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A linear connection associated with any second order differential equation field

  • M. Crampin
Part of the Mathematics and Its Applications book series (MAIA, volume 350)

Abstract

There are several problems concerning systems of second-order ordinary differential equations

x= i (t,x j x j )

which can be treated geometrically: for example,
  • is the system trivial, i.e. is there a change of coordinates y i = y i (t,x j ) such that the system becomes ÿi = 0?

  • is the system linear, i.e. is there a change of coordinates which makes the right-hand side linear in yi

  • is the system separable, i.e. is there a change of coordinates such that the system separates into independent systems of lower dimension?

  • is the system derivable from a Lagrangian (the inverse problem of the calculus of variations)?

Keywords

Vector Field Vector Bundle Linear Connection Horizontal Structure Local Trivialization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. Crampin, E. Martínez and W. Sarlet. Linear connections for systems of second-order ordinary differential equations. Preprint The Open University, Milton Keynes, UK (1994).Google Scholar
  2. 2.
    M. Crampin, G. E. Prince and G. Thompson. A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics. J. Phys. A: Math. Gen. 17 (1984), 1437–1447.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    M. Crampin, W. Sarlet, E. Martínez, G. Byrnes and G. E. Prince. Towards a geometrical understanding of Douglas’s solution of the inverse problem of the calculus of variations. Inverse Problems 10 (1994), 245–260.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    E. Martínez, J. F. Cariñena. Linear connection induced by an Ehresmann connection on the tangent bundle. Linearization of second-order differential equations. Preprint University of Zaragoza, Spain (1993).Google Scholar
  5. 5.
    W. Sarlet, A. Vandecasteele, F.Cantrijn and E.Martínez. Derivations of forms along a map: the framework for time-dependent second-order equations. Diff. Geom. Appl., in press.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • M. Crampin
    • 1
  1. 1.Department of Applied MathematicsThe Open University Walton HallMilton KeynesUK

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