As I hinted a couple of posts back, I am interested in discussing the application of the calculus of variations to differential geometry. So fix a Riemannian manifold with metric denoted either by or , and let be a smooth path in . Define the energy as
The energy integral is closely related to the length function, though it is easier to deal with. Now we are interested in studying a variation of the curve and how the energy integral behaves with respect to . Recall that is a smooth map with for all . The last two conditions mean that is a family of curves that keep the endpoints fixed.
Define as the curve , and consider the function of ,
Ultimately, we are interested in curves that minimize the energy integral, at least locally. This means that for any variation as above, should have a local minimum at . So we will compute , and, eventually, the second derivative too. The evaluations involve nothing more than a rehash of many standard tricks we have repeated already.
If we differentiate under the integral sign, legal because of all the smoothness, we get
By the formula for differentiating inner products and the symmetry of the Levi-Civita connection, this becomes
If we let be the variation vector field and set , we find that
which using similar identities becomes
(Note that by definition.) The left integral is , since for a variation keeping endpoints fixed. Hence we have the following:
Theorem 1 (First Variation Formula)For variations fixing the endpoints
In particular, since can be really chosen arbitrary with , we see that a curve locally minimizes the energy only if it is a geodesic. There is a reason for this. First, by Cauchy-Schwarz, we have
There is equality precisely when moves at constant speed, i.e. is constant. In particular,
with equality holding for a geodesic. If is not a geodesic, we can always find a shorter geodesic path between that necessarily makes smaller by the above inequality (and equality).