By Frederick S. And Frederick H. Bailey Woods

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Moreover, for a given differential equation, such transformations can be found algorithmically. ] This follows from the properties of such transformations and, in particular, their characterization by infinitesimal generators [see Chapter 2]. 31 References on dimensional analysis specific to various fields include: de Jong (1967) [economics]; Sedov (1982), Birkhoff (1950), Barenblatt (1979, 1987, 1996), and Zierep (1971) [mechanics, elasticity, and hydrodynamics]; Venikov (1969) [electrical engineering]; Taylor (1974) [mechanical engineering]; Becker (1976) [chemical engineering]; Haynes (1982) [geography]; Kurth (1972) [astrophysics]; Murota (1985) [systems analysis]; Schepartz (1980) and Barenblatt (1987) [biomedical sciences].

70a,b) in terms of t = ò G(e ¢) de ¢. 0 2. 71a) xy . 71a,b). 71a,b). 71a,b) in terms of its Lie series developed from ξ (x). 3. 93), find the infinitesimal generator, explicitly integrate out the initial value problem for the infinitesimals, and find canonical coordinates: (a) in (x, t)-space; and (b) in (x, t, u)-space. 51 4. Find the one-parameter groups of transformations and canonical coordinates corresponding to the infinitesimal generators: ¶ ¶ (a) X 1 = x + y ; ¶y ¶x ¶ ¶ (b) X 2 = x - y ; and ¶y ¶x ¶ ¶ (c) X 3 = x 2 + y2 .

6) becomes Y= Proof. We have Y = åi =1h i (y ) n ¶ ¶yi ¶ . 54) . 51) it follows that h i (y ) = Xyi = 0, i = 1,2, K , n - 1, h n (y ) = Xyn = 1. 54). 55b) with infinitesimal generator Y= ¶ . 56b) the infinitesimal generator is given by X = x ¶ ¶x + 2y ¶ ¶y . The canonical coordinate r(x, y) satisfies Xr = x ¶r ¶r + 2y = 0. 58) with the general solution given by r ( x, y ) = y = const. 59) The canonical coordinate s(x, y) satisfies Xs = x ¶s ¶s + 2y = 1. 60) is given by s(x, y) = s(x) satisfying ds 1 = .

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