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

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.

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.