scholarly journals A SURVEY OF LAGRANGIAN MECHANICS AND CONTROL ON LIE ALGEBROIDS AND GROUPOIDS

2006 ◽  
Vol 03 (03) ◽  
pp. 509-558 ◽  
Author(s):  
JORGE CORTÉS ◽  
MANUEL DE LEÓN ◽  
JUAN C. MARRERO ◽  
D. MARTÍN DE DIEGO ◽  
EDUARDO MARTÍNEZ
Keyword(s):  
Control Systems ◽  
Classical Field Theory ◽  
Nonholonomic Constraints ◽  
Classical Field ◽  
Lie Algebroids ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Geometric Description ◽  
Lagrangian And Hamiltonian Mechanics ◽  
And Control

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

scholarly journals Virial theorem in quasi-coordinates and Lie algebroid formalism

2014 ◽  
Vol 11 (09) ◽  
pp. 1450055 ◽  
Author(s):  
José F. Cariñena ◽  
Irina Gheorghiu ◽  
Eduardo Martínez ◽  
Patrícia Santos
Keyword(s):  
Virial Theorem ◽  
Mechanical Systems ◽  
Geometric Approach ◽  
Point Of View ◽  
Lie Algebroids ◽  
Hamiltonian Mechanics ◽  
Lie Algebroid ◽  
Lagrangian And Hamiltonian Mechanics ◽  
Generalized Virial Theorem

In this paper, the geometric approach to the virial theorem (VT) developed in [J. F. Cariñena, F. Falceto and M. F. Rañada, A geometric approach to a generalized virial theorem, J. Phys. A: Math. Theor. 45 (2012) 395210, 19 pp.] is written in terms of quasi-velocities (see [J. F. Cariñena, J. Nunes da Costa and P. Santos, Quasi-coordinates from the point of view of Lie algebroid structures, J. Phys. A: Math. Theor. 40 (2007) 10031–10048]). A generalization of the VT for mechanical systems on Lie algebroids is also given, using the geometric tools of Lagrangian and Hamiltonian mechanics on the prolongation of the Lie algebroid.

scholarly journals Gauging Newton's law

2007 ◽  
Vol 85 (4) ◽  
pp. 307-344 ◽  
Author(s):  
James T Wheeler
Keyword(s):  
Gauge Theory ◽  
Gauge Theories ◽  
Measurement Theory ◽  
Hamiltonian Mechanics ◽  
Newtonian Mechanics ◽  
Lagrangian Mechanics ◽  
Newton's Law ◽  
Newton’S Law ◽  
Lagrangian And Hamiltonian Mechanics ◽  
Systematic Development

We derive both Lagrangian and Hamiltonian mechanics as gauge theories of Newtonian mechanics. Systematic development of the distinct symmetries of dynamics and measurement suggest that gauge theory may be motivated as a reconciliation of dynamics with measurement. Applying this principle to Newton's law with the simplest measurement theory leads to Lagrangian mechanics, while use of conformal measurement theory leads to Hamiltonian mechanics. PACS Nos.: 45.20.Jj, 11.25.Hf, 45.10.–b

scholarly journals Conformal elasticity of mechanism-based metamaterials

2022 ◽  
Vol 13 (1) ◽  
Author(s):  
Michael Czajkowski ◽  
Corentin Coulais ◽  
Martin van Hecke ◽  
D. Zeb Rocklin
Keyword(s):  
Field Theory ◽  
Classical Field Theory ◽  
Classical Field ◽  
Approximate Symmetry ◽  
Analytic Method ◽  
Conformal Maps ◽  
Deformation Control ◽  
Unique Method ◽  
Nonlinear Deformations ◽  
And Control

AbstractDeformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries. However, flexible metamaterials often contain an additional approximate symmetry due to the presence of a designer soft strain pathway. Here we show that low energy deformations of designer dilational metamaterials will be governed by a scalar field theory, conformal elasticity, in which the nonuniform, nonlinear deformations observed under generic loads correspond with the well-studied—conformal—maps. We validate this approach using experiments and finite element simulations and further show that such systems obey a holographic bulk-boundary principle, which enables an analytic method to predict and control nonuniform, nonlinear deformations. This work both presents a unique method of precise deformation control and demonstrates a general principle in which mechanisms can generate special classes of soft deformations.

scholarly journals Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

2016 ◽  
Vol 13 (06) ◽  
pp. 1650072 ◽  
Author(s):  
Václav Zatloukal
Keyword(s):  
Field Theory ◽  
Configuration Space ◽  
Equations Of Motion ◽  
Classical Field Theory ◽  
Classical Field ◽  
Hamiltonian Constraint ◽  
Hamiltonian Mechanics ◽  
Weyl Scalar ◽  
Hamilton Jacobi Equation ◽  
General Method

Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined by a (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton–Jacobi equation applicable in the field theory then follows readily. The general method is illustrated with three examples: non-relativistic Hamiltonian mechanics, De Donder–Weyl scalar field theory, and string theory.

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