American Institute of Mathematical Sciences

Advanced
March  2012, 4(1): 99-110. doi: 10.3934/jgm.2012.4.99

Lagrangian dynamics of submanifolds. Relativistic mechanics

Gennadi Sardanashvily 1,

1. 

Department of Theoretical Physics, Moscow State University, Moscow, Russian Federation

Received  September 2011 Revised  January 2012 Published  April 2012

Geometric formulation of Lagrangian relativistic mechanics in the terms of jets of one-dimensional submanifolds is generalized to Lagrangian theory of submanifolds of arbitrary dimension.
Keywords: Lagrangian theory, jet manifold, relativistic mechanics..
Mathematics Subject Classification: Primary: 58A20, 70H40; Secondary: 83A0.
Citation: Gennadi Sardanashvily. Lagrangian dynamics of submanifolds. Relativistic mechanics. Journal of Geometric Mechanics, 2012, 4 (1) : 99-110. doi: 10.3934/jgm.2012.4.99
References:
[1]

A. Echeverría Enríquez, M. Muñoz Lecanda and N. Román Roy, Geometrical setting of time-dependent regular systems. Alternative models,, Reviews in Mathematical Physica, 3 (1991), 301. doi: 10.1142/S0129055X91000114. Google Scholar

[2]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory,", World Scientific Publishing Co., (1997). Google Scholar

[3]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory,, Journal of Mathematical Physics, 50 (2009). Google Scholar

[4]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, "Advanced Classical Field Theory,", World Scientific Publishing Co. Pte. Ltd., (2009). Google Scholar

[5]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, "Geometric Formulation of Classical and Quantum Mechanics,", World Scientific Publishing Co. Pte. Ltd., (2010). Google Scholar

[6]

I. Krasil'shchik, V. Lychagin and A. Vinogradov, "Geometry of Jet Spaces and Nonlinear Partial Differential Equations,", Gordon and Breach, (1985). Google Scholar

[7]

M. De León and P. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North-Holland, (1989). Google Scholar

[8]

L. Mangiarotti and G. Sardanashvily, "Gauge Mechanics,", World Scientific Publishing Co., (1998). Google Scholar

[9]

M. Modugno and A. Vinogradov, Some variations on the notion of connections,, Annali di Matematica Pura ed Applicata, CLXVII (1994), 33. doi: 10.1007/BF01760328. Google Scholar

[10]

J. Polchinski, "String Theory,", Cambridge University Press, (1998). Google Scholar

[11]

G. Sardanashvily, Hamiltonian time-dependent mechanics,, Journal of Mathematical Physics, 39 (1998), 2714. doi: 10.1063/1.532416. Google Scholar

[12]

G. Sardanashvily, Relativistic mechanics in a general setting,, International Journal of Geometric Methods in Modern Physics, 7 (2010), 1307. doi: 10.1142/S0219887810004804. Google Scholar

show all references

References:
[1]

A. Echeverría Enríquez, M. Muñoz Lecanda and N. Román Roy, Geometrical setting of time-dependent regular systems. Alternative models,, Reviews in Mathematical Physica, 3 (1991), 301. doi: 10.1142/S0129055X91000114. Google Scholar

[2]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, "New Lagrangian and Hamiltonian Methods in Field Theory,", World Scientific Publishing Co., (1997). Google Scholar

[3]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory,, Journal of Mathematical Physics, 50 (2009). Google Scholar

[4]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, "Advanced Classical Field Theory,", World Scientific Publishing Co. Pte. Ltd., (2009). Google Scholar

[5]

G. Giachetta, L. Mangiarotti and G. Sardanashvily, "Geometric Formulation of Classical and Quantum Mechanics,", World Scientific Publishing Co. Pte. Ltd., (2010). Google Scholar

[6]

I. Krasil'shchik, V. Lychagin and A. Vinogradov, "Geometry of Jet Spaces and Nonlinear Partial Differential Equations,", Gordon and Breach, (1985). Google Scholar

[7]

M. De León and P. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North-Holland, (1989). Google Scholar

[8]

L. Mangiarotti and G. Sardanashvily, "Gauge Mechanics,", World Scientific Publishing Co., (1998). Google Scholar

[9]

M. Modugno and A. Vinogradov, Some variations on the notion of connections,, Annali di Matematica Pura ed Applicata, CLXVII (1994), 33. doi: 10.1007/BF01760328. Google Scholar

[10]

J. Polchinski, "String Theory,", Cambridge University Press, (1998). Google Scholar

[11]

G. Sardanashvily, Hamiltonian time-dependent mechanics,, Journal of Mathematical Physics, 39 (1998), 2714. doi: 10.1063/1.532416. Google Scholar

[12]

G. Sardanashvily, Relativistic mechanics in a general setting,, International Journal of Geometric Methods in Modern Physics, 7 (2010), 1307. doi: 10.1142/S0219887810004804. Google Scholar

[1]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421
[2]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic & Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345
[3]

Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67
[4]

Juan Carlos Marrero, David Martín de Diego, Ari Stern. Symplectic groupoids and discrete constrained Lagrangian mechanics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 367-397. doi: 10.3934/dcds.2015.35.367
[5]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[6]

Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249
[7]

Eduard Feireisl, Šárka Nečasová, Reimund Rautmann, Werner Varnhorn. New developments in mathematical theory of fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : i-ii. doi: 10.3934/dcdss.2014.7.5i
[8]

Benjamin Seibold, Rodolfo R. Rosales, Jean-Christophe Nave. Jet schemes for advection problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1229-1259. doi: 10.3934/dcdsb.2012.17.1229
[9]

Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375
[10]

Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317
[11]

Weinan E, Jianfeng Lu. Mathematical theory of solids: From quantum mechanics to continuum models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5085-5097. doi: 10.3934/dcds.2014.34.5085
[12]

Jean-Marie Souriau. On Geometric Mechanics. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595
[13]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139
[14]

Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic & Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014
[15]

Gianne Derks. Book review: Geometric mechanics. Journal of Geometric Mechanics, 2009, 1 (2) : 267-270. doi: 10.3934/jgm.2009.1.267
[16]

Andrew D. Lewis. The physical foundations of geometric mechanics. Journal of Geometric Mechanics, 2017, 9 (4) : 487-574. doi: 10.3934/jgm.2017019
[17]

Jean-Claude Zambrini. Stochastic deformation of classical mechanics. Conference Publications, 2013, 2013 (special) : 807-813. doi: 10.3934/proc.2013.2013.807
[18]

Vieri Benci. Solitons and Bohmian mechanics. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 303-317. doi: 10.3934/dcds.2002.8.303
[19]

Paul Popescu, Cristian Ida. Nonlinear constraints in nonholonomic mechanics. Journal of Geometric Mechanics, 2014, 6 (4) : 527-547. doi: 10.3934/jgm.2014.6.527
[20]

Jamie Cruz, Miguel Gutiérrez. Spiral motion in classical mechanics. Conference Publications, 2009, 2009 (Special) : 191-197. doi: 10.3934/proc.2009.2009.191

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences