# Variation of a path

From Diffgeom

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## Definition

Let be a differential manifold and a path in . A **variation** of is a map such that .

For a given

## Properties

### Smooth variation

`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.