I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced covariant derivatives (specifically regarding maps of type $ \phi\colon M\to S $ between Riemannian manifolds $ \left(M,g\right) $ and $ \left(S,h\right) $, where $ M $ may be required to be compact, and the relevant energy functionals are action functionals having Lagrangian depending on $ T\phi\in\Gamma\left(\phi^* TS\otimes T^{*}M\right) $ and covariant derivatives of $ T\phi $).

Of all the non-elementary references I've found, the exposition is restricted to the energy functional whose critical points are harmonic maps. I am in the process of finding all the relevant papers by James Eells, as was suggested to me by Arthur Fischer, though the papers I've found so far deal exclusively with the harmonic map case. The monograph “Variational Problems In Geometry” by Seiki Nishikawa covers a lot of relevant material, but is mainly based on local coordinate calculations, and also only addresses the harmonic map case. I have found other relevant papers and textbooks along the same lines.

My question is: What are the best references for this type of calculus of variations? I wish to know what has been already published regarding global formulation using vector bundles with covariant derivatives/linear Ehresmann connections (geared mainly towards calculating the first and second variations), so that I may tie my work in with known results and identify particular areas that could be expanded/improved upon.

For reference, the texts/papers I've so far found relevant to or inspiring my work include: Marden and Hughes “Mathematical Foundations of Elasticity”, Jeff M Lee “Manifolds and Differential Geometry”, Yuanlong Xin “Geometry of Harmonic Maps”, David Bleecker “Gauge Theory and Variational Principles”, Richard Palais “Foundations of Global Nonlinear Analysis”.

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    $\begingroup$ For an approach using exterior differential systems, see amazon.com/… $\endgroup$ – Deane Yang Nov 27 '11 at 1:03
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    $\begingroup$ You can try Ian Anderson's survey and book about the variational bicomplex. Survey: digitalcommons.usu.edu/cgi/…, Book: math.uni.lu/~michel/data/VARIATIONNAL%20BICOMPLEX.pdf $\endgroup$ – Dmitri Pavlov Nov 27 '11 at 9:12
  • $\begingroup$ In what sense do you mean global? As in working with analysis on infinite dimensional spaces of maps, like $C^\infty(M,S)$? Or as in taking the topologies of $M$ and $S$ into account, rather than working in individual charts? $\endgroup$ – Igor Khavkine Nov 27 '11 at 10:11
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    $\begingroup$ Global in the sense of avoiding local coordinate expressions. In this case, it is achieved using higher level properties of the relevant objects, such as product rules, symmetries of various tensors, etc. To be careful about the analysis will require talking about the infinite dimensional spaces of maps as you mentioned. $\endgroup$ – Victor Dods Nov 28 '11 at 0:59

Anderson's book should also be my recomendation. Although it is not finished (and there are some minor mistakes in it), it covers many different topics, and collects many results (and bibliography) that are hard to find elsewhere.

As for the Czech school, there is Krupka's review Global variational theory in fibred spaces, (2010), and Krupkova's Variational Equations on Manifolds (2009), which include a complete list of references inside.

Another interesting reference is the section devoted to variational caluclus in Aldrovandi's book An introduction to geometrical physics.

  • $\begingroup$ Anderson also has a much shorter survey about variational bicomplexes, which I mentioned in my comment to the main post. $\endgroup$ – Dmitri Pavlov Jan 2 '12 at 17:05
  • $\begingroup$ You may be interested in contributing to a proposal Spanish language version of math stackexchange; it could use some input from fluent professors: area51.stackexchange.com/proposals/64529/… $\endgroup$ – Brian Rushton Feb 1 '14 at 3:11

Under certain nondegeneracy conditions, a Lagrangian $L$ on a manifold $M$, i.e., a function on the total space $TM$ of the tangent bundle of $M$ defines a diffeomorphism

$$\Psi_L: TM\to T^*M$$

known as Legendre transform determined by $L$. The cotangent bundle is a equipped with a natural symplectic structure. The lagrangian $L$ induces via $\Psi_L$ a function $H$ on $T^*M$, the Hamiltonian of the variational problem and via $\Psi_L$, the extremal curves of the variational problem correspond to curves on $T^*M$ which are integral curves of the symplectic gradient of $H$. This is as invariant a description of ($1$-dimensional) variational calculus as it gets.

When is the Legendre transform well defined? For example, when the restriction of $L$ to any tangent space is strictly convex. This happens for the lagrangians arising in classical mechanics, or the Lagrangians in Finsler geometry.

For multi-dimensional variational calculus, things are more complicated and personally I find it more productive to deal with each individual variational problem separately.


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