×

Homogeneous systems of higher-order ordinary differential equations. (English) Zbl 1244.34010

This works starts by surveying the geometric theory of bundles of \(n\)-velocities, \(\text{T}^n\mathbb{R}^m\), and systems of differential equations of the form
\[ \frac{d^{n+1}y^i}{dx^{n+1}}=f^i\left(y,\frac{dy}{dx},\ldots,\frac{d^{n}y}{dx^{n} }\right),\qquad i=1,\ldots,m, \]
where \(f^1,\ldots,f^m\) are arbitrary functions. In particular, one of the approaches to the covariant derivative is presented, the horizontal distribution and the Jacobi endomorphism associated to these equations, see [M. Crampin and D. J. Saunders, “On the geometry of higher-order ordinary differential equations and the Wünschmann invariant,” Groups, geometry and physics. Zaragoza: Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza. Monografías de la Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza 29, 79–92 (2006; Zbl 1123.53012)].
The author extends the notion of homogeneous second-order systems to higher-order systems and proves that such systems admit analogous properties to second-order ones. A concept of strongly homogeneous systems of higher-order differential equations is suggested and their characteristics are discussed. It is demonstrated that there exists no strongly homogeneous differential equation on the whole \(\text{T}^2\mathbb{R}\). However, on the submanifold \[ \mathcal{S}=\left\{\left(y,\frac{dy}{dx},\frac{d^2y}{dx^2}\right)\in \text{T}^2\mathbb{R}\bigg|\frac{dy}{dx}>0\right\}, \] every strong homogeneous third-order differential equation is a Kummer–Schwarz equation \[ \frac{d^3y}{dx^3}=\frac 32 \left(\frac{dy}{dx}\right)^{-1}\frac{d^2y}{dx^2}+\kappa(y)\left(\frac{dy}{dx} \right)^3, \] where \(\kappa\) is an arbitrary smooth function depending on \(y\).

MSC:

34A26 Geometric methods in ordinary differential equations
53B15 Other connections

Citations:

Zbl 1123.53012
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] Anderson, I., Thompson, G.: The inverse problem of the calculus of variations for ordinary differential equations. Mem. Amer. Math. Soc. 98 1992 No. 473 · Zbl 0760.49021
[2] de Andrés, L.C., de León, M., Rodrigues, P.R.: Canonical connections associated with regular Lagrangians of higher order. Geom. Dedicata 39 1991 17-28 · Zbl 0729.53030
[3] Antonelli, P., Bucataru, I.: KCC theory of a system of second order differential equations. Handbook of Finsler Geometry Vol. 1 , Antonelli (ed.)Kluwer 2003 83-174 · Zbl 1105.53017
[4] Bucataru, I., Constantinescu, O., Dahl, M.F.: A geometric setting for systems of ordinary differential equations. preprint: arXiv:1011.5799 [math.DG] · Zbl 1286.34018 · doi:10.1142/S0219887811005701
[5] Crampin, M.: Connections of Berwald type. Publ. Math. Debrecen 57 2000 455-473 · Zbl 0980.53031
[6] Crampin, M., Sarlet, W., Cantrijn, F.: Higher-order differential equations and higher-order Lagrangian mechanics. Math. Proc. Cam. Phil. Soc. 99 1986 565-587 · Zbl 0609.58049 · doi:10.1017/S0305004100064501
[7] Crampin, M., Saunders, D.J.: Affine and projective transformations of Berwald connections. Diff. Geom. Appl. 25 2007 235-250 · Zbl 1158.53055 · doi:10.1016/j.difgeo.2007.02.001
[8] Crampin, M., Saunders, D.J.: On the geometry of higher-order ordinary differential equations and the Wuenschmann invariant. Groups, Geometry and Physics , Clemente-Gallardo and Martínez (eds.)Monografía 29, Real Academia de Ciencias de Zaragoza 2007 79-92 · Zbl 1123.53012
[9] Fritelli, S., Kozameh, C., Newman, E.T.: Differential geometry from differential equations. Comm. Math. Phys. 223 2001 383-408 · Zbl 1027.53080 · doi:10.1007/s002200100548
[10] Godliński, M., Nurowski, P.: Third order ODEs and four-dimensional split signature Einstein metrics. J. Geom. Phys. 56 2006 344-357 · Zbl 1098.34005 · doi:10.1016/j.geomphys.2005.01.011
[11] Godliński, M., Nurowski, P.: Geometry of third-order ODEs. preprint: arXiv:0902.4129v1 [math.DG]
[12] Saunders, D.J.: On the inverse problem for even-order ordinary differential equations in the higher-order calculus of variations. Diff. Geom. Appl. 16 2002 149-166 · Zbl 1048.34019 · doi:10.1016/S0926-2245(02)00065-7
[13] Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Kluwer 2001 · Zbl 1009.53004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.