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 {g} or {\left \langle \cdot, \cdot \right \rangle}, and let {c: I \rightarrow M} be a smooth path in {M}. Define the energy as

\displaystyle E(c) := \frac{1}{2} \int g(c',c').  

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 {H} of the curve {c} and how the energy integral behaves with respect to {H}. Recall that {H: I \times(-\epsilon,\epsilon) \rightarrow M} is a smooth map with {H(t,0)=c(t), H(a,u)=c(a), H(b,u)=c(b)} for all {t,u}. The last two conditions mean that {H} is a family of curves that keep the endpoints fixed.

Define {H_u} as the curve {J(\cdot, u)}, and consider the function of {u},

\displaystyle E(u) := E(H_u) = \frac{1}{2} \int_I g\left( \frac{\partial}{\partial t} H(t,u) , \frac{\partial}{\partial t} H(t,u) \right).  

Ultimately, we are interested in curves that minimize the energy integral, at least locally. This means that for any variation {H} as above, {E(u)} should have a local minimum at {u=0}. So we will compute {\frac{d}{du} H(u)|_{u=0}}, 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

\displaystyle \frac{d}{du} E(u) = \frac{1}{2} \int_I \frac{\partial}{\partial u} g\left( \frac{\partial}{\partial t} H(t,u) , \frac{\partial}{\partial t} H(t,u) \right)   

By the formula for differentiating inner products and the symmetry of the Levi-Civita connection, this becomes  

\displaystyle \int_I g\left( \frac{D}{du} \frac{\partial}{\partial t} H(t,u), \frac{\partial}{\partial t} H(t,u) \right) = \int_I g\left( \frac{D}{dt} \frac{\partial}{\partial u} H(t,u), \frac{\partial}{\partial t} H(t,u) \right)   

If we let {V(t)} be the variation vector field and set {u=0}, we find that  

\displaystyle \frac{d}{du} E(u) |_{u=0} = \int_I g\left( \frac{D}{dt} V(t), \dot{c}(t) \right),  

which using similar identities becomes  

\displaystyle \int_I \frac{d}{d t} g\left( V(t), \dot{c}(t) \right) - \int_I g\left( V(t), \ddot{c}(t) \right)

 (Note that {\ddot{c}(t) := \frac{D}{dt} \dot{c}(t)} by definition.) The left integral is {g(V(b), \dot{c}(b)) - g(V(a), \dot{c}(a)) = 0}, since {V(b)=V(a)=0} for a variation keeping endpoints fixed. Hence we have the following:  

Theorem 1 (First Variation Formula)

For variations fixing the endpoints
 
\displaystyle \boxed{ \frac{d}{du}E(u)|_{u=0} = -\int_I g\left( V(t), \ddot{c}(t) \right) .}  

 

In particular, since {V(t)} can be really chosen arbitrary with {V(a)=V(b)=0}, we see that a curve locally minimizes the energy {E} only if it is a geodesic. There is a reason for this. First, by Cauchy-Schwarz, we have

\displaystyle l(c) \leq \sqrt{E(c)} \sqrt{length(I)}.  

There is equality precisely when {c} moves at constant speed, i.e. {|c'|} is constant. In particular,  

\displaystyle \left(\frac{l(c)}{\sqrt{length(I)}}\right)^2 \leq E(c)   

with equality holding for a geodesic. If {c} is not a geodesic, we can always find a shorter geodesic path between {c(a),c(b)} that necessarily makes {E} smaller by the above inequality (and equality).

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