I have come across two different occurrences of energy minimization problems which find an interpretation using notions from algebraic topology, and I was wondering whether analogous situations have been studied in a systematic way somewhere in the literature.
The first is a result by Cartan in Riemannian Geometry states that, given a compact Riemannian manifold $M$, every homotopy class of loops in the path space of $M$ contains a closed geodesic.
The second is a result in Hodge theory stating the existence of an isomorphism: $$ H^k_\Delta(M) \xrightarrow{\sim} H^k_{deR}(M) $$ between the $k$th de Rham cohomology group of $M$ and the vector space of harmonic $k$ forms on $M$.
In both instances, objects which are defined as infimums of sorts (closed geodesics and harmonic forms) can be identified as representatives of homotopy/cohomology classes related to the underlying manifold. Is there a body of work in which the above two results are brought to a common ground?
Closed geodesics are special loops. Namely, the closed loops are exactly the loops that extremize the energy. Let $\Lambda M$ be the space of all loops in $M$. The closed geodesics are the critical points of the function(al) $E:\Lambda M\rightarrow \mathbb R$
$$ E(\gamma)=\int_{S^1}\Vert \dot\gamma(t)\Vert dt $$
Finding all closed geodesics amounts to finding all the critical points. It is well known that critical points of a function are closely related to the topology of the domain, which can be studied with algebraic topological methods.
This paradigm is available in many geometric situations. Very often the objects of interest extremize some function defined on some space, which then can be studied with algebraic topology. Some keywords that might get you into this are calculus of variations and Morse theory.