American Institute of Mathematical Sciences

Advanced
March  2012, 4(1): 27-47. doi: 10.3934/jgm.2012.4.27

Homogeneity and projective equivalence of differential equation fields

Mike Crampin 1, and  David Saunders 2,

1. 

Department of Mathematics, Ghent University, Krijgslaan 281, B-9000 Gent, Belgium
2. 

Department of Mathematics, Faculty of Science, The University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic

Received  September 2011 Revised  March 2012 Published  April 2012

We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are regular (in a natural sense that we define) form a projective equivalence class of homogeneous systems. We show further that the geodesics, or base integral curves, of projectively equivalent homogeneous differential equation fields are the same apart from orientation-preserving reparametrization; that is, homogeneous differential equation fields determine systems of paths.
Keywords: reparametrization., homogeneity, Ordinary differential equations, parametric Lagrangian, projective equivalence.
Mathematics Subject Classification: 34A26, 70H03, 70H5.
Citation: 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
References:
[1]

I. Bucataru, O. A. Constantinescu and M. F. Dahl, A geometric setting for systems of ordinary differential equations, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1291-1327. doi: 10.1142/S0219887811005701.  Google Scholar

[2]

M. Crampin, Homogeneous systems of higher-order ordinary differential equations, Communications in Mathematics, 18 (2010), 37-50.  Google Scholar

[3]

M. Crampin and D. J. Saunders, The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems, Houston J. Math., 30 (2004), 657-689.  Google Scholar

[4]

F. Faà di Bruno, Sullo sviluppo delle Funzioni, Annali di Scienze Matematiche e Fisiche, 6 (1855), 479-480. Google Scholar

[5]

I. Kolář, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, 1993.  Google Scholar

[6]

B. S. Kruglikov and V. V. Lychagin, Geometry of differential equations, in "Handbook of Global Analysis" (eds. D. Krupka and D. J. Saunders), 1214, Elsevier Sci. B. V., Amsterdam, (2008), 725-771.  Google Scholar

[7]

R. Ya. Matsyuk, Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'Zitterbewegung' in general relativity,, preprint, ().   Google Scholar

[8]

J. Muñoz Masqué and I. M. Pozo Coronado, Parameter-invariant second-order variational problems in one varaiable, J. Phys. A, 31 (1998), 6225-6242. doi: 10.1088/0305-4470/31/29/014.  Google Scholar

[9]

J. J. Stoker, "Differential Geometry," Pure and Applied Mathematics, Vol. XX, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969.  Google Scholar

show all references

References:
[1]

I. Bucataru, O. A. Constantinescu and M. F. Dahl, A geometric setting for systems of ordinary differential equations, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1291-1327. doi: 10.1142/S0219887811005701.  Google Scholar

[2]

M. Crampin, Homogeneous systems of higher-order ordinary differential equations, Communications in Mathematics, 18 (2010), 37-50.  Google Scholar

[3]

M. Crampin and D. J. Saunders, The Hilbert-Carathéodory and Poincaré-Cartan forms for higher-order multiple-integral variational problems, Houston J. Math., 30 (2004), 657-689.  Google Scholar

[4]

F. Faà di Bruno, Sullo sviluppo delle Funzioni, Annali di Scienze Matematiche e Fisiche, 6 (1855), 479-480. Google Scholar

[5]

I. Kolář, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, 1993.  Google Scholar

[6]

B. S. Kruglikov and V. V. Lychagin, Geometry of differential equations, in "Handbook of Global Analysis" (eds. D. Krupka and D. J. Saunders), 1214, Elsevier Sci. B. V., Amsterdam, (2008), 725-771.  Google Scholar

[7]

R. Ya. Matsyuk, Higher order variational origin of the Dixon's system and its relation to the quasi-classical 'Zitterbewegung' in general relativity,, preprint, ().   Google Scholar

[8]

J. Muñoz Masqué and I. M. Pozo Coronado, Parameter-invariant second-order variational problems in one varaiable, J. Phys. A, 31 (1998), 6225-6242. doi: 10.1088/0305-4470/31/29/014.  Google Scholar

[9]

J. J. Stoker, "Differential Geometry," Pure and Applied Mathematics, Vol. XX, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969.  Google Scholar

[1]

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
[2]

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
[3]

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
[4]

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
[5]

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
[6]

Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036
[7]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39
[8]

Hongwei Lou, Weihan Wang. Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Mathematical Control & Related Fields, 2015, 5 (3) : 517-527. doi: 10.3934/mcrf.2015.5.517
[9]

Alex Bihlo, James Jackaman, Francis Valiquette. On the development of symmetry-preserving finite element schemes for ordinary differential equations. Journal of Computational Dynamics, 2020, 7 (2) : 339-368. doi: 10.3934/jcd.2020014
[10]

Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283
[11]

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
[12]

Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91
[13]

David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, 2021, 29 (4) : 2791-2817. doi: 10.3934/era.2021014
[14]

William Guo. The Laplace transform as an alternative general method for solving linear ordinary differential equations. STEM Education, 2021, 1 (4) : 309-329. doi: 10.3934/steme.2021020
[15]

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
[16]

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
[17]

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, 2020, 16 (3) : 1481-1502. doi: 10.3934/jimo.2019012
[18]

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
[19]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229
[20]

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

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (135)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy