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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 |
References:
[1] |
J. C. Álvarez Paiva, Symplectic geometry and Hilbert's fourth problem,, Journal of Differential Geometry, 69 (2005), 353.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
I. Bucataru and Z. Musznay, Projective metrizability and formal integrability,, Symmetry, 7 (2011).
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).
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.
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.
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.
|
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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.
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.
doi: 10.5802/aif.282. |
[20] |
I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).
|
[21] |
O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, I. Regularity,, Archivum Mathematicum (Brno), 22 (1986), 97.
|
[22] |
O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, II. Inverse problems,, Archivum Mathematicum (Brno), 23 (1987), 155.
|
[23] |
O. Krupková, "The Geometry of Ordinary Variational Equations,", Springer-Verlag, (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.
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., (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.
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.
|
[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).
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.
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.
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).
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.
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.
doi: 10.1016/S0926-2245(02)00065-7. |
[37] |
D. J. Saunders, Projective metrizability in Finsler geometry,, Communications in Mathematics, 20 (2012), 63.
|
[38] |
Z. Shen, "Differential Geometry of Spray and Finsler Spaces,", Springer, (2001).
|
[39] |
J. Szilasi, A setting for spray and Finsler geometry,, in, 2 (2003), 1183.
|
[40] |
J. Szilasi, Calculus along the tangent bundle projection and projective metrizability,, in, (2008), 539.
doi: 10.1142/9789812790613_0045. |
[41] |
J. Szilasi and Sz. Vattamány, On the Finsler-metrizabilities of spray manifolds,, Periodica Mathematica Hungarica, 44 (2002), 81.
doi: 10.1023/A:1014928103275. |
[42] |
W. M. Tulczyjew, The Lagrange differential,, Bulletin de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, 24 (1976), 1089.
|
[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. |
show all references
References:
[1] |
J. C. Álvarez Paiva, Symplectic geometry and Hilbert's fourth problem,, Journal of Differential Geometry, 69 (2005), 353.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
I. Bucataru and Z. Musznay, Projective metrizability and formal integrability,, Symmetry, 7 (2011).
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).
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.
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.
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.
|
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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.
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.
doi: 10.5802/aif.282. |
[20] |
I. Kolář, P. W. Michor and J. Slovák, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).
|
[21] |
O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, I. Regularity,, Archivum Mathematicum (Brno), 22 (1986), 97.
|
[22] |
O. Krupková, Lepagean $2$-forms in higher-order Hamiltonian dynamics, II. Inverse problems,, Archivum Mathematicum (Brno), 23 (1987), 155.
|
[23] |
O. Krupková, "The Geometry of Ordinary Variational Equations,", Springer-Verlag, (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.
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., (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.
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.
|
[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).
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.
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.
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).
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.
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.
doi: 10.1016/S0926-2245(02)00065-7. |
[37] |
D. J. Saunders, Projective metrizability in Finsler geometry,, Communications in Mathematics, 20 (2012), 63.
|
[38] |
Z. Shen, "Differential Geometry of Spray and Finsler Spaces,", Springer, (2001).
|
[39] |
J. Szilasi, A setting for spray and Finsler geometry,, in, 2 (2003), 1183.
|
[40] |
J. Szilasi, Calculus along the tangent bundle projection and projective metrizability,, in, (2008), 539.
doi: 10.1142/9789812790613_0045. |
[41] |
J. Szilasi and Sz. Vattamány, On the Finsler-metrizabilities of spray manifolds,, Periodica Mathematica Hungarica, 44 (2002), 81.
doi: 10.1023/A:1014928103275. |
[42] |
W. M. Tulczyjew, The Lagrange differential,, Bulletin de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, 24 (1976), 1089.
|
[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. |
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