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

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