Extremal fields and global minimals

Abstract

The two-dimensional torus has the standard representation ${{\mathbb{T}}^{2}} = {{\mathbb{R}}^{2}}/{{\mathbb{Z}}^{2}}$. We often will work on its covering surface ℝ² where everything is invariant under its fundamental group ℤ². In this chapter we deal with the variational principle ∫F(t,x,p) dt on ℝ², whereF is assumed to satisfy the following properties:

• i) F ∈ C²(${{\mathbb{T}}^{2}} \times {{\mathbb{R}}^{2}}$):

  $$\begin{array}{*{20}{c}} {a)F \in {{C}^{2}}({{\mathbb{R}}^{3}}),} \hfill \\ {b)F(t + 1,x,p) = F(t,x + 1,p) = F(t,x,p).} \hfill \\ \end{array}$$

  (2.1)

• ii) F has quadratic growth: There exist δ > 0, c > 0 such that

  $$\begin{array}{*{20}{c}} {c) \delta \leqslant {{F}_{{pp}}} \leqslant {{\delta }^{{ - 1}}},} \hfill \\ {d)|{{F}_{x}}| \leqslant c(1 + {{p}^{2}}),} \hfill \\ {e) |{{F}_{{tp}}}| + |{{F}_{{px}}}| \leqslant c(1 + |p|).} \hfill \\ \end{array}$$

  (2.2)