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Lagrangian mechanics on manifolds

  • V. I. Arnold
Part of the Graduate Texts in Mathematics book series (GTM, volume 60)

Abstract

In this chapter we introduce the concepts of a differentiable manifold and its tangent bundle. A lagrangian function, given on the tangent bundle, defines a lagrangian “holonomic system” on a manifold. Systems of point masses with holonomic constraints (e.g., a pendulum or a rigid body) are special cases.

Keywords

Tangent Vector Tangent Bundle Configuration Space Lagrangian Function Constraint Force 
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. 35.
    The proof is based on the fact that, due to the conservation of energy, a moving point cannot move further from y than cN-½, which approaches zero as N → ∞.Google Scholar
  2. 36.
    By differentiable here we mean r times continuously differentiable; the exact value of r (1 ≤ r ≤ ∞)is immaterial (we may take r = ∞, for example).Google Scholar
  3. 37.
    A manifold is connected if it cannot be divided into two disjoint open subsets.Google Scholar
  4. 38.
    The authors of several textbooks mistakenly assert that the converse is also true, i.e., that if hs takes solutions to solutions, then h s preserves L.Google Scholar
  5. 39.
    Strictly speaking, in order to define a variation δφ, one must define on the set of curves near x on M the structure of a region in a vector space. This can be done using coordinates on M; however, the property of being a conditional extremal does not depend on the choice of a coordinate system.Google Scholar
  6. 40.
    The distance of the points x(t) + ξ(t) from M is small of second-order compared with ξ(t).Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • V. I. Arnold
    • 1
  1. 1.Department of MathematicsUniversity of MoscowMoscowUSSR

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