The Elements of the Calculus of Variations

  • B. A. Dubrovin
  • A. T. Fomenko
  • S. P. Novikov
Part of the Graduate Texts in Mathematics book series (GTM, volume 93)


In §29.2 we defined the geodesics x i = x i (t) relative to a given connexion Γ jk i by means of the equation
$${{\nabla }_{T}}\left( T \right) = 0$$
, where T i = dx i /dt is the velocity vector of the curve; in other words the geodesics are the solutions of the equations
$$\frac{{{d^2}{x^i}}}{{d{t^2}}} + \Gamma _{jk}^i\frac{{d{x^j}}}{{dt}}\frac{{d{x^k}}}{{dt}} = 0$$


Phase Space Vector Field Hamiltonian System Poisson Bracket Canonical Transformation 
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Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • B. A. Dubrovin
    • 1
  • A. T. Fomenko
    • 2
  • S. P. Novikov
    • 3
  1. 1.c/o VAAP-Copyright Agency of the U.S.S.R.MoscowUSSR
  2. 2.3 Ya KaracharavskayaMoscowUSSR
  3. 3.L. D. Landau Institute for Theoretical PhysicsAcademy of Sciences of the U.S.S.R.MoscowUSSR

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