# Lagrangian mechanics on manifolds

Chapter

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## 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
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## References

- 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 - 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 - 37.A manifold is connected if it cannot be divided into two disjoint open subsets.Google Scholar
- 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 - 39.Strictly speaking, in order to define a variation
*δ*φ, 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 - 40.The distance of the points x(
*t*) +*ξ*(*t*) from M is small of second-order compared with*ξ*(*t*).Google Scholar

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© Springer Science+Business Media New York 1978