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

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.

scholarly journals Higher Order Hamiltonian Systems with Generalized Legendre Transformation

Mathematics ◽  
2018 ◽  
Vol 6 (9) ◽  
pp. 163
Author(s):  
Dana Smetanová
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Higher Order ◽  
Second Order ◽  
Legendre Transformation ◽  
Lagrange Equations ◽  
Hamilton Equations ◽  
Strongly Regular

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition.

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.

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