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

scholarly journals ROUTH REDUCTION FOR SINGULAR LAGRANGIANS

2010 ◽  
Vol 07 (08) ◽  
pp. 1451-1489 ◽  
Author(s):  
BAVO LANGEROCK ◽  
MARCO CASTRILLÓN LÓPEZ
Keyword(s):  
Kinetic Energy ◽  
Original System ◽  
Regularity Conditions ◽  
Lagrangian Systems ◽  
Lagrange Equations ◽  
Reduction Procedure ◽  
Momentum Map ◽  
Routh Reduction ◽  
Regular Value ◽  
Singular Lagrangians

This paper concerns the Routh reduction procedure for Lagrangians systems with symmetry. It differs from the existing results on geometric Routh reduction in the fact that no regularity conditions on either the Lagrangian L or the momentum map JL are required apart from the momentum being a regular value of JL. The main results of this paper are: the description of a general Routh reduction procedure that preserves the Euler–Lagrange nature of the original system and the presentation of a presymplectic framework for Routh reduced systems. In addition, we provide a detailed description and interpretation of the Euler–Lagrange equations for the reduced system. The proposed procedure includes Lagrangian systems with a non-positively definite kinetic energy metric.

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.

Poincaré–Birkhoff results for Lagrangian systems with subquadratic potential

1995 ◽  
Vol 125 (6) ◽  
pp. 1169-1177
Author(s):  
Silvia Cingolani ◽  
Lorenzo Pisani
Keyword(s):  
Periodic Solutions ◽  
Morse Theory ◽  
Time Dependent ◽  
Lagrangian Systems ◽  
Infinitely Many Solutions ◽  
Minimal Period ◽  
Lagrangian Equation ◽  
Asymptotically Flat ◽  
Alternative Result ◽  
Existence Of Periodic Solutions

In this paper, using a recent generalisation of Morse Theory, we study the existence of periodic solutions of the Lagrangian equation (1.1) with subquadratic potential and asymptotically flat, nonconstant, time-dependent metric on ℝN. In Section 3, we get also an ‘alternative result’ about the minimal period or the existence of infinitely many solutions.

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