This chapter links the first half of our treatise to the second by preparing the transition from the Euler—Largrange formalism of the calculus of variations to the canonical formalism of Hamilton—Jacobi, which in some sense is the dual picture of the first. The duality transformation transforming one formalism into the other is the so-called Legendre transformation derived from the Lagrangian F of the variational problem that we are to consider. This transformation yields a global diffeomorphism and is therefore particularly powerful if F(x, z, p) is elliptic (i.e. uniformly convex) with respect to p. Thus the central themes of this chapter are duality and convexity.
Weitere Kapitel dieses Buchs durch Wischen aufrufen
- Legendre Transformation, Hamiltonian Systems, Convexity, Field Theories
- Springer Berlin Heidelberg
- Chapter 7
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