a connection is unique if a symmetry condition is imposed. In the foliated case the local expression of this unique connection is obtained. The notion of Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundle projection and as one-form the Cartan form.

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