A setting for higher order differential equation fields and higher order Lagrange and Finsler spaces

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September 2013, 5(3): 257-279. doi: 10.3934/jgm.2013.5.257

A setting for higher order differential equation fields and higher order Lagrange and Finsler spaces

Ioan Bucataru ^1,

1.	Faculty of Mathematics, University Alexandru Ioan Cuza, Iaşi, 700506, Romania

Received January 2013 Published September 2013

Abstract
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We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order $2k$ in terms of a semi-basic $1$-form of order $k$. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [15] to formulate and discuss the projective metrizability problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projective metrizability in terms of homogeneous semi-basic $1$-forms. Such a semi-basic $1$-form is the Poincaré-Cartan $1$-form of a higher order Finsler function, while the potential of such semi-basic $1$-form is a higher order Finsler function.

Keywords: higher order Finsler spaces., projective metrizability, Ordinary differential equations, homogeneity.

Mathematics Subject Classification: Primary: 34A26, 70H50; Secondary: 70H03, 53C6.

Citation: Ioan Bucataru. A setting for higher order differential equation fields and higher order Lagrange and Finsler spaces. Journal of Geometric Mechanics, 2013, 5 (3) : 257-279. doi: 10.3934/jgm.2013.5.257

References:

[1]	J. C. Álvarez Paiva, Symplectic geometry and Hilbert's fourth problem, Journal of Differential Geometry, 69 (2005), 353-378.
[2]	I. Anderson and G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Memoirs of the American Mathematical Society, 98 (1992), 1-110. doi: 10.1090/memo/0473.
[3]	L. C. de Andrés, M. de León and P. R. Rodrigues, Connections on tangent bundles of higher order associated to regular Lagrangians, Geometriae Dedicata, 39 (1991), 17-28. doi: 10.1007/BF00147300.
[4]	I. Bucataru, O. Constantinescu and M. F. Dahl, A geometric setting for systems of ordinary differential equations, International Journal of Geometric Methods in Modern Physics, 8 (2011), 1291-1327. doi: 10.1142/S0219887811005701.
[5]	I. Bucataru and M. F. Dahl, Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations, Journal of Geometric Mechanics (JGM), 1 (2009), 159-180. doi: 10.3934/jgm.2009.1.159.
[6]	I. Bucataru and R. Miron, The geometry of systems of third order differential equations induced by second order regular Lagrangians, Mediterranean Journal of Mathematics, 6 (2009), 483-500. doi: 10.1007/s00009-009-0020-9.
[7]	I. Bucataru and Z. Musznay, Projective metrizability and formal integrability, Symmetry, Integrability, and Geometry: Methods and Applications (SIGMA), 7 (2011), 114, 22 pages. doi: 10.3842/SIGMA.2011.114.
[8]	I. Bucataru and Z. Musznay, Projective and Finsler metrizability: parameterization-rigidity of the geodesics, International Journal of Mathematics, 23 (2012), 1250099, 15 pages. doi: 10.1142/S0129167X12500991.
[9]	R. Caddeo, S. Montaldo, C. Oniciuc and P. Piu, The Euler-Lagrange method for biharmonic curves, Mediterranean Journal of Mathematics, 3 (2006), 449-465. doi: 10.1007/s00009-006-0090-x.
[10]	M. Crampin, On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics, Journal of Physics A: Mathematical and General, 14 (1981), 2567-2575. doi: 10.1088/0305-4470/14/10/012.
[11]	M. Crampin, Some remarks on the Finslerian version of Hilbert's fourth problem, Houston Journal of Mathematics, 37 (2011), 369-391.
[12]	M. Crampin, T. Mestdag and D. J. Saunders, The multiplier approach to the projective Finsler metrizability problem, Differential Geometry and its Applications, 30 (2012), 604-621. doi: 10.1016/j.difgeo.2012.07.004.
[13]	M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian Mechanics, Mathematical Proceedings of the Cambridge Philosophical Society, 99 (1986), 565-587. doi: 10.1017/S0305004100064501.
[14]	M. Crampin and D. J. Saunders, The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems, Houston Journal of Mathematics, 30 (2004), 657-689.
[15]	M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields, Journal of Geometric Mechanics (JGM), 4 (2012), 27-47. doi: 10.3934/jgm.2012.4.27.
[16]	A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Nederlandse Akademie van Wetenschappen. Proceedings. Series A. Indagationes Mathematicae, 18 (1956), 338-359.
[17]	J. Grifone and Z. Muzsnay, "Variational Principles for Second-Order Differential Equations," World-Scientific, 2000. doi: 10.1142/9789812813596.
[18]	A. Kawaguchi, Theory of connections in a Kawaguchi space of higher order, Proceedings of the Imperial Academy, 13 (1937), 237-240. doi: 10.3792/pia/1195579892.
[19]	J. Klein and A. Voutier, Formes extérieures génératrices de sprays, Annales de L'Institut Fourier (Grenoble), 18 (1968), 241-260. doi: 10.5802/aif.282.
[20]	I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, 1993.
[21]	O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, I. Regularity, Archivum Mathematicum (Brno), 22 (1986), 97-120.
[22]	O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, II. Inverse problems, Archivum Mathematicum (Brno), 23 (1987), 155-170.
[23]	O. Krupková, "The Geometry of Ordinary Variational Equations," Springer-Verlag, Berlin, 1997.
[24]	M. de León and D. M de Diego, Symmetries and constants of the motion for higher-order Lagrangian systems, Journal of Mathematical Physics, 36 (1995), 4138-4161. doi: 10.1063/1.530952.
[25]	M. de León and P. R. Rodrigues, "Generalized Classical Mechanics and Field Theory. A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives," North-Holland Publishing Co., Amsterdam, 1985.
[26]	M. de León and P. R. Rodrigues, The inverse problem of Lagrangian dynamics for higher-order differential equations: A geometrical approach, Inverse Problems, 8 (1992), 525-540. doi: 10.1088/0266-5611/8/4/006.
[27]	R. L. Lovas, A note on Finsler-Minkowski norms, Houston Journal of Mathematics, 33 (2007), 701-707.
[28]	M. Matsumoto, "Foundations of Finsler Geometry and Special Finsler Spaces," Kaiseisha Press, 1986.
[29]	R. Y. Matsyuk, Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'Zitterbewegung' in General Relativity, Differential Geometry and its Applications, 29 (2011), S149-S155. doi: 10.1016/j.difgeo.2011.04.020.
[30]	R. Miron, Noether theorem in higher-order Lagrangian mechanics, International Journal of Theoretical Physics, 34 (1994), 1123-1146. doi: 10.1007/BF00671371.
[31]	R. Miron, "The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics," Kluwer Academic Publishers, 1997.
[32]	G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Physics Reports, 188 (1990), 147-284. doi: 10.1016/0370-1573(90)90137-Q.
[33]	P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 385203 (35pp). doi: 10.1088/1751-8113/44/38/385203.
[34]	W. Sarlet, The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics, Journal of Physics A: Mathematical and Theoretical, 15 (1982), 1503-1517. doi: 10.1088/0305-4470/15/5/013.
[35]	D. J. Saunders, "The Geometry of Jet Bundles," Cambridge University Press, 1989. doi: 10.1017/CBO9780511526411.
[36]	D. J. Saunders, On the inverse problem for even-order ordinary differential equations in the higher-order calculus of variations, Differential Geometry and its Applications, 16 (2002), 149-166. doi: 10.1016/S0926-2245(02)00065-7.
[37]	D. J. Saunders, Projective metrizability in Finsler geometry, Communications in Mathematics, 20 (2012), 63-68.
[38]	Z. Shen, "Differential Geometry of Spray and Finsler Spaces," Springer, 2001.
[39]	J. Szilasi, A setting for spray and Finsler geometry, in "Handbook of Finsler Geometry" (ed. P.L. Antonelli), Kluwer Acad. Publ., Dordrecht, 2 (2003), 1183-1426.
[40]	J. Szilasi, Calculus along the tangent bundle projection and projective metrizability, in "Differential Geometry and its Applications," Proceedings of the 10th International Conference on DGA2007, World Scientific (2008), 539-558. doi: 10.1142/9789812790613_0045.
[41]	J. Szilasi and Sz. Vattamány, On the Finsler-metrizabilities of spray manifolds, Periodica Mathematica Hungarica, 44 (2002), 81-100. doi: 10.1023/A:1014928103275.
[42]	W. M. Tulczyjew, The Lagrange differential, Bulletin de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, Astronomiques et Physiques , 24 (1976), 1089-1096.
[43]	Z. Urban and D. Krupka, The Zermelo conditions and higher order homogeneous functions, Publicationes Mathematicae Debrecen, 82 (2013), 59-76. doi: 10.5486/PMD.2013.5500.

show all references

References:

[1]	J. C. Álvarez Paiva, Symplectic geometry and Hilbert's fourth problem, Journal of Differential Geometry, 69 (2005), 353-378.
[2]	I. Anderson and G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Memoirs of the American Mathematical Society, 98 (1992), 1-110. doi: 10.1090/memo/0473.
[3]	L. C. de Andrés, M. de León and P. R. Rodrigues, Connections on tangent bundles of higher order associated to regular Lagrangians, Geometriae Dedicata, 39 (1991), 17-28. doi: 10.1007/BF00147300.
[4]	I. Bucataru, O. Constantinescu and M. F. Dahl, A geometric setting for systems of ordinary differential equations, International Journal of Geometric Methods in Modern Physics, 8 (2011), 1291-1327. doi: 10.1142/S0219887811005701.
[5]	I. Bucataru and M. F. Dahl, Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations, Journal of Geometric Mechanics (JGM), 1 (2009), 159-180. doi: 10.3934/jgm.2009.1.159.
[6]	I. Bucataru and R. Miron, The geometry of systems of third order differential equations induced by second order regular Lagrangians, Mediterranean Journal of Mathematics, 6 (2009), 483-500. doi: 10.1007/s00009-009-0020-9.
[7]	I. Bucataru and Z. Musznay, Projective metrizability and formal integrability, Symmetry, Integrability, and Geometry: Methods and Applications (SIGMA), 7 (2011), 114, 22 pages. doi: 10.3842/SIGMA.2011.114.
[8]	I. Bucataru and Z. Musznay, Projective and Finsler metrizability: parameterization-rigidity of the geodesics, International Journal of Mathematics, 23 (2012), 1250099, 15 pages. doi: 10.1142/S0129167X12500991.
[9]	R. Caddeo, S. Montaldo, C. Oniciuc and P. Piu, The Euler-Lagrange method for biharmonic curves, Mediterranean Journal of Mathematics, 3 (2006), 449-465. doi: 10.1007/s00009-006-0090-x.
[10]	M. Crampin, On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics, Journal of Physics A: Mathematical and General, 14 (1981), 2567-2575. doi: 10.1088/0305-4470/14/10/012.
[11]	M. Crampin, Some remarks on the Finslerian version of Hilbert's fourth problem, Houston Journal of Mathematics, 37 (2011), 369-391.
[12]	M. Crampin, T. Mestdag and D. J. Saunders, The multiplier approach to the projective Finsler metrizability problem, Differential Geometry and its Applications, 30 (2012), 604-621. doi: 10.1016/j.difgeo.2012.07.004.
[13]	M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian Mechanics, Mathematical Proceedings of the Cambridge Philosophical Society, 99 (1986), 565-587. doi: 10.1017/S0305004100064501.
[14]	M. Crampin and D. J. Saunders, The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems, Houston Journal of Mathematics, 30 (2004), 657-689.
[15]	M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields, Journal of Geometric Mechanics (JGM), 4 (2012), 27-47. doi: 10.3934/jgm.2012.4.27.
[16]	A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Nederlandse Akademie van Wetenschappen. Proceedings. Series A. Indagationes Mathematicae, 18 (1956), 338-359.
[17]	J. Grifone and Z. Muzsnay, "Variational Principles for Second-Order Differential Equations," World-Scientific, 2000. doi: 10.1142/9789812813596.
[18]	A. Kawaguchi, Theory of connections in a Kawaguchi space of higher order, Proceedings of the Imperial Academy, 13 (1937), 237-240. doi: 10.3792/pia/1195579892.
[19]	J. Klein and A. Voutier, Formes extérieures génératrices de sprays, Annales de L'Institut Fourier (Grenoble), 18 (1968), 241-260. doi: 10.5802/aif.282.
[20]	I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, 1993.
[21]	O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, I. Regularity, Archivum Mathematicum (Brno), 22 (1986), 97-120.
[22]	O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, II. Inverse problems, Archivum Mathematicum (Brno), 23 (1987), 155-170.
[23]	O. Krupková, "The Geometry of Ordinary Variational Equations," Springer-Verlag, Berlin, 1997.
[24]	M. de León and D. M de Diego, Symmetries and constants of the motion for higher-order Lagrangian systems, Journal of Mathematical Physics, 36 (1995), 4138-4161. doi: 10.1063/1.530952.
[25]	M. de León and P. R. Rodrigues, "Generalized Classical Mechanics and Field Theory. A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives," North-Holland Publishing Co., Amsterdam, 1985.
[26]	M. de León and P. R. Rodrigues, The inverse problem of Lagrangian dynamics for higher-order differential equations: A geometrical approach, Inverse Problems, 8 (1992), 525-540. doi: 10.1088/0266-5611/8/4/006.
[27]	R. L. Lovas, A note on Finsler-Minkowski norms, Houston Journal of Mathematics, 33 (2007), 701-707.
[28]	M. Matsumoto, "Foundations of Finsler Geometry and Special Finsler Spaces," Kaiseisha Press, 1986.
[29]	R. Y. Matsyuk, Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'Zitterbewegung' in General Relativity, Differential Geometry and its Applications, 29 (2011), S149-S155. doi: 10.1016/j.difgeo.2011.04.020.
[30]	R. Miron, Noether theorem in higher-order Lagrangian mechanics, International Journal of Theoretical Physics, 34 (1994), 1123-1146. doi: 10.1007/BF00671371.
[31]	R. Miron, "The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics," Kluwer Academic Publishers, 1997.
[32]	G. Morandi, C. Ferrario, G. Lo Vecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Physics Reports, 188 (1990), 147-284. doi: 10.1016/0370-1573(90)90137-Q.
[33]	P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 385203 (35pp). doi: 10.1088/1751-8113/44/38/385203.
[34]	W. Sarlet, The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics, Journal of Physics A: Mathematical and Theoretical, 15 (1982), 1503-1517. doi: 10.1088/0305-4470/15/5/013.
[35]	D. J. Saunders, "The Geometry of Jet Bundles," Cambridge University Press, 1989. doi: 10.1017/CBO9780511526411.
[36]	D. J. Saunders, On the inverse problem for even-order ordinary differential equations in the higher-order calculus of variations, Differential Geometry and its Applications, 16 (2002), 149-166. doi: 10.1016/S0926-2245(02)00065-7.
[37]	D. J. Saunders, Projective metrizability in Finsler geometry, Communications in Mathematics, 20 (2012), 63-68.
[38]	Z. Shen, "Differential Geometry of Spray and Finsler Spaces," Springer, 2001.
[39]	J. Szilasi, A setting for spray and Finsler geometry, in "Handbook of Finsler Geometry" (ed. P.L. Antonelli), Kluwer Acad. Publ., Dordrecht, 2 (2003), 1183-1426.
[40]	J. Szilasi, Calculus along the tangent bundle projection and projective metrizability, in "Differential Geometry and its Applications," Proceedings of the 10th International Conference on DGA2007, World Scientific (2008), 539-558. doi: 10.1142/9789812790613_0045.
[41]	J. Szilasi and Sz. Vattamány, On the Finsler-metrizabilities of spray manifolds, Periodica Mathematica Hungarica, 44 (2002), 81-100. doi: 10.1023/A:1014928103275.
[42]	W. M. Tulczyjew, The Lagrange differential, Bulletin de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, Astronomiques et Physiques , 24 (1976), 1089-1096.
[43]	Z. Urban and D. Krupka, The Zermelo conditions and higher order homogeneous functions, Publicationes Mathematicae Debrecen, 82 (2013), 59-76. doi: 10.5486/PMD.2013.5500.

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