ۥ- Pi^ODDDDDDDDDDDD D D 9O^^^^^^^^45^4i^i^=\time \@ "MMMM d, rrrr"stycze 7 Geometrical symmetries of first order ordinary differential equations Abstract The general description of continuous 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 continuos 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 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 equivalent in a subset in the domain of the system on which its uniquely defined solution exist and when considering only diffeomorphic transformations(1). The second definition allows one to relax the assumption of differentiability and invertiblity of the symmetry transformation as it is not necessary to make substitution into the system. The most straightforward 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 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 non-linear one and very difficult deal with. This is also a reason why this method really appears in applications. The less general bat very efficacious 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 local Lie group of transformations(2) and are considered only infinitesimally close to the identity. In this case they are also transformations of variables but additionally depend on at least one parameter. The dependence on parameter allows one to apply the infinitesimal invariance criterion for the system which provides conditions for a symmetry group generator. In comparison with the method based on direct substitution in this approach one needs to solve only linear system of partial differential equations for components of the generator. The knowledge of generators provides then several tools for examining properties of the equation, simplifying its form or even integrating it by quadratures. The method of Lie has become the most important one for symmetry analyse of partial differential equations but in case of ordinary equations its usefulness in case of ordinary differential equations is often questionable 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 lattice where solutions may be constructed by non-linear transformations on solutions of two ordinary differential 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 space osad Equation.3 , with coordinates osad Equation.3 . In this section the system osad Equation.3  is assumed (required to possess) osad Equation.3  first integrals osad Equation.3 , such that the mapping osad Equation.3  in which osad Equation.3  is a local 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 continuos partial derivatives and that the matrix osad Equation.3  has non-zero determinant at the point osad Equation.3 . If these conditions are met (then) also the mapping osad Equation.3  is continuously differentiable and possess nondegenerated Jacobian matrix at osad Equation.3  because we have chosen osad Equation.3 . 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.3 . 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 this change turns the system osad Equation.3  into the following one osad Equation.3  Because osad Equation.3  and osad Equation.3  are joined by diffeomorphism osad Equation.3  and osad Equation.3  the following equality hold osad Equation.3  Functions osad Equation.3 , as first integrals of osad Equation.3 , have to satisfy the partial differential equation osad Equation.3  and substituting it to the last equality shows that osad Equation.3  The sum over osad Equation.3  here equals to osad Equation.3  as it follows again from properties of partial derivatives osad Equation.3 , and we have osad Equation.3  .Hence the transformation of the total derivative of osad Equation.3  over osad Equation.3  is osad Equation.3 . components of the system osad Equation.3  in new variables takes a form osad Equation.3  osad Equation.3  but as we have shown osad Equation.3  hence we obtain osad Equation.3  osad Equation.3  or equivalently, as the matrix osad Equation.3  is not - degenerated. osad Equation.3  osad Equation.3  which are components of the system osad Equation.3 . 2. Geometrical symmetries of first order ordinary differential equation systems The most suitable for our considerations in this section is the following definition of symmetry of osad Equation.3  A symmetry 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 variables 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 pass to the variables osad Equation.3  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 in original variables osad Equation.3 system using transformation osad Equation.3 . Firstly, we find the condition that must be satisfied when the following change of variables osad Equation.3 ???? is a symmetry transformation for osad Equation.3 . Accordingly to chosen definition of symmetry in new variables osad Equation.3  the system osad Equation.3  should be in a form 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 transformation osad Equation.3  do not depend explicite on osad Equation.3 . As we have mentioned earlier, having a symmetry transformation osad Equation.3  for osad Equation.3  we may construct also the symmetry osad Equation.3  for osad Equation.3  by means of superposition osad Equation.3  It is possible to do so if domains and images of the appropriate mappings above overlaps. Continuos parameterised symmetries Consider a family osad Equation.3  of diffeomorphisms osad Equation.3  in which osad Equation.3  plays a role of parameter. We assume that osad Equation.3  is an identity transformation for osad Equation.3  Transformations osad Equation.3 are symmetries of osad Equation.3  as functions osad Equation.3  does not depend on osad Equation.3 . By linearization for infinitesimal values of osad Equation.3  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 infinitesimal generator of osad Equation.3  or symmetry generator of the system osad Equation.3 . Notice that the same shape of symmetry generator is obtained by infinitesimal invariance criterion for osad Equation.3 . That is the coefficient about osad Equation.3  does not depend on osad Equation.3 explicate. Let osad Equation.3  osad Equation.3  be two symmetry generators. Then their Lie bracket is a vector field osad Equation.3  with coefficients 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 appears to have interesting and surprising algebraic structure of Virasoso Algebras Discrette symetries 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 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] _____________________________ 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 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. _____________________________________________________________________ Thiss passage is given only loccaly???? 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"7 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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