Families of lift and contact transformations

Abstract

A study of the contact transformation is carried out, with special emphasis on the lift transformation. The latter is the more basic concept, because it applies to a single curve, rather than to pairs of them, in the first instance. From that viewpoint the lift transformation can also transform an unfamiliar system of partial differential equations to a familiar one, without reference to contact. We illustrate this by showing how the semi-geostrophic equations of meteorology transform to hamiltonian form. General and special lift transformations are distinguished. New and very wide families of such transformations are derived. These give a desirable perspective to particular cases such as the Legendre transformation and the geostrophic transformation. A brief study of generalized duality is described. The distinction between regular and singular lifted curves is discussed in general terms, and their participation in a contact transformation is illustrated by an accessible physical example.

The mathematical structure of theories of semigeostrophic type

“…Then, if we refrain from assigning the function F (s i ), equation (7.29) becomes an equation Salmon (1985), Magnusdottir & Schubert (1990 and Purser (1988Purser ( , 1993. Future work will be needed to study them in the context of the guidance offered by the lift structure explained here and in Sewell & Roulstone (1994).…”

“…There are, after all, many different types of lift transformation. Sewell & Roulstone (1994) proved how those in (3.9) are particular members of a class deducible from a simple algebraic starting point, namely by requiring the X i and Ψ to be quadratic functions of the x i , ψ and z i . Here we explore briefly a different starting point.…”

“…(In lift geometry there is a sense in which a hypothesis like ψ = ψ(x i ) confines discussion to regular points only (see (5) of Sewell & Roulstone 1994) where auxiliary parameters are not needed to describe the surface. Non-regular points where dψ = z i dx i still holds can be of intrinsic interest, but we shall not digress to discuss them here.…”

Holomorphic structures in hydrodynamical models of nearly geostrophic flow

“…Finally, we show how the contact structure of the semi-geostrophic equations, which is perhaps the most important mathematical property of these equations because it facilitates practical solution strategies (see Purser (1993) and references therein; see also Roulstone & Sewell (1997) and Sewell & Roulstone (1994)), generalizes to other near-local balanced models.…”