Newtonian Dynamics - Embedding and The Induced Riemannian Metric

Embedding and The Induced Riemannian Metric

Geometrically, the vector-function (7) implements an embedding of the comfiguration space of the constrained Newtonian dynamical system into the -dimensional flat comfiguration space of the unconstrained Newtonian dynamical system (3). Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold . The components of the metric tensor of this induced metric are given by the formula

$\displaystyle g_{ij}=\left(\frac{\partial\mathbf r}{\partial q^i},\frac{\partial\mathbf r}{\partial q^j}\right)$ ,

(11)

where is the scalar product associated with the Euclidean structure (4).

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