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I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally:

In the following piece-wise smooth means smooth on each set of a closed covering, implying continuity and boundedness of 1-sided derivatives.

For any closed Riemannian manifold $M$ define $\Lambda M$ as the space of piece-wise smooth maps from $S^1=I/\{0,1\}$ to M. define the energy of $\gamma \in \Lambda M$ by

$E(\gamma)=\int_{S^1} \mid\mid \gamma'(t)\mid\mid^2dt$