Variation of a path

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Definition

Let M be a differential manifold and \omega a path in M. A variation of \omega is a map \alpha: (-\epsilon, \epsilon) \times [0,1] \to M such that \alpha(0,t) = \omega(t) \forall t.

For a given

Properties

Smooth variation

Further information: Smooth variation

A smooth variation is a variation for which the map \alpha is smooth from the product manifold to M. Note that only a smooth path can admit a smooth variation,

Piecewise smooth variation

Further information: Piecewise smooth variation

A piecewise smooth variation is a variation for which we can find o = s_1 < s_2 < \ldots < s_r such that \alpha restricted to (-\epsilon,\epsilon) \times [s_{i-1},s_i] is smooth for 1 \le i \le r. Note that only a piecewise smooth path can admit a piecewise smooth variation.