scholarly journals Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967–3034]

2011 ◽  
Vol 261 (2) ◽  
pp. 542-589 ◽  
Author(s):  
Guangcun Lu
Keyword(s):  
Hamiltonian Systems ◽  
Cotangent Bundle ◽  
Lagrangian Systems ◽  
Conley Conjecture ◽  
Funct Anal

scholarly journals Non-Smooth Geodesic Flows and Classical Mechanics

1969 ◽  
Vol 12 (2) ◽  
pp. 209-212 ◽  
Author(s):  
J. E. Marsden
Keyword(s):  
Phase Space ◽  
Kinetic Energy ◽  
Hamiltonian Systems ◽  
Configuration Space ◽  
Smooth Function ◽  
Cotangent Bundle ◽  
Classical Mechanics ◽  
Riemannian Metric ◽  
Geodesic Flows ◽  
Image Position

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by

scholarly journals Symmetries, pseudosymmetries and conservation laws in Lagrangian and Hamiltonian k-symplectic formalisms

2016 ◽  
Vol 13 (03) ◽  
pp. 1650026
Author(s):  
Florian Munteanu
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Systems ◽  
Conservation Law ◽  
Type Theorem ◽  
Natural Extension ◽  
Lagrangian Systems ◽  
Symmetry And Conservation Law

In this paper, we will present Lagrangian and Hamiltonian [Formula: see text]-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will obtain new kinds of conservation laws for [Formula: see text]-symplectic Hamiltonian systems and [Formula: see text]-symplectic Lagrangian systems.

On the number of caustics for invariant tori of Hamiltonian systems with two degrees of freedom

1991 ◽  
Vol 11 (2) ◽  
pp. 273-278 ◽  
Author(s):  
Misha Bialy
Keyword(s):  
Hamiltonian Systems ◽  
Cotangent Bundle ◽  
Degrees Of Freedom ◽  
Invariant Tori ◽  
Hamiltonian Function ◽  
Canonical Projection ◽  
Classical Type ◽  
Two Dimensional ◽  
Hamiltonian Flow ◽  
Connected Manifold

Let X be a two-dimensional orientable connected manifold without boundary, H: T*X → ℝ a smooth hamiltonian function denned on the cotangent bundle. We will assume that H is of a ‘classical type’ that is convex and even on each fibre Tx*X. The goal of this paper is to describe the set Γ of all singular points of the projection Θ|L where ι: L → T*X is a smooth embedded 2-torus invariant under the hamiltonian flow h1, Θ: T*X → X is the canonical projection.

scholarly journals ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS

1995 ◽  
Vol 10 (04) ◽  
pp. 579-610 ◽  
Author(s):  
V. MUKHANOV ◽  
A. WIPF
Keyword(s):  
String Theory ◽  
Hamiltonian Systems ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Linear Constraints ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Diffeomorphism Invariance ◽  
Symmetry Transformations ◽  
Local Symmetries

In this paper we show how the well-known local symmetries of Lagrangian systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangian system. The non-linear constraints (which we have, for instance, in gravity, supergravity and string theory) generate the dynamics of the corresponding Lagrangian system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We show the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems, in particular those which are diffeomorphism-invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian and Lagrangian formalisms is found. The possible applications of our results are discussed.

scholarly journals Singular Lagrangians and precontact Hamiltonian systems

2019 ◽  
Vol 16 (10) ◽  
pp. 1950158 ◽  
Author(s):  
Manuel de León ◽  
Manuel Lainz Valcázar
Keyword(s):  
Hamiltonian Systems ◽  
Contact Geometry ◽  
Lagrangian Systems ◽  
Dirac Bracket ◽  
Jacobi Bracket ◽  
Singular Lagrangians

In this paper, we discuss the singular Lagrangian systems on the framework of contact geometry. These systems exhibit a dissipative behavior in contrast with the symplectic scenario. We develop a constraint algorithm similar to the presymplectic one studied by Gotay and Nester (the geometrization of the well-known Dirac–Bergmann algorithm). We also construct the Hamiltonian counterpart and prove the equivalence with the Lagrangian side. A Dirac–Jacobi bracket is constructed similar to the Dirac bracket.

scholarly journals GEOMETRIC HAMILTON–JACOBI THEORY

2006 ◽  
Vol 03 (07) ◽  
pp. 1417-1458 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
XAVIER GRÀCIA ◽  
GIUSEPPE MARMO ◽  
EDUARDO MARTÍNEZ ◽  
MIGUEL C. MUÑOZ-LECANDA ◽  
...  
Keyword(s):  
Rigid Body ◽  
Tangent Bundle ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Relativistic Particle ◽  
Lagrangian Systems ◽  
Singular Lagrangians

The Hamilton–Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron–monopole system.

scholarly journals The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

2017 ◽  
Vol 17 (1) ◽  
Author(s):  
Alberto Abbondandolo ◽  
Luca Asselle ◽  
Gabriele Benedetti ◽  
Marco Mazzucchelli ◽  
Iskander A. Taimanov
Keyword(s):  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Cotangent Bundle ◽  
Energy Levels ◽  
High Energy ◽  
Intermediate Range ◽  
Magnetic Flows ◽  
Almost All

AbstractWe consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range

Construction of the formulation of Hamiltonian systems from Lagrangian systems with Legendre transformation

2018 ◽  
Vol 11 (91) ◽  
pp. 4525-4531 ◽  
Author(s):  
Diana Marcela Devia Narvaez ◽  
German Correa Velez ◽  
Diego Fernando Devia Narvaez
Keyword(s):  
Hamiltonian Systems ◽  
Legendre Transformation ◽  
Lagrangian Systems
Sign in / Sign up

Export Citation Format

Share Document