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December  2014, 6(4): 503-526. doi: 10.3934/jgm.2014.6.503

Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings

Janusz Grabowski 1, , Katarzyna Grabowska 2, and  Paweł Urbański 2,

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland
2. 

Division of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland, Poland

Received  January 2014 Revised  May 2014 Published  December 2014

The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is $T M$, is based on the existence of canonical symplectic isomorphisms of double vector bundles $T^* TM$, $T^*T^* M$, and $TT^* M$, where the symplectic structure on $TT^* M$ is the tangent lift of the canonical symplectic structure $T^* M$. We show that there exists an analogous picture in the dynamics of objects for which the configuration space is $\wedge^n T M$, if we make use of certain structures of graded bundles of degree $n$, i.e. objects generalizing vector bundles (for which $n=1$). For instance, the role of $TT^*M$ is played in our approach by the manifold $\wedge^nT M\wedge^nT^*M$, which is canonically a graded bundle of degree $n$ over $\wedge^nT M$. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.
Keywords: minimal surfaces., Tulczyjew triples, Lagrange formalism, Hamiltonian formalism, double vector bundles, variational calculus.
Mathematics Subject Classification: Primary: 70S05, 53C80; Secondary: 53D17, 58A20, 58A32, 70G4.
Citation: Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503
References:
[1]

C. Buttin, Théorie des opérateurs différentiels gradués sur les formes différentielles, Bull. Soc. Math. France, 102 (1974), 49-73.  Google Scholar

[2]

F. Cantrijn, L. A. Ibort and M. De Leon, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236.  Google Scholar

[3]

C. M. Campos, E. Guzmán and J. C. Marrero, Classical field theories of first order and lagrangian submanifolds of premultisymplectic manifolds, J. Geom. Mech., 4 (2012), 1-26. doi: 10.3934/jgm.2012.4.1.  Google Scholar

[4]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order theories, Differential Geom. Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y.  Google Scholar

[5]

A. Echeverría-Enríquez and M. C. Muñoz-Lecanda, Geometry of multisymplectic Hamiltonian first order theory, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075.  Google Scholar

[6]

L. E. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[7]

M. Forger, C. Paufler and H. Römer, A general construction of Poisson brackets on exact multisymplectic manifolds, Rep. Math. Phys., 51 (2003), 187-195. doi: 10.1016/S0034-4877(03)80012-5.  Google Scholar

[8]

M. Forger, C. Paufler and H. Römer, Hamiltonian multivector fields and Poisson forms in multisymplectic field theories, J. Math. Phys., 46 (2005), 112903-112932. doi: 10.1063/1.2116320.  Google Scholar

[9]

M. Forger and L. G. Gomes, Multisymplectic and polysymplectic structures on fiber bundles, Rev. Math. Phys., 25 (2013), 1350018, 47pp. doi: 10.1142/S0129055X13500189.  Google Scholar

[10]

K. Gawędzki, On the geometrization of the canonical formalism in the classical field theory, Rep. Math. Phys., 3 (1972), 307-326. doi: 10.1016/0034-4877(72)90014-6.  Google Scholar

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G. Giachetta and L. Mangiarotti, Constrained Hamiltonian Systems and Gauge Theories, Int. J. Theor. Phys., 34 (1995), 2353-2371. doi: 10.1007/BF00670772.  Google Scholar

[12]

G. Giachetta, L. Mangiarotti and G. A. Sardanashvili, Advanced Classical Field Theory, World Scientific, Singapore, 2009. doi: 10.1142/9789812838964.  Google Scholar

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M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part I: Covariant field theory,, preprint, ().   Google Scholar

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M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part II: Canonical analysis of field theories,, preprint, ().   Google Scholar

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K. Grabowska, The Tulczyjew triple for classical fields, J. Phys. A, 45 (2012), 145207-145242. doi: 10.1088/1751-8113/45/14/145207.  Google Scholar

[16]

K. Grabowska, Lagrangian and Hamiltonian formalism in field theory: A simple model, J. Geom. Mech., 2 (2010), 375-395. doi: 10.3934/jgm.2010.2.375.  Google Scholar

[17]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204 (25pp). doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[18]

K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253. doi: 10.1016/j.geomphys.2011.06.018.  Google Scholar

[19]

K. Grabowska and J. Grabowski, Tulczyjew triples: From statics to field theory, J. Geom. Mech., 5 (2013), 445-472. doi: 10.3934/jgm.2013.5.445.  Google Scholar

[20]

K. Grabowska, J. Grabowski and P. Urbański, Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.  Google Scholar

[21]

J. Grabowski, Brakets, Int. J. Geom. Methods Mod. Phys. 10 (2013), 1360001, 45 pp. doi: 10.1142/S0219887813600013.  Google Scholar

[22]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305. doi: 10.1016/j.geomphys.2009.06.009.  Google Scholar

[23]

J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J.Geom. Phys., 62 (2012), 21-36. doi: 10.1016/j.geomphys.2011.09.004.  Google Scholar

[24]

J. Grabowski and P. Urbański, Tangent lifts of Poisson and related structures, J. Phys. A, 28 (1995), 6743-6777. doi: 10.1088/0305-4470/28/23/024.  Google Scholar

[25]

J. Grabowski and P. Urbański, Algebroids - general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141. doi: 10.1016/S0393-0440(99)00007-8.  Google Scholar

[26]

C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations. I. The local case, J. Differential Geom., 25 (1987), 23-53.  Google Scholar

[27]

J. Kijowski and W. Szczyrba, A canonical structure for classical field theories, Commun. Math. Phys., 46 (1976), 183-206. doi: 10.1007/BF01608496.  Google Scholar

[28]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lecture Notes in Physics, 107. Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[29]

I. Kolář and J. Tomáš, Gauge-natural transformations of some cotangent bundles, Acta Univ. M. Belii ser. Mathematics, 5 (1997), 3-9.  Google Scholar

[30]

K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95.  Google Scholar

[31]

O. Krupková, Hamiltonian field theory, J. Geom. Phys. 43 (2002), 93-132. doi: 10.1016/S0393-0440(01)00087-0.  Google Scholar

[32]

M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew's triples and lagrangian submanifolds in classical field theories, in Applied Differential Geometry and Mechanics (eds. W. Sarlet and F. Cantrijn), Univ. of Gent, Gent, Academia Press, (2003), 21-47. Google Scholar

[33]

P. Libermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics, Translated from the French by Bertram Eugene Schwarzbach. Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[34]

D. Lüst and S. Theisen, Lectures on String Theory, Lecture Notes in Physics, 346, Springer-Verlag, Berlin, 1989.  Google Scholar

[35]

M. Łukasik, Rachunek Wariacyjny Niezale.zny od Parametryzacji. Przypadek Jednowymiarowy (Polish), PhD Thesis, University of Warsaw, 2012. Google Scholar

[36]

G. Martin, A Darboux theorem for multi-symplectic manifolds, Lett. Math. Phys., 16 (1988), 133-138. doi: 10.1007/BF00402020.  Google Scholar

[37]

G. Pidello and W. Tulczyjew, Derivations of differential forms on jet bundles, Ann. Mat. Pura Appl., 147 (1987), 249-265. doi: 10.1007/BF01762420.  Google Scholar

[38]

J. Pradines, Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris, série A, 278 (1974), 1523-1526.  Google Scholar

[39]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson brackets and beyond (Manchester, 2001),Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 169-185. doi: 10.1090/conm/315/05479.  Google Scholar

[40]

G. Sardanashvily, Lagrangian dynamics of submanifolds. Relativistic mechanics, J. Geom. Mech., 4 (2012), 99-110. doi: 10.3934/jgm.2012.4.99.  Google Scholar

[41]

P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, Travaux mathématiques, Univ. Luxemb., 16 (2005), 121-137.  Google Scholar

[42]

W. Tulczyjew, Hamiltonian systems, Lagrangian systems, and the Legendre transformation, Symposia Math., 14 (1974), 247-258.  Google Scholar

[43]

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.  Google Scholar

[44]

W. M. Tulczyjew, A symplectic framework for linear field theories, Ann. Mat. Pura Appl., 130 (1982), 177-195. doi: 10.1007/BF01761494.  Google Scholar

[45]

W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Naples, 1989.  Google Scholar

[46]

W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting (Warsaw,1998), Acta Phys. Polon. B, 30 (1999), 2909-2978.  Google Scholar

[47]

L. Vitagliano, Partial differential Hamiltonian systems, Cand. J. Math., 65 (2013), 1164-1200. doi: 10.4153/CJM-2012-055-0.  Google Scholar

[48]

P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Matem. Torino, 54 (1996), 405-421.  Google Scholar

[49]

T. T. Voronov, Graded manifolds and Drienfeld doubles for Lie bialgebroids, in Quantization, Poisson brackets and beyond (Manchester, 2001), Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 131-168. doi: 10.1090/conm/315/05478.  Google Scholar

[50]

Y. Xin, Minimal Submanifolds and Related Topics, Nankai Tracts in Mathematics 8, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812564382.  Google Scholar

show all references

References:
[1]

C. Buttin, Théorie des opérateurs différentiels gradués sur les formes différentielles, Bull. Soc. Math. France, 102 (1974), 49-73.  Google Scholar

[2]

F. Cantrijn, L. A. Ibort and M. De Leon, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236.  Google Scholar

[3]

C. M. Campos, E. Guzmán and J. C. Marrero, Classical field theories of first order and lagrangian submanifolds of premultisymplectic manifolds, J. Geom. Mech., 4 (2012), 1-26. doi: 10.3934/jgm.2012.4.1.  Google Scholar

[4]

J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order theories, Differential Geom. Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y.  Google Scholar

[5]

A. Echeverría-Enríquez and M. C. Muñoz-Lecanda, Geometry of multisymplectic Hamiltonian first order theory, J. Math. Phys., 41 (2000), 7402-7444. doi: 10.1063/1.1308075.  Google Scholar

[6]

L. E. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[7]

M. Forger, C. Paufler and H. Römer, A general construction of Poisson brackets on exact multisymplectic manifolds, Rep. Math. Phys., 51 (2003), 187-195. doi: 10.1016/S0034-4877(03)80012-5.  Google Scholar

[8]

M. Forger, C. Paufler and H. Römer, Hamiltonian multivector fields and Poisson forms in multisymplectic field theories, J. Math. Phys., 46 (2005), 112903-112932. doi: 10.1063/1.2116320.  Google Scholar

[9]

M. Forger and L. G. Gomes, Multisymplectic and polysymplectic structures on fiber bundles, Rev. Math. Phys., 25 (2013), 1350018, 47pp. doi: 10.1142/S0129055X13500189.  Google Scholar

[10]

K. Gawędzki, On the geometrization of the canonical formalism in the classical field theory, Rep. Math. Phys., 3 (1972), 307-326. doi: 10.1016/0034-4877(72)90014-6.  Google Scholar

[11]

G. Giachetta and L. Mangiarotti, Constrained Hamiltonian Systems and Gauge Theories, Int. J. Theor. Phys., 34 (1995), 2353-2371. doi: 10.1007/BF00670772.  Google Scholar

[12]

G. Giachetta, L. Mangiarotti and G. A. Sardanashvili, Advanced Classical Field Theory, World Scientific, Singapore, 2009. doi: 10.1142/9789812838964.  Google Scholar

[13]

M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part I: Covariant field theory,, preprint, ().   Google Scholar

[14]

M. J. Gotay, J. Isenberg and J. E. Marsden, Momentum maps and classical relativistic fields, Part II: Canonical analysis of field theories,, preprint, ().   Google Scholar

[15]

K. Grabowska, The Tulczyjew triple for classical fields, J. Phys. A, 45 (2012), 145207-145242. doi: 10.1088/1751-8113/45/14/145207.  Google Scholar

[16]

K. Grabowska, Lagrangian and Hamiltonian formalism in field theory: A simple model, J. Geom. Mech., 2 (2010), 375-395. doi: 10.3934/jgm.2010.2.375.  Google Scholar

[17]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204 (25pp). doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[18]

K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253. doi: 10.1016/j.geomphys.2011.06.018.  Google Scholar

[19]

K. Grabowska and J. Grabowski, Tulczyjew triples: From statics to field theory, J. Geom. Mech., 5 (2013), 445-472. doi: 10.3934/jgm.2013.5.445.  Google Scholar

[20]

K. Grabowska, J. Grabowski and P. Urbański, Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575. doi: 10.1142/S0219887806001259.  Google Scholar

[21]

J. Grabowski, Brakets, Int. J. Geom. Methods Mod. Phys. 10 (2013), 1360001, 45 pp. doi: 10.1142/S0219887813600013.  Google Scholar

[22]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305. doi: 10.1016/j.geomphys.2009.06.009.  Google Scholar

[23]

J. Grabowski and M. Rotkiewicz, Graded bundles and homogeneity structures, J.Geom. Phys., 62 (2012), 21-36. doi: 10.1016/j.geomphys.2011.09.004.  Google Scholar

[24]

J. Grabowski and P. Urbański, Tangent lifts of Poisson and related structures, J. Phys. A, 28 (1995), 6743-6777. doi: 10.1088/0305-4470/28/23/024.  Google Scholar

[25]

J. Grabowski and P. Urbański, Algebroids - general differential calculi on vector bundles, J. Geom. Phys., 31 (1999), 111-141. doi: 10.1016/S0393-0440(99)00007-8.  Google Scholar

[26]

C. Günther, The polysymplectic Hamiltonian formalism in field theory and calculus of variations. I. The local case, J. Differential Geom., 25 (1987), 23-53.  Google Scholar

[27]

J. Kijowski and W. Szczyrba, A canonical structure for classical field theories, Commun. Math. Phys., 46 (1976), 183-206. doi: 10.1007/BF01608496.  Google Scholar

[28]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lecture Notes in Physics, 107. Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[29]

I. Kolář and J. Tomáš, Gauge-natural transformations of some cotangent bundles, Acta Univ. M. Belii ser. Mathematics, 5 (1997), 3-9.  Google Scholar

[30]

K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95.  Google Scholar

[31]

O. Krupková, Hamiltonian field theory, J. Geom. Phys. 43 (2002), 93-132. doi: 10.1016/S0393-0440(01)00087-0.  Google Scholar

[32]

M. de León, D. Martín de Diego and A. Santamaría-Merino, Tulczyjew's triples and lagrangian submanifolds in classical field theories, in Applied Differential Geometry and Mechanics (eds. W. Sarlet and F. Cantrijn), Univ. of Gent, Gent, Academia Press, (2003), 21-47. Google Scholar

[33]

P. Libermann and C. M. Marle, Symplectic Geometry and Analytical Mechanics, Translated from the French by Bertram Eugene Schwarzbach. Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[34]

D. Lüst and S. Theisen, Lectures on String Theory, Lecture Notes in Physics, 346, Springer-Verlag, Berlin, 1989.  Google Scholar

[35]

M. Łukasik, Rachunek Wariacyjny Niezale.zny od Parametryzacji. Przypadek Jednowymiarowy (Polish), PhD Thesis, University of Warsaw, 2012. Google Scholar

[36]

G. Martin, A Darboux theorem for multi-symplectic manifolds, Lett. Math. Phys., 16 (1988), 133-138. doi: 10.1007/BF00402020.  Google Scholar

[37]

G. Pidello and W. Tulczyjew, Derivations of differential forms on jet bundles, Ann. Mat. Pura Appl., 147 (1987), 249-265. doi: 10.1007/BF01762420.  Google Scholar

[38]

J. Pradines, Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris, série A, 278 (1974), 1523-1526.  Google Scholar

[39]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson brackets and beyond (Manchester, 2001),Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 169-185. doi: 10.1090/conm/315/05479.  Google Scholar

[40]

G. Sardanashvily, Lagrangian dynamics of submanifolds. Relativistic mechanics, J. Geom. Mech., 4 (2012), 99-110. doi: 10.3934/jgm.2012.4.99.  Google Scholar

[41]

P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, Travaux mathématiques, Univ. Luxemb., 16 (2005), 121-137.  Google Scholar

[42]

W. Tulczyjew, Hamiltonian systems, Lagrangian systems, and the Legendre transformation, Symposia Math., 14 (1974), 247-258.  Google Scholar

[43]

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. H. Poincaré Sect. A (N.S.), 27 (1977), 101-114.  Google Scholar

[44]

W. M. Tulczyjew, A symplectic framework for linear field theories, Ann. Mat. Pura Appl., 130 (1982), 177-195. doi: 10.1007/BF01761494.  Google Scholar

[45]

W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Naples, 1989.  Google Scholar

[46]

W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting (Warsaw,1998), Acta Phys. Polon. B, 30 (1999), 2909-2978.  Google Scholar

[47]

L. Vitagliano, Partial differential Hamiltonian systems, Cand. J. Math., 65 (2013), 1164-1200. doi: 10.4153/CJM-2012-055-0.  Google Scholar

[48]

P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Matem. Torino, 54 (1996), 405-421.  Google Scholar

[49]

T. T. Voronov, Graded manifolds and Drienfeld doubles for Lie bialgebroids, in Quantization, Poisson brackets and beyond (Manchester, 2001), Contemp. Math. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 131-168. doi: 10.1090/conm/315/05478.  Google Scholar

[50]

Y. Xin, Minimal Submanifolds and Related Topics, Nankai Tracts in Mathematics 8, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812564382.  Google Scholar

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Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015

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