Lagrangian mechanics on manifolds
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.
KeywordsTangent Vector Tangent Bundle Configuration Space Lagrangian Function Constraint Force
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- 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 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
- 40.The distance of the points x(t) + ξ(t) from M is small of second-order compared with ξ(t).Google Scholar