I'm trying to understand how the calculus of variations works when you set smooth manifolds. The texts I read usually change from Euclidean space to manifolds, as if there is no difference, but I am someone who (at least once) has to do things formally.

**The case of Euclidean space:** $ $ To let $ I = (a, b) subset mathbb {R} $ and let it go $ q_0: I to mathbb {R} ^ n $ be a curve. We define a *deformation* from $ q_0 $ with fixed endpoints as another curve $ q: I times (- epsilon, epsilon) to mathbb {R} ^ n $ satisfying:

- $ q (t, 0) = q_0 (t) $ for all $ t in I $,
- $ q (a, s) = q_0 (a) $ and $ q (b, s) = q_0 (b) $ for all $ s in (- epsilon, epsilon) $,

The variation of $ q_0 $ in terms of deformation $ q $designated $ Delta q $, is defined by:

$$

Left. Delta q_0 (t) right. = left. frac { partial} { partial s} q (t, s) right | _ {s = 0}

$$

**Smooth distributor:** The definitions of curves and thus deformations work essentially the same and only replace them $ mathbb {R} ^ n $ from a few varied ones $ Q $, To define the variation, you can switch to local charts and then glue the pieces together carefully.

partition wall $ (a, b) $ as $ {a = t_0, t_1, cdots, t_ {n-1}, t_n = b } $ so the limitation of $ q $ to the subinterval $ (t_i, t_ {i + 1}) $ is contained in the local table $ (U_i, phi_i) $, In each diagram, the previous definition of $ Delta q_0 $ is well defined:

$$

Left. delta ( phi circ q_0) (t) right. = left. frac { partial} { partial s} ( phi circ q) (t, s) right | _ {s = 0}

$$

It can also be shown that these "local" deformations coincide with each other at the intersection of diagrams, so that conceptually there should be no problem in sticking the parts together again. I'm not sure though *from where* This object lives or why, so I do not feel comfortable going on.

Should the deformation be a different curve $ Q $? A curve continues $ TQ $? Does anyone know a clearer way to do this?