An Introduction to Riemannian Geometry by Leonor Godinho & José Natário

Author:Leonor Godinho & José Natário
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

We can look for the global minima (or maxima) of the action by considering curves on .

Definition 5.2

A variation of is a map (for some ) such that and the map given by is differentiable. The curve is said to be a critical point of the action if

for any variation of .

Notice that the global minima (or maxima) of must certainly be attained at critical points. However, a critical point is not necessarily a point of minimum (or maximum). It turns out that the critical points of the action are solutions of second-order ODEs.

Theorem 5.3

The curve is a critical point of the action determined by the Lagrangian if and only if it satisfies the Euler–Lagrange equations

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