Contact Transformations. Legendre and Euler Transformations

S.2.3. Contact Transformations. Legendre and Euler Transformations

S.2.3-1. General form of contact transformations. Consider functions of two variables w = w(x, y). A common property of contact transformations is

the dependence of the original variables on the new variables and their first derivatives:

w=W ξ, η, u, , . (16)

∂ξ ∂η The functions

X, Y , and W cannot be arbitrary and are selected so as to ensure that the first derivatives of the original variables depend only on the transformed variables and, possibly, their first derivatives,

∂ξ ∂η Contact transformations (16)–(17) do not increase the order of the equations to which they are

We now outline the procedure for finding the functions U and V in (17) and the relations that must hold for the functions X, Y , and W in (16). Let us differentiate the first and second expressions in (16) with respect to x and y as composite functions taking into account that u = u(ξ, η). Thus, we obtain the following four relations:

∂q = 1, ∂y ∂η ∂u q+ ∂p q ξ ∂q q η ∂y where p= ∂u ∂ξ , q= ∂u ∂η , and p η = q ξ ; the subscripts ξ and η denote the corresponding partial derivatives. The first two

∂u p+ ∂p p ξ +

relations in (18) constitute a system of linear algebraic equations for ∂ξ and ∂η , and the other two relations form a system of linear algebraic equations for ∂ξ

∂y and ∂η ∂y . Having solved these systems, we find the derivatives: ∂ξ ∂x = A, ∂η ∂x = B,

∂x

∂x

∂ξ ∂y = C, ∂η ∂y = D. Then, differentiating the third relation in (16) with respect to x and y, we express U = ∂w ∂x and V= ∂w ∂y in terms of the new variables to obtain U=A ∂W + ∂W p+ ∂W p + ∂W p

∂p ξ ∂q Relations (17) require that U and W should be independent of the second derivatives, i.e.,

which results in additional relations for the functions

X, Y , W .

In general, a contact transformation (16)–(17) reduces a second-order equation in two indepen- dent variables

∂x ∂y ∂x ∂x∂y ∂y

to an equation of the form

In some cases, equation (20) turns out to be more simple than (19). If u = u(ξ, η) is a solution of equation (20), then formulas (16) define the corresponding solution of equation (19) in parametric form.

S.2.3-2. Legendre transformation. An important special case of contact transformations is the Legendre transformation defined by the

w(x, y) + u(ξ, η) = xξ + yη, x=

where u is the new dependent variable and ξ, η are the new independent variables. Differentiating the first relation in (21) with respect to x and y and taking into account the other two relations, we obtain the first derivatives:

With (21)–(22), we find the second derivatives

2 = J 2 ∂x , ∂η ∂x∂y ∂y∂x ∂ξ∂η ∂y ∂ξ where

=− J

J=

∂ξ 2 ∂η 2 ∂ξ∂η The Legendre transformation (21), with J ≠ 0, allows us to rewrite a general second-order equation with two independent variables

∂x 2 ∂y 2 ∂x∂y

∂w ∂w ∂ 2 w ∂ 2 w ∂ 2 w

(23) in the form

F x, y, w,

∂x ∂y ∂x ∂x∂y ∂y 2

∂u ∂u ∂u

∂u

= 0. (24) Sometimes equation (24) may be simpler than (23).

Let u = u(ξ, η) be a solution of equation (24). Then the formulas

u(ξ, η), x=

define the corresponding solution of equation (23) in parametric form. Remark. The Legendre transformation may result in the loss of solutions for which J = 0. Example 6. The Legendre transformation (21) reduces the nonlinear equation

∂y 2 =0 to the following linear equation with variable coefficients:

2 − g(ξ, η) ∂ u + h(ξ, η) ∂ u ∂η = 0. ∂ξ∂η ∂ξ 2

f (ξ, η) ∂ 2 u

S.2.3-3. Euler transformation. The Euler transformation belongs to the class of contact transformations and is defined by the

relations

∂u

w(x, y) + u(ξ, η) = xξ, x=

Differentiating the first relation in (25) with respect to x and y and taking into account the other two relations, we find that

∂w =

∂w

=− ∂u .

∂x

∂y

The subscripts indicate the corresponding partial derivatives. The Euler transformation (25)–(27) is employed in finding solutions and linearization of certain nonlinear partial differential equations. The Euler transformation (25) allows us to reduce a general second-order equation with two independent variables

∂x ∂y ∂x ∂x∂y ∂y

to the equation

In some cases, equation (29) may become simpler than equation (28). Let u = u(ξ, η) be a solution of equation (29). Then formulas (25) define the corresponding solution of equation (28) in parametric form.

Example 7. The equation

can be linearized with the help of the Euler transformation (25)–(27) to obtain

Example 8. The equation

can be linearized by the Euler transformation (25)–(27) to obtain

References for Subsection S.2.3: M. G. Kurenskii (1934), N. H. Ibragimov (1985, 1994), H. Stephani (1989), B. J. Cant- well (2002), A. D. Polyanin and V. F. Zaitsev (2002).