Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T14:37:25.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher-order differential equations and higher-order lagrangian mechanics

Published online by Cambridge University Press:  24 October 2008

M. Crampin ,
W. Sarlet  and
F. Cantrijn
M. Crampin
Affiliation:
Faculty of Mathematics, The Open University, Milton Keynes M K7 6AA, U.K.
W. Sarlet
Affiliation:
Instituut voor Theoretische Mechanika, Rijksuniversiteit Gent, B-9000 Gent, Belgium
F. Cantrijn
Affiliation:
Instituut voor Theoretische Mechanika, Rijksuniversiteit Gent, B-9000 Gent, Belgium
Get access Rights & Permissions [Opens in a new window]

Extract

The study of higher-order mechanics, by various geometrical methods, in the framework of the theory of higher-order tangent bundles or jet spaces, has been undertaken by a number of authors recently: for example, Tulczyjew [16, 17], Rodrigues [14, 15] de León [8], Krupka and Musilova [11, and references therein]. In this article we wish to complement these studies by approaching the subject from a new point of view, one which we developed for second-order differential equation fields and first-order Lagrangian mechanics in [19]. In particular, our aim is to show that many of the results we obtained there may be extended to the higher-order case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 3 , May 1986 , pp. 565 - 587
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Catz, G.. Sur le fibré tangent d’ordre 2. C. R. Acad. Sci. Paris 278 (1974), 277–280.Google Scholar
[2]Crampin, M.. On horizontal distributions on the tangent bundle of a differentiable manifold. J. London Math. Soc. 3 (1971), 178–182.CrossRefGoogle Scholar
[3]Crampin, M.. Tangent bundle geometry for lagrangian dynamics. J. Phys. A: Math. Gen. 16 (1983), 3755–3772.CrossRefGoogle Scholar
[4]Grifone, J.. Structure presque -tangent et connexions I. Ann. Inst. Fourier (Grenoble) 22 (1) (1972), 287–334.CrossRefGoogle Scholar
[5]Klein, J.. Espaces variationnels et mècanique. Ann. Inst. Fourier (Grenoble) 12 (1962), 1–124.CrossRefGoogle Scholar
[6]Klein, J.. Structures symplectiques ou J-symplectiques homogènes sur l’espace tangent à une variètè. Sympos. Math. 14 (1974), 181–192.Google Scholar
[7]Klein, J.. Geometry of sprays. Lagrangian case. Principle of least curvature. In Proc. IUTAM-ISIMM symposium on modern developments in analytical mechanics, Vol. I (Benenti, Francaviglia, and Lichnerowicz, , eds.) (Torino, 1983), 177–196.Google Scholar
[8]De León, M.. Systèmes lagrangiens réguliers d’ordre supérieur. C. R. Acad. Sci. Paris 294 (1982), 451–453.Google Scholar
[9]De León, M. and Rodrigues, P.. Generalized Classical Mechanics and Field Theory (North-Holland, 1985).Google Scholar
[10]De León, M. and Villaverde, C.. Calcul différentiel sur les fibrés tangents d’ordre Supérieur. C. R. Acad. Sci. Paris 292 (1981), 881–884.Google Scholar
[11]Krupka, D. and Musilova, J.. Hamilton extremals in higher order mechanics. Arch. Math. (Brno) 20 (1984), 21–30.Google Scholar
[12]Marmo, G. and Saletan, E. J.. Ambiguities in the lagrangian and hamiltonian formalism: transformation properties. Nuovo Cimento B 40 (1977), 67–89.CrossRefGoogle Scholar
[13]Prince, G.. Toward a classification of dynamical symmetries in classical mechanics. Bull. Austral. Math. Soc. 27 (1983), 53–71.CrossRefGoogle Scholar
[14]Rodrigues, P.. Sur les systèmes mècaniques lagrangiens homogènes d’ordre supèrieur. C. R. Acad. Sci. Paris 281 (1975), 643–646.Google Scholar
[15]Rodrigues, P.. Sur les systèmes mécaniques généralisés. C. R. Acad. Sci. Paris 282 (1976), 1307–1309.Google Scholar
[16]Tulczyjew, W. M.. Sur la différentielle de Lagrange. C. R. Acad. Sci. Paris 280 (1975), 1295–1298.Google Scholar
[17]Tulczyjew, W. M.. The Lagrange differential. Bull. Acad. Polon. Sci. 24 (1976), 1089–1096.Google Scholar
[18]Sarlet, W.. Symmetries and alternative lagrangians in higher-order mechanics. Phys. Lett. A 108 (1985), 14–18.CrossRefGoogle Scholar
[19]Sarlet, W., Cantrijn, F. and Crampin, M.. A new look at second-order equations and lagrangian mechanics. J. Phys. A: Math. Gen. 17 (1984), 1999–2009.CrossRefGoogle Scholar
[20]Yano, K. and Ishihara, S.. Tangent and Cotangent Bundles (Marcel Dekker, 1973).Google Scholar