Good reference for globally formulated calculus of variations on Riemannian manifolds? - MathOverflow

Good reference for globally formulated calculus of variations on Riemannian manifolds?

2011-11-27T00:04:09Z

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”.

Answer by Jose Navarro for Good reference for globally formulated calculus of variations on Riemannian manifolds?

2011-12-30T10:49:23Z

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.

Answer by Liviu Nicolaescu for Good reference for globally formulated calculus of variations on Riemannian manifolds?

2012-01-08T11:58:42Z

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.