dif8-page2 - 2 two-dimensional sphere S iii The...

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2 two-dimensional sphere S . iii) The configuration space of a planar double pendulum is the direct 2 1 1 product of two circles, i.e. the torus T = S x S . iv) The configuration space of a spherical double pendulum is the direct 2 2 product of two spheres S x S . v) A rigid segment in the plane has for its configuration space the direct 2 1 product R R x S , which is homeomorphic to the open solid torus. As we see in the above examples, the configuration space of a mechanical system is not necessarily homeomorphic to a linear space, but in each case the points of the configuration space have a neighborhood homeomorphic to an open ball. In the following chain of definitions we fix a positive integer n. Definition. Let X be an arbitrary set. A local parameterization of X is an -------------------------------------------------- -------------------------------------------------------------------------------------------------------------- n injective mapping v : W ----------L X from an open subset
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