By B. Abraham-Shrauner, A. Guo (auth.), N. H. Ibragimov, M. Torrisi, A. Valenti (eds.)

On the get together of the one hundred and fiftieth anniversary of Sophus Lie, a world paintings­ store "Modern staff research: complex analytical and computational equipment in mathematical physics" has been geared up in Acireale (Catania, Sicily, October 27­ 31, 1992). The Workshop was once aimed to enlighten the current kingdom ofthis speedily increasing department of utilized arithmetic. major themes of the convention have been: • classical Lie teams utilized for developing invariant options and conservation legislation; • conditional (partial) symmetries; • Backlund alterations; • approximate symmetries; • crew research of finite-difference equations; • difficulties of crew type; • software program programs in crew research. The good fortune of the Workshop used to be as a result of participation of many specialists in crew research from assorted nations. This ebook includes chosen papers awarded on the Workshop. we wish to thank the medical Committee for the beneficiant help of recommending invited lectures and choosing the papers for this quantity, in addition to the contributors of the Organizing Committee for his or her aid. The Workshop was once made attainable by means of the monetary help of numerous sponsors which are indexed less than. it's also a excitement to thank our colleague Enrico Gregorio for his worthwhile support of this volume.

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Extra resources for Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics: Proceedings of the International Workshop Acireale, Catania, Italy, October 27–31, 1992

Sample text

The left hand side of / is an invariant, Q the Euler-Lagrange equations (10). A and 1/ are arbitrary. With respect to case i), we find from (17) (18) where E can be calculated by quadrature to be (19) = = (; The definition of momenta by PP Pand Pc shows that E is nothing but the energy in the relative coordinates of the two ions in the Paul trap. This result is more or less trivial since we started from a Lagrangian. A ±1 and v 0, we already know that f is an invariant of motion. The question now is what happens if the z-component of f is = = 50 G.

P .. 14) 3. 2) and we require the fixed function V({) to be an even degree polynomial in {, V ' ({)/{ ~ 0, V"({) ~ 0 for all { E R, and the fixed function y(t) satisfies y(t) + V' (y(t)) = O. 3) ,= Here we use the notation d/dy. 5). 5) is, in fact, a Bateman pair. 2) is given explicitly by LB = u·(u + V'(u)). 5) is given explicitly by PI .. V'(qy). 8) is time-independent, hence a constant of the motion. 9) The constants of the motion Hand h are obviously functionally independent. What was a pleasant surprise is that they are in involution.

_ _1)" ddt: ~ denotes the varia- = &q~"J tional derivative or the Euler operator. At this preliminary stage of our examination, we know the equations of motion and probably the physical interpretation contained in the equations. It is, however, unknown to us under which conditions solutions for the system (1) can be found. 1. =O 0, k 1,2, ... , N to determine symmetries of ODE's does not provide all symmetries of the equations. In turn, we have to use a generalized method to uncover the 'hidden' symmetries.

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