scholarly journals The Madelung transform as a momentum map

2017 ◽  
Vol 9 (2) ◽  
pp. 157-165 ◽  
Author(s):  
Daniel Fusca ◽  
Keyword(s):  
Momentum Map

scholarly journals The Optimal Momentum Map

2006 ◽  
pp. 329-362 ◽  
Author(s):  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
Momentum Map

scholarly journals The momentum map in Poisson geometry

2009 ◽  
Vol 131 (5) ◽  
pp. 1261-1310 ◽  
Author(s):  
Rui Loja Fernandes ◽  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
Poisson Geometry ◽  
Momentum Map

scholarly journals Uniqueness of the momentum map

2016 ◽  
Vol 106 ◽  
pp. 342-351 ◽  
Author(s):  
Chiara Esposito ◽  
Ryszard Nest
Keyword(s):  
Momentum Map

scholarly journals The tropical momentum map: a classification of toric log symplectic manifolds

2016 ◽  
Vol 367 (3-4) ◽  
pp. 1217-1258 ◽  
Author(s):  
Marco Gualtieri ◽  
Songhao Li ◽  
Álvaro Pelayo ◽  
Tudor S. Ratiu
Keyword(s):  
Symplectic Manifolds ◽  
Momentum Map

scholarly journals Covariant momentum map thermodynamics for parametrized field theories

2020 ◽  
pp. 2150018
Author(s):  
Goffredo Chirco ◽  
Marco Laudato ◽  
Fabio Maria Mele
Keyword(s):  
Phase Space ◽  
Gibbs State ◽  
Gravitational Interaction ◽  
Thermodynamic Description ◽  
Theoretical Physics ◽  
Field Theories ◽  
Covariant Formulation ◽  
Extended Phase ◽  
Momentum Map ◽  
Space Formulation

A general-covariant statistical framework capable of describing classical fluctuations of the gravitational field is a thorny open problem in theoretical physics, yet ultimately necessary to understand the nature of the gravitational interaction, and a key to quantum gravity. Inspired by Souriau’s symplectic generalization of the Maxwell–Boltzmann–Gibbs equilibrium in Lie group thermodynamics, we investigate a space–time-covariant formulation of statistical mechanics for parametrized first-order field theories, as a simplified model sharing essential general covariant features with canonical general relativity. Starting from a covariant multi-symplectic phase space formulation, we define a general-covariant notion of Gibbs state in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space. We show how such a covariant notion of equilibrium encodes the whole information about symmetry, gauge and dynamics carried by the theory, associated with a canonical spacetime foliation, where the covariant choice of a reference frame reflects in a Lie algebra-valued notion of local temperature. We investigate how physical equilibrium, hence time evolution, emerges from such a state and the role of the gauge symmetry in the thermodynamic description.

scholarly journals Covariant momentum map for non-Abelian topological BF field theory

2019 ◽  
Vol 36 (24) ◽  
pp. 245003
Author(s):  
Alberto Molgado ◽  
Ángel Rodríguez-López
Keyword(s):  
Field Theory ◽  
Momentum Map

Damping-Induced Self Recovery Phenomenon in Mechanical Systems With an Unactuated Cyclic Variable

2012 ◽  
Vol 135 (2) ◽  
Author(s):  
Dong Eui Chang ◽  
Soo Jeon
Keyword(s):  
Inverted Pendulum ◽  
Mechanical Systems ◽  
Conserved Quantity ◽  
The Self ◽  
Underactuated Mechanical Systems ◽  
Damping Force ◽  
Momentum Map ◽  
Time Integral ◽  
Conservation Of Momentum ◽  
Globally Attractive

Conservation of momentum is often used in controlling underactuated mechanical systems with symmetry. If a symmetry-breaking force is applied to the system, then the momentum is not conserved any longer in general. However, there exist forces linear in velocity such as the damping force that break the symmetry but induce a new conserved quantity in place of the original momentum map. This paper formalizes a new conserved quantity which can be constructed by combining the time integral of a general damping force and the original momentum map associated with the symmetry. From the perspective of stability theories, the new conserved quantity implies the corresponding variable possesses the self recovery phenomenon, i.e., it will be globally attractive to the initial condition of the variable. We discover that what is fundamental in the damping-induced self recovery is not the positivity of the damping coefficient but certain properties of the time integral of the damping force. The self recovery effect and theoretical findings are demonstrated by simulation results using the two-link planar manipulator and the torque-controlled inverted pendulum on a passive cart. The results in this paper will be useful in designing and controlling mechanical systems with underactuation.

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.

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