Time-dependent Lagrangian systems: a geometric approach using semibasic forms

1990 ◽  
Vol 23 (15) ◽  
pp. 3475-3482 ◽  
Author(s):  
M F Ranada
Keyword(s):  
Geometric Approach ◽  
Time Dependent ◽  
Lagrangian Systems

Geometric approach to dynamics obtained by deformation of time-dependent Lagrangians

2016 ◽  
Vol 86 (2) ◽  
pp. 1285-1291 ◽  
Author(s):  
José F. Cariñena ◽  
José Fernández Núñez
Keyword(s):  
Geometric Approach ◽  
Time Dependent

scholarly journals A note on Hybrid Routh reduction for time-dependent Lagrangian systems

2020 ◽  
Vol 12 (2) ◽  
pp. 309-321
Author(s):  
Leonardo J. Colombo ◽  
◽  
María Emma Eyrea Irazú ◽  
Eduardo García-Toraño Andrés ◽  
◽  
...  
Keyword(s):  
Time Dependent ◽  
Lagrangian Systems ◽  
Routh Reduction

A geometric approach to Noether's Second Theorem in time-dependent lagrangian mechanics

1991 ◽  
Vol 23 (1) ◽  
pp. 51-63 ◽  
Author(s):  
José F. Cariñena ◽  
José Fernández-Núñez ◽  
Eduardo Martínez
Keyword(s):  
Geometric Approach ◽  
Time Dependent ◽  
Lagrangian Mechanics

A Geometric Approach to Constraint Stabilization for Holonomic Lagrangian Systems

2007 ◽  
Author(s):  
Hiroaki Yoshimura
Keyword(s):  
Mechanical Systems ◽  
Geometric Approach ◽  
Geometric Constraint ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Numerical Verification ◽  
Linkage Mechanism ◽  
Constraint Stabilization ◽  
D’Alembert Principle ◽  
Configuration Manifold

In this paper, we develop a geometric approach to constraint stabilization for holonomic mechanical systems in the context of Lagrangian formulation. We first show that holonomic mechanical systems, for the case in which a given Lagrangian is hyperregular, can be formulated by using the Lagrangian two-form, namely, a symplectic structure on the tangent bundle of a configuration manifold that is induced from the cotangent bundle via the Legendre transformation. Then, we present an idea of geometric constraint stabilization and we show that a holonomic Lagrangian system with geometric constraint stabilization can be formulated by the Lagrange-d’Alembert principle, together with its local coordinate expression for the sake of numerical computations. Finally, we illustrate the numerical verification that the proposed method enables to stabilize constraint violations effectively in comparison with the Baumgarte and Gear–Gupta–Leimkuhler methods together with an example of a linkage mechanism.

GEOMETRIC ASPECTS OF TIME-DEPENDENT SINGULAR DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 02 (04) ◽  
pp. 597-618 ◽  
Author(s):  
XAVIER GRÀCIA ◽  
RUBÉN MARTÍN
Keyword(s):  
Differential Equations ◽  
Autonomous System ◽  
Time Dependent ◽  
Lagrangian Systems ◽  
Singular Matrix ◽  
Singular Case ◽  
Jet Bundle ◽  
Geometric Framework ◽  
Affine Spaces ◽  
Autonomous Case

A geometric framework for describing and solving time-dependent implicit differential equations F(t,x,x′) = 0 is studied, paying special attention to the linearly singular case, where F is affine in the velocities: A(t,x)x′ = b(t,x). This framework is based on the jet bundle of a time-dependent configuration space, and is an extension of the geometric framework of the autonomous case. When A is a singular matrix, the solutions can be obtained by means of constraint algorithms, either directly or through an equivalent autonomous system that can be constructed using the vector hull functor of affine spaces. As applications, we consider the jet bundle description of time-dependent lagrangian systems and the Skinner–Rusk formulation of time-dependent mechanics.

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