INFINITELY MANY EVEN PERIODIC SOLUTIONS OF LAGRANGIAN SYSTEMS ON ANY RIEMANNIAN TORI WITH EVEN POTENTIAL IN TIME

2009 ◽  
Vol 11 (02) ◽  
pp. 309-335 ◽  
Author(s):  
GUANGCUN LU ◽  
MINGYAN WANG
Keyword(s):  
Periodic Solutions ◽  
Periodic Potential ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Integer Multiple ◽  
Smooth Potential

In this paper, we prove that the Lagrangian system on any Riemannian torus with C3-smooth even and τ-periodic potential in time possesses infinitely many different periodic contractible even solutions with integer multiple periods of τ. As a consequence, we get that the same conclusion holds for any τ > 0 and the Lagrangian system on any Riemannian torus with C3-smooth potential independent of time.

scholarly journals Stability from Index Estimates for Periodic Solutions of Lagrangian Systems

1994 ◽  
Vol 108 (1) ◽  
pp. 170-189 ◽  
Author(s):  
G. Dellantonio ◽  
B. Donofrrio ◽  
I. Ekeland
Keyword(s):  
Periodic Solutions ◽  
Lagrangian Systems

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.

Morse Index of Approximating Periodic Solutions for the Billiard Problem. Application to Existence Results

1998 ◽  
Vol 50 (3) ◽  
pp. 497-524
Author(s):  
Philippe Bolle
Keyword(s):  
Periodic Solutions ◽  
Critical Points ◽  
Morse Index ◽  
Approximate Solutions ◽  
Existence Results ◽  
Lagrangian Systems ◽  
Potential Well ◽  
Regular Solutions ◽  
Open Set

AbstractThis paper deals with periodic solutions for the billiard problem in a bounded open set of ℝN which are limits of regular solutions of Lagrangian systems with a potential well. We give a precise link between the Morse index of approximate solutions (regarded as critical points of Lagrangian functionals) and the properties of the bounce trajectory to which they converge.

On Periodic Solutions to Constrained Lagrangian System

2019 ◽  
Vol 63 (1) ◽  
pp. 242-255
Author(s):  
Oleg Zubelevic
Keyword(s):  
Periodic Solutions ◽  
Configuration Space ◽  
Compact Manifold ◽  
Lagrangian System

AbstractA Lagrangian system is considered. The configuration space is a non-compact manifold that depends on time. A set of periodic solutions has been found.

scholarly journals OPTIMAL CONTROL REALIZATIONS OF LAGRANGIAN SYSTEMS WITH SYMMETRY

2011 ◽  
Vol 08 (07) ◽  
pp. 1627-1651 ◽  
Author(s):  
M. DELGADO-TÉLLEZ ◽  
A. IBORT ◽  
T. RODRÍGUEZ DE LA PEÑA
Keyword(s):  
Optimal Control ◽  
Control Systems ◽  
Optimal Control Problems ◽  
Lagrangian System ◽  
Control Problems ◽  
Lagrangian Systems ◽  
Explicit Realization ◽  
Control Equation ◽  
And Control

A new relation among a class of optimal control systems and Lagrangian systems with symmetry is discussed. It will be shown that a family of solutions of optimal control systems whose control equation are obtained by means of a group action are in correspondence with the solutions of a mechanical Lagrangian system with symmetry. This result also explains the equivalence of the class of Lagrangian systems with symmetry and optimal control problems discussed in [1, 2]. The explicit realization of this correspondence is obtained by a judicious use of Clebsch variables and Lin constraints, a technique originally developed to provide simple realizations of Lagrangian systems with symmetry. It is noteworthy to point out that this correspondence exchanges the role of state and control variables for control systems with the configuration and Clebsch variables for the corresponding Lagrangian system. These results are illustrated with various simple applications.

scholarly journals Homoclinics for singular strong force Lagrangian systems

2019 ◽  
Vol 9 (1) ◽  
pp. 644-653 ◽  
Author(s):  
Marek Izydorek ◽  
Joanna Janczewska ◽  
Jean Mawhin
Keyword(s):  
Singular Point ◽  
Homoclinic Solutions ◽  
Global Maximum ◽  
Lagrangian Systems ◽  
Infinite Depth ◽  
Force Condition ◽  
Strong Force ◽  
Action Integral ◽  
Smooth Potential ◽  
Single Well

Abstract We study the existence of homoclinic solutions for a class of Lagrangian systems $\begin{array}{} \frac{d}{dt} \end{array} $(∇Φ(u̇(t))) + ∇uV(t, u(t)) = 0, where t ∈ ℝ, Φ : ℝ2 → [0, ∞) is a G-function in the sense of Trudinger, V : ℝ × (ℝ2 ∖ {ξ}) → ℝ is a C1-smooth potential with a single well of infinite depth at a point ξ ∈ ℝ2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u± : ℝ → ℝ2 ∖ {ξ}.

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