Mechanics Systems on Para-Kaehlerian Manifolds of Constant J-Sectional Curvature (original) (raw)
Differential geometry and mathematical physics deal with metrical notions on manifolds that it began as the study of curves and surfaces using the methods of calculus. The most important class of Hermitian manifolds are Kähler manifolds. An almost Hermitian manifold whose associated (1,1)-form is closed is naturally called an almost Kähler manifold. A geodesic is the shortest route between two points on the Earth's surface and it is called as a curve. A geodesic is a generalization of the notion of a straight line to curved spaces at differential geometry. Classical mechanics has a large working area on geodesics. On the other hand, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange and the Hamilton equations. In this study, we obtained dynamics equations by using the Euler-Lagrange and the Hamilton mechanical equations as a representive of the object motion on an almost Kähler manifolds for geodesics. Also, implicit solutions of the differential equations found in this study are solved by Maple computation program and the geodesic graphs are drawn.
Geometrization of the scleronomic Riemannian mechanical systems
The paper is devoted to the geometrical theory, on the phase space, of the classical concept of scleronomic Riemannian mechanical systems in the general case when the external forces depend on the material points and their velocities. We discuss the canonical semispray, the nonlinear connection, the metrical connection, the electromagnetic field and the almost Hermitian model of the mentioned mechanical system. Based on the methods of Lagrange geometry we prepare here the framework for the investigation of the geometrical theory of Riemannian mechanical systems whose external forces depend on the accelerations of order k ≥ 1.
Mechanical Systems on Almost Para/Pseudo-Kähler–Weyl Manifolds
International Journal of Geometric Methods in Modern Physics, 2013
The paper aims to introduce Lagrangian and Hamiltonian formalism for mechanical systems using para/pseudo-Kähler manifolds, representing an interesting multidisciplinary field of research. Moreover, the geometrical, relativistical, mechanical and physical results related to para/pseudo-Kähler mechanical systems are given, too.
Hamiltonian Equations of Kähler-Einstein Manifolds with Equal Kähler Angles
The paper aims to introduce Hamiltonian formalism for mechanical systems using Kähler-Einstein manifolds with equal Kähler angles, which represent an interesting multidisciplinary field of research. Also, solutions of these equations will be made using the computer program Maple and the geometrical-physical results related to on Kähler-Einstein mechanical systems are also to be issued.
The main purpose of the present paper is to study almost complex structures Euler-Lagrangian Equations on Walker 4-manifold with Walker metric. In this study, routes of bodies moving in space will be modeled mathematically Euler-Lagrangian Equations on Walker 4-manifold with Walker metric that these are time- dependent partial di¤erential equations. Here we present complex analogues of Euler-Lagrangian Equations on Walker 4-manifold with Walker metric. Also, the geometrical-physical results related to complex mechan- ical systems are also discussed for Euler-Lagrangian Equations on Walker 4-manifold with Walker metric for dynamical systems and solution of the motion equations using symbolic computational program will be made.
We consider Euler-Lagrange equations on almost paracontact manifolds. It is well known that the geometry of almost contact manifolds is a natural extension in the odd dimensional case of almost Hermitian geometry. A Hermitian manifold is a complex manifold with a Hermitian metric on its holomorphic tangent space. Also, the contact geometry as symplectic geometry has large and comprehensive applications in physics, geometrical optics, classical mechanics, thermodynamics, geometric quantization, and applied mathematics. On the other hand, one way of solving problems in classical and analytical mechanics is through use of the Euler-Lagrange equations. The purpose of the present paper is to solve the problems of classical mechanics with 3-dimensional real number space on an almost paracontact manifold by using Euler-Lagrange equations. ZEKI KASAP 2
This article relates to the equations of moving objects in space. Hence Lagrangian formalism of mechanical systems on Flat manifolds that represent an interesting multidisciplinary field of research. We, as a result modeling obtained of partial differential equations, have be solved by the symbolic computational program. Also, the geometrical-physical results related to on Flat manifolds of mechanical systems.
Hamilton Mechanical Equations with Four Almost Complex Structures on Riemannian Geometry
Abstract: The study concerns the Hamilton Mechanical Equations with Four Almost Complex Structures on Riemannian geometry. The four complex structures have been derived also important applications of Hamiltonian mechanical systems of the notions of Riemannian mentioned. Finally achieved that Almost complex structures have this systems in Mechanics and Physical Fields as well as in differential geometry
A geometrical approach to the Hamilton-Jacobi form of dynamics and its generalizations
La Rivista del Nuovo Cimento, 1990
Introduction. 2. Geometrical background --Lagrangian structures. 2"1. Subspaces of a symplectic linear space. 2"2. Submanifolds of a symplectic manifold, descriptions using constraints. 2"3. Mappings and Lagrangian submanifolds. 2"4. Transversality and caustics. 2"5. Closed-form graphs as Lagrangian submanifolds. 2"6. The Poisson bracket and the restriction operation. 3. Action of canonical transformations on Lagrangian structures. 4. Time-independent Hamilton~lacobi theory. 4"1. Conventional Hamflton-Jacobi theory and the complete integral in geometrical form. 4"2. Covariance group and implementable symmetries for the Hamilton-Jacobi equation. 4"3. Examples and limitations of conventional Hamilton-Jacobi theory. 4"4. The generalized Hamilton~acobi problem. Simultaneous systems of Hamilton~lacobi problems. 5. Time-dependent Hamilton~lacobi theory. 5"1. Extended phase space, time-dependent Hamiltonian and symmetries. 5"2. Conventional time-dependent Hamilton~lacobi theory in geometrical form. Implementable symmetries. 5"3. Generalized time-dependent Hamilton~lacobi problem. 6. The Hamilton~lacobi problem on a Lie-group. 6"1. Canonical G action on (J~, ~). (*) Jawaharlal Nehru Fellow. 2 G. MARMO, G. MORANDI and N. MUKUNDA 63 65 68 7. 71 8.