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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), 301-330. 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., Inc., River Edge, NJ, 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), 012903, 19 pp.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

L. Mangiarotti and G. Sardanashvily, "Gauge Mechanics," World Scientific Publishing Co., Inc., River Edge, NJ, 1998.  Google Scholar

[9]

M. Modugno and A. Vinogradov, Some variations on the notion of connections, Annali di Matematica Pura ed Applicata, CLXVII (1994), 33-71. doi: 10.1007/BF01760328.  Google Scholar

[10]

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

[11]

G. Sardanashvily, Hamiltonian time-dependent mechanics, Journal of Mathematical Physics, 39 (1998), 2714-2729. 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), 1307-1319. 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), 301-330. 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., Inc., River Edge, NJ, 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), 012903, 19 pp.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

L. Mangiarotti and G. Sardanashvily, "Gauge Mechanics," World Scientific Publishing Co., Inc., River Edge, NJ, 1998.  Google Scholar

[9]

M. Modugno and A. Vinogradov, Some variations on the notion of connections, Annali di Matematica Pura ed Applicata, CLXVII (1994), 33-71. doi: 10.1007/BF01760328.  Google Scholar

[10]

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

[11]

G. Sardanashvily, Hamiltonian time-dependent mechanics, Journal of Mathematical Physics, 39 (1998), 2714-2729. 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), 1307-1319. doi: 10.1142/S0219887810004804.  Google Scholar

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