This book studies a category of mathematical objects called Hamiltonians, which are dependent on both time and momenta. The authors address the development of the distinguished geometrization on dual 1-jet spaces for time-dependent Hamiltonians, in contrast with the time-independent variant on cotangent bundles. Two parts are presented to include both geometrical theory and the applicative models: Part One: Time-dependent Hamilton Geometry and Part Two: Applications to Dynamical Systems, Economy and Theoretical Physics. The authors present 1-jet spaces and their duals as appropriate fundamental ambient mathematical spaces used to model classical and quantum field theories. In addition, the authors present dual jet Hamilton geometry as a distinct metrical approach to various interdisciplinary problems.
Author(s): Mircea Neagu, Alexandru Oană
Series: Synthesis Lectures on Mathematics & Statistics
Publisher: Springer
Year: 2022
Language: English
Pages: 90
City: Cham
Preface
Contents
Part I Time-Dependent Hamilton Geometry
1 The Dual 1-Jet Space upper E Superscript asterisk Baseline equals upper J Superscript 1 asterisk Baseline left parenthesis double struck upper R comma upper M right parenthesisEast=J1ast(mathbbR,M)
1.1 Dual Jet Geometrical Objects of Momenta
1.2 Time-Dependent Semisprays of Momenta
1.3 Nonlinear Connections and Adapted Bases
2 upper NN-Linear Connections
2.1 Local Adapted Components
2.2 Torsion dd-Tensors
2.3 Curvature dd-Tensors
3 hh-Normal upper NN-Linear Connections
3.1 Local Adapted Components
3.2 Torsion and Curvature dd-Tensors
3.3 Ricci and Deflection dd-Tensor Identities
3.4 Local Bianchi Identities
4 Distinguished Geometrization of the Time-Dependent Hamiltonians of Momenta
4.1 Time-Dependent Hamiltonians of Momenta
4.2 Canonical Nonlinear Connections on upper H Superscript nHn-Spaces
4.3 Cartan Canonical Connection in upper H Superscript nHn-Spaces
4.4 dd-Torsions and dd-Curvatures
4.5 Momentum Field-Like Geometrical Models
4.5.1 Geometrical Momentum Maxwell-Like Equations
4.5.2 Geometrical Momentum Einstein-Like Equations
Part II Applications to Dynamical Systems, Economy and Theoretical Physics
5 The Time-Dependent Hamiltonian of the Least Squares Variational Method
5.1 Hamiltonian d minusdd-Torsions and d minusdd-Curvatures of a Dynamical System
5.2 Hamilton Geometrization of an Economy Dynamical System
6 Time-Dependent Hamiltonian of Electrodynamics
6.1 Introduction
6.2 The Time-Dependent Hamilton Space of Electrodynamics
6.3 Momentum Electromagnetic-Like Geometrical Model
6.4 Momentum Gravitational-Like Geometrical Model
7 The Geometry of Conformal Hamiltonian of the Time-Dependent Coupled Harmonic Oscillators
7.1 Introduction
7.2 The Canonical Nonlinear Connection
7.3 Cartan Canonical Connection. d minusdd-Torsions and d minusdd-Curvatures
7.4 From Hamiltonian of Time-Dependent Coupled Oscillators to Field-Like Geometrical Models
7.4.1 Gravitational-Like Geometrical Model of Momenta
7.4.2 Geometrical Momentum Electromagnetic-Like 2-Form
8 On the Dual Jet Conformal Minkowski Hamiltonian
8.1 Introduction
8.2 The Canonical Nonlinear Connection
8.3 Cartan Canonical Connection. d minusdd-Torsions and d minusdd-Curvatures
8.4 Momentum Field-Like Geometrical Models
8.4.1 The Gravitational-Like Geometrical Model
8.4.2 The Electromagnetic-Like Geometrical Model
References
Index
