Variation of a path
For a given
Further information: Smooth variation
A smooth variation is a variation for which the map is smooth from the product manifold to . 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 such that restricted to is smooth for . Note that only a piecewise smooth path can admit a piecewise smooth variation.