ۥ- a[Y!}}}}}}}   4+&J$key words first integrals of \time \@ "d MMMM, rrrr"4 stycznia, rrrr Geometrical symmetries of first order ordinary differential equations Abstract The general description of continous and discrete symmetries of first order ordinary differential equations is given here. We analyse algebraic structures as well as geometric properties associated with this kind of symmetries. We show that Lie algebra of continous symmetries contains system of the Virasoro algebras A geometrical symmetry of a system of differential equation may be defined as any change of its dependent and independent variables which preserves a form of the system. More precisely (such a change) a function defining the change is a symmetry transformation if the system can be in new variables got by the same mathematical formulae like in the old ones. It is also possible to define symmetries as a transformations in space of variables of the system which map solution to its solution. Booth definitions are equvalent in a subset in the domain of the system on which uniquely defined solution of the system exist and when considering only invertible transformations(1). The second definition allows one to relax the assumption of differentiability and invertiblity of the symmetry transformation as it is not neccesary to make substitution into the system. The most straighforward method of determining symmetries is to substitute an unknown, but assumed to be invertible and smooth enough, function of variables directly into the system and then impose the condition of the first definition above. As a result one obtains system of partial differential equation for substituted function whose solution are wanted symmetry transformations. The great advantage of this method is that a very general description of various kind of symmetries, including discrete too, can be uniformly obtained in this way. As it will be shown it is particularly beneficial for first order ordinary differential equations and their systems. The major difficulty in applications of this method is that the system of equation which has to be solved to find symmetries is, in many particular cases, a nonlinear one and very difficult to solve. This is also a reason why this method rearly appears in applications. The less general bat very efficacious(co mniej pochlebnego) is the method invented by Sophus Lie around 1870 [Lie, Olv, Ovs, BluKu, Vin, Steph]. Namely, for many systems of equations the question of determining symmetries become simpler when symmetries form a Lie or locall Lie group of transformations(2). and are considered only infinithesimaly. In this case they are also transformations of variables but additionaly depend on at least one parameter. If moreover whey are limited to act only infinitesimaly one can define them in terms of, symmetry group generator. In comparision with the method (based on) of direct substitution in this approach one needs to solve only linear partial system of differential equations for (components)coordinates of the generator. The knowledge of generators provides then several tools for examining properties of the equation, symmplifying its form or even integrating it by quadratures. However, difficulties connected with the problem of solving differential equations are inevitably present in Lie's method, it has been shown that it is very useful and posses a wide range of, booth physical and mathemetical, applications. ___________________________________________________________________________ (wstopce na pierwszej stronie) (1) In order to change variables in differential equations by means of some transformations one needs also its inverse. ___________________________________________________________________________ Because of limitations mentioned above and as we shall to provide possibly general description of symmetries of first order ordinary differential equations we have to cast aside infinithesemal methods. To motivate this step let as show a simple example Example Consider a family osad Equation.3  of transformations osad Equation.3  osad Equation.3 . defined for each osad Equation.3  and osad Equation.3 . The infinitesimal "generator" of this family is osad Equation.3 . In other words the family osad Equation.3  is infinitesimaly constant but it is obviously a family of symmetries of diferential equation osad Equation.3  This kind of symmetries can not be succesfuly analised with help of infinithesemal method. By the way this family is not a Lie group of transformations of osad Equation.3  even localy. We will use a method of direct substitutions which allows one to investigate the symmetries given in example above too. The price which is paid is that the symmetries will be here formulated in terms of first integrals Although we deal here only with ordinary differential equations the method presented here is also useful for partial differential equations whose solutions may be obtained by solving ordinary differential equations. An interesting example here is two dimensional generalised toda latice whre solutions may be constructed by nonlinear transformations on solutions of two ordinary diferential equation systems. 1. First order ordinary differential equations and first integrals We shall begin with system of ordinary differential equations osad Equation.3  defined on a subset osad Equation.3  of osad Equation.3  dimensional euclidean space osad Equation.3 , with coordinates osad Equation.3 . In this section the system osad Equation.3  is asssumed (required to possess) osad Equation.3  first integrals osad Equation.3 , such that the mapping osad Equation  in which osad Equation.3  is a locall diffeomorphisms from some set osad Equation.3  onto set osad Equation.3  with its inverse diffeomorphism osad Equation.3  where osad Equation.3  from osad Equation.3  onto osad Equation.3 . _______________________________________________________________________ Remark. To satisfy the requirement above it is enough to demand that in a neighbourhood of a point osad Equation.3  each functions osad Equation.3  posses continous partial derivatives and that the matrix osad Equation  has non-zero determinant at the point osad Equation.3 . If these conditions are met (then) also the mapping osad Equation.3  is continously differentiable and possess nondegenerated Jacobian matrix at osad Equation.3  because we have chossen osad Equation . Then (with use of the inverse function theorem), by inverse function theorem [CWD p. ], osad Equation.3  is invertible on some set containing osad Equation . This set is here denoted by osad Equation.3  ________________________________________________________________________ Now we let the integrals osad Equation.3  be the new dependent variables of the system osad Equation.3  and osad Equation.3  its new independent one. We check that in new variables the system osad Equation.3  is of the form (We check that this change turns the system (1.1) into the following one???) osad Equation.3  Because osad Equation.3  and osad Equation.3  are (diffeomorphicaly connected) joined by (diffeomorphism) formulae osad Equation.3  and osad Equation.3  the following equality hold Firstly we have by properties of partial derivatives osad Equation.3  Functions osad Equation.3 , as first integrals of osad Equation.3 , are have to satisfy the partial differential equation osad Equation.3  and substituting to the last equality we obtain (and substitution of it into the last equality above provides) osad Equation.3  but the sum over osad Equation.3  here equals to osad Equation.3  as it follows again from properties of partial derivatives osad Equation.3 , Hence we have osad Equation.3  The transformation of total derivatives is osad Equation.3 . Hence for each i-th component of the systmem osad Equation.3  in new variables is osad Equation.3  but osad Equation.3  what we have proved previously hence we obtain osad Equation.3  ??? jak to wyglda osad Equation.3  ??? which is equivalent to osad Equation.3  as the matrix osad Equation.3  is not - degenerated. ____________________________________________________________________ as first integrals have to satisfy the partial differential equation osad Equation.3 , osad Equation.3 , ____________________________________________________________________ 2. Geometrical symmetries of first order ordinary differential equation systems Geometrical symmetries of systems of type (1.1) may be defined on several ways [Olv, BluKu, Steph,Vin, Ovs]. The most suitable for our considerations in this section is the following one A symmmetry of system of equations osad Equation.3  is any change of its dependent variables osad Equation.3  and independent one osad Equation.3  osad Equation.3 , which leaves its form invariant. More precise, after the symmetry transformation, in new wariables osad Equation.3  the system osad Equation.3  can be brought to exactly the same form osad Equation.3 . The definition given above is independent of the choice of variables. This implies that when we are looking for symmetries or examine their properties we may (przej do zmiennych s J ??? by means of transformations osad Equation.3  and consider system osad Equation.3  instead of osad Equation.3 . Then one can always get symmetries back for original system using transformation osad Equation.3  Thiss passage is given only loccaly???? ________________________________________________________________ Conversely On the some way can we get them back for original system (1.1). in coordinates so that the system (1.1) takes the form (1.6) (which is easer to analyse?????). deal with , turns into _________________________________________________________________ Firstly, we find the condition that must be satisfied when the following change of variables (2.3) osad Equation.3  is a symmetry transformation for (1.6). Accordingly to choosen definition of symmetry in new variables osad Equation.3  the system (1.6) should be in a form (2.4) osad Equation.3 , osad Equation.3  As we have osad Equation.3  it is equal to zero when osad Equation.3  or in other words when functions osad Equation.3  in (2.3) do not depend explicite on osad Equation.3 . Geometrically %We consider the system (1.6) defined on whole space Discrette symetries AFINE??????????????????????????????? Continous parametrised symmetries Consider a family osad Equation.3  of diffeomorphisms osad Equation.3  in which osad Equation.3  plays a role of parameter. Transformations osad Equation.3 are symmetries of (1.6) as functions osad Equation.3  does not depend on osad Equation.3  explicite. We will assume also that osad Equation.3 . By linearization about osad Equation.3  (the identity transformation) we obtain osad Equation.3  osad Equation.3  in which osad Equation.3 , osad Equation.3 , osad Equation.3  The vector field osad Equation.3  is called an infinithesimal generator of osad Equation.3  or symmetry generator of (for) (1.6). Notice that the same shape of symmetry generator is obtained by infinithesimal invariance criterion for (1.6). That is the cooeficient about osad Equation.3  does not depend on osad Equation.3 explicite. Let osad Equation.3  osad Equation.3  be two symmetry generators. Then their Lie bracket is a vector field osad Equation.3  with coeeficients osad Equation.3  osad Equation.3  and as functions osad Equation.3  depend only on osad Equation.3  it is also a symmetry generator If generators are fields of class osad Equation.3  then the set osad Equation.3  of them the structure of Lie algebra may be given with multiplication defined by Lie bracket. Consider a set in this algebra (base) osad Equation.3 , osad Equation.3  osad Equation.3 , osad Equation.3 , osad Equation.3  In this base they obey the commutation rules osad Equation.3 ,, osad Equation.3 , osad Equation.3  osad Equation.3 , osad Equation.3 , osad Equation.3  They apears to have interesting and surprising algebraic structure of Virasoso Algebras Example 1. Equation on osad Equation.3  with coordinates osad Equation.3  osad Equation.3  first integral osad Equation.3  change of variables osad Equation.3 , osad Equation.3  jacobian matrices osad Equation.3 , osad Equation.3  in new coordinates osad Equation.3 , osad Equation.3  Hence equation 1 has a form osad Equation.3  or equivalently osad Equation.3  Algebraic structure for family of solution and for discrette transformation first integrals finding methods (2.2) osad Equation.3  examples of finding methods 1) by solution of equation in space of dependent and independent variables osad Equation.3  2) first integrals defined by general solution of an equation system osad Equation.3  in which osad Equation.3 , osad Equation.3  are osad Equation.3  matrices and for some natural?? number osad Equation.3  satisfy the condition osad Equation.3 . A general solution to this equation may be obtained by Picard Method [Zill] in a form of a finite series osad Equation.3  where osad Equation.3  3) first integral by guessing from the equation bk w Maurinie lub n-body problem 4) first integral by trace method for equations of type (t4) osad Equation.3  equations in Lax type representations included here. Integrals given as osad Equation.3  because of the property osad Equation.3  example - one dimensional generalised Toda latice Let osad Equation.3 , osad Equation.3  are generators of a simple Lie algebra osad Equation.3  of rank osad Equation.3  which satisfy the following commutation rules osad Equation.3 , osad Equation.3 , osad Equation.3 , osad Equation.3 , (1.2.8), where osad Equation.3  is the Cartan matrix of osad Equation.3 . The generators osad Equation.3  are called Chevalley generators of osad Equation.3 . Let osad Equation.3 , osad Equation.3  are operators on osad Equation.3  in which functions osad Equation.3 , osad Equation.3 , osad Equation.3  osad Equation.3  depend solely on variable osad Equation.3 . The differential equation of type (t4) osad Equation.3  or equivalently more Lax like type osad Equation.3  leads to the following first order ordinary differential equation system for functions osad Equation.3 , osad Equation.3 , osad Equation.3  osad Equation.3  osad Equation.3 , (t5) osad Equation.3 , osad Equation.3 , Taking osad Equation.3 , osad Equation.3 , osad Equation.3  from (t5) equation (t5) we obtain generalized one dimensional Toda latice equation system osad Equation.3 , osad Equation.3 , osad Equation.3  5) by continous parametrised symmetries of the equation sprawdzi [BluKu, Steph]???? functionaly independent exist test when functionaly independent such that (1.5) osad Equation.3 , osad Equation.3 , osad Equation.3 , explicite solution For simplicity we will call such kind of systems an explicite integrable ones LAX REPR Symmetries By a symmetry of an ordinary differential equation is meant here a transformation of it's dependent and independent variables which preserves the form of equation osad Equation.3  mappings osad Equation.3  problem jak transformuje si s the some form of the equation osad Equation.3  in new wariables can be discrette and continous and mixed discrette continous if continous -> generator and parameter osad Equation.3  check it for sure!!!!!!!!!!!!!!!!!!!!!!!!!!!! continous with at least on parameter - algebraic structure Virasoro, Katz Mody Virasoro Ovsjanikov analyse of algebraic structure Example method of integrating generalised Toda Latices is clearly described in [LezSav] _____________________________________________________________________ The Jacobian matrices are given for mappings (1.4) and (1.5) are given respectively (1.6) osad Equation.3 , osad Equation.3  and satisfy the condition (1.7) osad Equation.3  in which osad Equation.3 is osad Equation.3  unit matrix. _____________________________________________________________________ (1.9) osad Equation.3 , (1.9') osad Equation.3  Hence (1.8) and (1.8') give (1.9) osad Equation.3 , (1.9') osad Equation.3  from (1.3) we have osad Equation.3 , osad Equation.3 , _____________________________ Now we find a form of the system (1.1) when adopting the integrals osad Equation.3  as its new dependent variables and osad Equation.3  as a new independent one. The Virasoro algebras are upon exploits promote properties may be inevitably(w sposb nieunikniony) transfered to in a sequel _______________ _______________________________________________________________________ and algebraic structures as well as geometric properties associated wit them is our aim here. implies symmetries Symetries arises as solutions of differential equations in theory of ordinary and partial Unfortunately it turns out that this method range of applications brought to the it is given Inspite of this virtues Are not suitable for our ends Diffeomorphism f iff f and its inverse continously diferentaiable (i.e. of class C1) For the sake of review Substituting (1.8) in (1.7), yields Substituting smth into (1), gives Actualy substituting (3) into (2) and summing over n, we obtain xxx Thence we have ???Notice that the same definition can be appplied also to higher order ODEs or PDE.??? This method allows as to consider discrette and continous parametrised symetries simultaneously. _____________________________________________________________________ is of the form Since S and S' are isometric, it follows [cf. 10, 15] from these stipulations that Zrezygnujemy tutaj z rozwaania symetrii jako ciglych transformacji Symetrie rwna rniczkowych take mog posiada algebraiczn by rozpatrywane jako lokalne grupy Liego [Olv, Ovs????] jest to jednak \time \@ "d MMMM, rrrr"4 stycznia, rrrr method presented here do not use infinitesemal invariance criterion the prise is paid it is in a language of first integrals The method presented here is freed from this disadvantage INTRODUCTION 1) SYMMETRIES ussual way assumptions to consider is that solution exist ode ->infinitesemal invariance->symmetry->solution, symmetry = tool for finding solution explicite described in [olv, ovs,bluku] by integration strategies [bluku, steph] which leads to another ODE , we may hope that it will be easier to solve our way assumptions \symbol SYMBOL \f "Symbol" \s 10 \h solution exist and is given explicite with first integrals \symbol SYMBOL \f "Symbol" \s 10 \h which may be found by infinitisemal techniques as well is freed from infinitisemal invariance criterion and Stephani Although symmetries that do not form a (locall) Lie group can be very useful in studying differential equations, there is no practicable way to find them. condition in terms of its infinitesemal generator exist for such kind of symmetries exist Although clear conditions for existence and uniqueness of a solution of ODE are given [??] the question of finding it in an explicite form still remains open. The system is required to posses uniquelly defined solution (1.2) osad Equation.3 , such that osad Equation.3 . for each point osad Equation.3  of some set osad Equation.3 , ___________________________________________________ (existence and uniquenes in osad Equation.3  [Mau p. 209-]) If these conditions are satisfied K. Maurin Analiza 270- function osad Equation.3  is bounded continous and locally satisfy Lipschitz condition dependent independent ___________________________________________________ Namely, osad Equation.3  for all admissible values of osad Equation.3  osad Equation.3  (which is also a neccesary and suufficient condition for function osad Equation.3  to be a first integral of (1.1) [Mau p. 270-brak dowodu]) References two problems of classical monodromy group theory for second order linear differential equations are treated a) to calculate the group when the differential equation is given; b) to determine all differential equations having a specified monodromy groups. This problem was also described by D. Hilbert in his lecture (1900) in Paris Hejhal D. A.,(1975), Monodromy groups and linearly polimorphic functions, Acta Math. 135, 1-55 P. J. Olver, (1977) Evolution Equations possesing Infinitely many symmetries. J. Math. Phys. 18., 1212-1215. Symmetry reduction of classical soliton equations lead to ODEs of Painleve type, meaning those whose movable singularities are only poles. [ARS] Ablowitz M.J, Ramani A., and Segur H., A connection between ninlinear evolution equations and ordinary differential equations of P-type, J. Math. Phys. 21 (1980), 715-721. Applications of associative, Jordan and other types of algebras to ODEs [Roh] Rohrl, H, Algebras and differential equations, Nagoya Math.J. 68 (1977), 59-122 [Wal] Walcher S, Algebras and Differential Equations, Hadronic Press, Palm Harbor, Fla., 1991 ???????? [Fok] Fokas A. P., A symmetry approach to exactly solvable evolution equations, J. Math Phys. 2 (1980), 1318-1325. P. D. 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