By Henri Hogbe-Nlend, Vincenzo Bruno Moscatelli
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Example text
44) t=0 the chain rule yields r vh · zi = ζji α j . j =1 Moreover for a differentiable function f defined on M we have (vh · f )(z1 , . . zn ) = i ∂f vh · zi = ∂zi ζji j i ∂f ∂zi . 14 For a differentiable group action G × M → M, with group parameters a1 , a2 , . . , ar near the identity element e, and z = (z1 , . . 45) α j vj . 2. 47) j vh · uαK = α φK,j αj . 45), for a prolonged action is vj = ξji i,α,K ∂ ∂ ∂ α + φ,jα α + φK,j . 14, x= ax + b , cx + d y = 6c(cx + d) + (cx + d)2 y, ad − bc = 1.
45) α j vj . 2. 47) j vh · uαK = α φK,j αj . 45), for a prolonged action is vj = ξji i,α,K ∂ ∂ ∂ α + φ,jα α + φK,j . 14, x= ax + b , cx + d y = 6c(cx + d) + (cx + d)2 y, ad − bc = 1. Take local coordinates near the identity to be (a, b, c) so that e = (1, 0, 0). 6. Hint: (α, β, γ ) = (α 1 , α 2 , α 3 ). 10 to the prolonged action is the first step of Sophus Lie’s algorithm for calculating the symmetry group of a differential equation. This algorithm is discussed in detail in textbooks, for example Bluman and Cole (1974), Ovsiannikov (1982), Bluman and Kumei (1989), Stephani (1989), Olver (1993), Hydon (2000) and Cantwell (2002), and we refer the interested reader to these.
It is well worth taking the time to calculate a selection of prolongations of infinitesimals, not only to be sure which index is which in the preferred notation, but then also to implement it in the preferred computer algebra system. The software will be needed to do the calculations in Chapter 4. 53). The input will be lists of dependent and independent variables, the infinitesimals ξi and φ α and an index of differentiation K. The output will be φKα . 22 Virtually every computer algebra system has a package that implements Lie’s algorithm to find symmetries of differential equations, and all these have, of necessity, implementations of the prolongation formulae buried in them.