# American Institute of Mathematical Sciences

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

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

Received  January 2013 Published  September 2013

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.
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.   Google Scholar [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.  doi: 10.1090/memo/0473.  Google Scholar [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.  doi: 10.1007/BF00147300.  Google Scholar [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.  doi: 10.1142/S0219887811005701.  Google Scholar [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.  doi: 10.3934/jgm.2009.1.159.  Google Scholar [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.  doi: 10.1007/s00009-009-0020-9.  Google Scholar [7] I. Bucataru and Z. Musznay, Projective metrizability and formal integrability,, Symmetry, 7 (2011).  doi: 10.3842/SIGMA.2011.114.  Google Scholar [8] I. Bucataru and Z. Musznay, Projective and Finsler metrizability: parameterization-rigidity of the geodesics,, International Journal of Mathematics, 23 (2012).  doi: 10.1142/S0129167X12500991.  Google Scholar [9] R. Caddeo, S. Montaldo, C. Oniciuc and P. Piu, The Euler-Lagrange method for biharmonic curves,, Mediterranean Journal of Mathematics, 3 (2006), 449.  doi: 10.1007/s00009-006-0090-x.  Google Scholar [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.  doi: 10.1088/0305-4470/14/10/012.  Google Scholar [11] M. Crampin, Some remarks on the Finslerian version of Hilbert's fourth problem,, Houston Journal of Mathematics, 37 (2011), 369.   Google Scholar [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.  doi: 10.1016/j.difgeo.2012.07.004.  Google Scholar [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.  doi: 10.1017/S0305004100064501.  Google Scholar [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.   Google Scholar [15] M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields,, Journal of Geometric Mechanics (JGM), 4 (2012), 27.  doi: 10.3934/jgm.2012.4.27.  Google Scholar [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.   Google Scholar [17] J. Grifone and Z. Muzsnay, "Variational Principles for Second-Order Differential Equations,", World-Scientific, (2000).  doi: 10.1142/9789812813596.  Google Scholar [18] A. Kawaguchi, Theory of connections in a Kawaguchi space of higher order,, Proceedings of the Imperial Academy, 13 (1937), 237.  doi: 10.3792/pia/1195579892.  Google Scholar [19] J. Klein and A. Voutier, Formes extérieures génératrices de sprays,, Annales de L'Institut Fourier (Grenoble), 18 (1968), 241.  doi: 10.5802/aif.282.  Google Scholar [20] I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).   Google Scholar [21] O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, I. Regularity,, Archivum Mathematicum (Brno), 22 (1986), 97.   Google Scholar [22] O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, II. Inverse problems,, Archivum Mathematicum (Brno), 23 (1987), 155.   Google Scholar [23] O. Krupková, "The Geometry of Ordinary Variational Equations,", Springer-Verlag, (1997).   Google Scholar [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.  doi: 10.1063/1.530952.  Google Scholar [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., (1985).   Google Scholar [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.  doi: 10.1088/0266-5611/8/4/006.  Google Scholar [27] R. L. Lovas, A note on Finsler-Minkowski norms,, Houston Journal of Mathematics, 33 (2007), 701.   Google Scholar [28] M. Matsumoto, "Foundations of Finsler Geometry and Special Finsler Spaces,", Kaiseisha Press, (1986).   Google Scholar [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).  doi: 10.1016/j.difgeo.2011.04.020.  Google Scholar [30] R. Miron, Noether theorem in higher-order Lagrangian mechanics,, International Journal of Theoretical Physics, 34 (1994), 1123.  doi: 10.1007/BF00671371.  Google Scholar [31] R. Miron, "The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics,", Kluwer Academic Publishers, (1997).   Google Scholar [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.  doi: 10.1016/0370-1573(90)90137-Q.  Google Scholar [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).  doi: 10.1088/1751-8113/44/38/385203.  Google Scholar [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.  doi: 10.1088/0305-4470/15/5/013.  Google Scholar [35] D. J. Saunders, "The Geometry of Jet Bundles,", Cambridge University Press, (1989).  doi: 10.1017/CBO9780511526411.  Google Scholar [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.  doi: 10.1016/S0926-2245(02)00065-7.  Google Scholar [37] D. J. Saunders, Projective metrizability in Finsler geometry,, Communications in Mathematics, 20 (2012), 63.   Google Scholar [38] Z. Shen, "Differential Geometry of Spray and Finsler Spaces,", Springer, (2001).   Google Scholar [39] J. Szilasi, A setting for spray and Finsler geometry,, in, 2 (2003), 1183.   Google Scholar [40] J. Szilasi, Calculus along the tangent bundle projection and projective metrizability,, in, (2008), 539.  doi: 10.1142/9789812790613_0045.  Google Scholar [41] J. Szilasi and Sz. Vattamány, On the Finsler-metrizabilities of spray manifolds,, Periodica Mathematica Hungarica, 44 (2002), 81.  doi: 10.1023/A:1014928103275.  Google Scholar [42] W. M. Tulczyjew, The Lagrange differential,, Bulletin de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, 24 (1976), 1089.   Google Scholar [43] Z. Urban and D. Krupka, The Zermelo conditions and higher order homogeneous functions,, Publicationes Mathematicae Debrecen, 82 (2013), 59.  doi: 10.5486/PMD.2013.5500.  Google Scholar

show all references

##### References:
 [1] J. C. Álvarez Paiva, Symplectic geometry and Hilbert's fourth problem,, Journal of Differential Geometry, 69 (2005), 353.   Google Scholar [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.  doi: 10.1090/memo/0473.  Google Scholar [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.  doi: 10.1007/BF00147300.  Google Scholar [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.  doi: 10.1142/S0219887811005701.  Google Scholar [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.  doi: 10.3934/jgm.2009.1.159.  Google Scholar [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.  doi: 10.1007/s00009-009-0020-9.  Google Scholar [7] I. Bucataru and Z. Musznay, Projective metrizability and formal integrability,, Symmetry, 7 (2011).  doi: 10.3842/SIGMA.2011.114.  Google Scholar [8] I. Bucataru and Z. Musznay, Projective and Finsler metrizability: parameterization-rigidity of the geodesics,, International Journal of Mathematics, 23 (2012).  doi: 10.1142/S0129167X12500991.  Google Scholar [9] R. Caddeo, S. Montaldo, C. Oniciuc and P. Piu, The Euler-Lagrange method for biharmonic curves,, Mediterranean Journal of Mathematics, 3 (2006), 449.  doi: 10.1007/s00009-006-0090-x.  Google Scholar [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.  doi: 10.1088/0305-4470/14/10/012.  Google Scholar [11] M. Crampin, Some remarks on the Finslerian version of Hilbert's fourth problem,, Houston Journal of Mathematics, 37 (2011), 369.   Google Scholar [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.  doi: 10.1016/j.difgeo.2012.07.004.  Google Scholar [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.  doi: 10.1017/S0305004100064501.  Google Scholar [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.   Google Scholar [15] M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields,, Journal of Geometric Mechanics (JGM), 4 (2012), 27.  doi: 10.3934/jgm.2012.4.27.  Google Scholar [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.   Google Scholar [17] J. Grifone and Z. Muzsnay, "Variational Principles for Second-Order Differential Equations,", World-Scientific, (2000).  doi: 10.1142/9789812813596.  Google Scholar [18] A. Kawaguchi, Theory of connections in a Kawaguchi space of higher order,, Proceedings of the Imperial Academy, 13 (1937), 237.  doi: 10.3792/pia/1195579892.  Google Scholar [19] J. Klein and A. Voutier, Formes extérieures génératrices de sprays,, Annales de L'Institut Fourier (Grenoble), 18 (1968), 241.  doi: 10.5802/aif.282.  Google Scholar [20] I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).   Google Scholar [21] O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, I. Regularity,, Archivum Mathematicum (Brno), 22 (1986), 97.   Google Scholar [22] O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, II. Inverse problems,, Archivum Mathematicum (Brno), 23 (1987), 155.   Google Scholar [23] O. Krupková, "The Geometry of Ordinary Variational Equations,", Springer-Verlag, (1997).   Google Scholar [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.  doi: 10.1063/1.530952.  Google Scholar [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., (1985).   Google Scholar [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.  doi: 10.1088/0266-5611/8/4/006.  Google Scholar [27] R. L. Lovas, A note on Finsler-Minkowski norms,, Houston Journal of Mathematics, 33 (2007), 701.   Google Scholar [28] M. Matsumoto, "Foundations of Finsler Geometry and Special Finsler Spaces,", Kaiseisha Press, (1986).   Google Scholar [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).  doi: 10.1016/j.difgeo.2011.04.020.  Google Scholar [30] R. Miron, Noether theorem in higher-order Lagrangian mechanics,, International Journal of Theoretical Physics, 34 (1994), 1123.  doi: 10.1007/BF00671371.  Google Scholar [31] R. Miron, "The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics,", Kluwer Academic Publishers, (1997).   Google Scholar [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.  doi: 10.1016/0370-1573(90)90137-Q.  Google Scholar [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).  doi: 10.1088/1751-8113/44/38/385203.  Google Scholar [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.  doi: 10.1088/0305-4470/15/5/013.  Google Scholar [35] D. J. Saunders, "The Geometry of Jet Bundles,", Cambridge University Press, (1989).  doi: 10.1017/CBO9780511526411.  Google Scholar [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.  doi: 10.1016/S0926-2245(02)00065-7.  Google Scholar [37] D. J. Saunders, Projective metrizability in Finsler geometry,, Communications in Mathematics, 20 (2012), 63.   Google Scholar [38] Z. Shen, "Differential Geometry of Spray and Finsler Spaces,", Springer, (2001).   Google Scholar [39] J. Szilasi, A setting for spray and Finsler geometry,, in, 2 (2003), 1183.   Google Scholar [40] J. Szilasi, Calculus along the tangent bundle projection and projective metrizability,, in, (2008), 539.  doi: 10.1142/9789812790613_0045.  Google Scholar [41] J. Szilasi and Sz. Vattamány, On the Finsler-metrizabilities of spray manifolds,, Periodica Mathematica Hungarica, 44 (2002), 81.  doi: 10.1023/A:1014928103275.  Google Scholar [42] W. M. Tulczyjew, The Lagrange differential,, Bulletin de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, 24 (1976), 1089.   Google Scholar [43] Z. Urban and D. Krupka, The Zermelo conditions and higher order homogeneous functions,, Publicationes Mathematicae Debrecen, 82 (2013), 59.  doi: 10.5486/PMD.2013.5500.  Google Scholar
 [1] Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595 [2] J. C. Alvarez Paiva and E. Fernandes. Crofton formulas in projective Finsler spaces. Electronic Research Announcements, 1998, 4: 91-100. [3] Mike Crampin, David Saunders. Homogeneity and projective equivalence of differential equation fields. Journal of Geometric Mechanics, 2012, 4 (1) : 27-47. doi: 10.3934/jgm.2012.4.27 [4] Yanfei Lu, Qingfei Yin, Hongyi Li, Hongli Sun, Yunlei Yang, Muzhou Hou. Solving higher order nonlinear ordinary differential equations with least squares support vector machines. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019012 [5] W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 [6] Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040 [7] Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91 [8] Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281 [9] Linda Beukemann, Klaus Metsch, Leo Storme. On weighted minihypers in finite projective spaces of square order. Advances in Mathematics of Communications, 2015, 9 (3) : 291-309. doi: 10.3934/amc.2015.9.291 [10] Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299 [11] Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029 [12] Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018 [13] Baruch Cahlon. Sufficient conditions for oscillations of higher order neutral delay differential equations. Conference Publications, 1998, 1998 (Special) : 124-137. doi: 10.3934/proc.1998.1998.124 [14] R.S. Dahiya, A. Zafer. Oscillation theorems of higher order neutral type differential equations. Conference Publications, 1998, 1998 (Special) : 203-219. doi: 10.3934/proc.1998.1998.203 [15] Carlos Durán, Diego Otero. The projective symplectic geometry of higher order variational problems: Minimality conditions. Journal of Geometric Mechanics, 2016, 8 (3) : 305-322. doi: 10.3934/jgm.2016009 [16] Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87 [17] Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353 [18] Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 [19] Mohammed Al Horani, Angelo Favini. Inverse problems for singular differential-operator equations with higher order polar singularities. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2159-2168. doi: 10.3934/dcdsb.2014.19.2159 [20] Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165

2018 Impact Factor: 0.525

## Metrics

• HTML views (0)
• Cited by (0)

• on AIMS