Dr. Walter Thirring, Dr. Evans Harrell (auth.)'s A Course in Mathematical Physics 1: Classical Dynamical PDF
By Dr. Walter Thirring, Dr. Evans Harrell (auth.)
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Additional info for A Course in Mathematical Physics 1: Classical Dynamical Systems
If N is a submanifold of M, and therefore T(N) is a submanifold of T(M), a nondegenerate g E ffg(M), g > 0, induces a Riemannian structure on N, because g also provides a nondegenerate mapping Yq(N) x Yq(N) ~ IR. The metric gik = Dik on IRm induces the usual metric on sn or Tn c IRn + 1. Since every m-dimensional manifold can be imbedded as a submanifold of 1R 2 m+ 1, it is always possible to find a Riemannian structure for any manifold. 3. 23). Up to a factor, this mapping T(M) x T(M) ~ IR is exactly the metric.
In order to progress from evaluating the derivative at a point to treating it as a function of q, we have to connect the tangent spaces at different points. " Yet within the domain of a single chart one could identify T(U) = UqeU 'Fq(M) with U x ~m, and then extend the mapping E>cCq) to E>c: T(U) --+ ~m X ~m, (q, v) --+ (cI>(q), E>cCq) . v). 10) It is possible to compare tangent vectors within this "tangent bundle" over U. The mapping E>c is plainly a bijection, and T(U) can be topologized so that it becomes a homeomorphism.
4 (x(u, v), y(u, v), z(u, v». If g-ll F is used as a chart, then the coordinate lines u = constant and v = constant are just sent to the two axes in ~2 by ec(q). t This notation is deprecated by the pedants, but is perfectly all right among friends. 6) 1. 4) may seem abstract, but it really only formalizes the intuitive notion of vectors in a tangent plane as arrows pointing in the directions of the curves passing through the point. This makes them elements of [Rn on the charts used to make M a manifold.
A Course in Mathematical Physics 1: Classical Dynamical Systems by Dr. Walter Thirring, Dr. Evans Harrell (auth.)