First Integrals and Hamiltonians of Some Classes of ODEs of Maximal Symmetry

An important class of linear ordinary differential equations (LODEs) consists of those equations having a symmetry algebra ofmaximal dimension.This is partly due to the simple characterization of such a class of equations [1, 2], according to which they are precisely the iterative equations, and equivalently they can be reduced by a point transformation to the canonical form y(n) = 0. Despite the fact that linear equations are the simplest types of differential equations, they are far from being completely understood and yet they frequently appear in the study of all other types of equations and in particular as canonical or reduced form of nonlinear or partial differential equations with more complex structures. Linear equations with maximal symmetry algebras have been studied in some recent papers [3–6], but in all such papers the emphasis has been placed more on the coefficient characterization and the elementary symmetry or the transformation properties of these equations. Nœther’s theorem [7, 8] is a powerful tool that allows us to associate each variational symmetry of the equationwith a conservation law, and, in the case of ordinary differential equations (ODEs), these conservation laws correspond to first integrals which in this instance are true constants of motion. The role of these first integrals is often crucial in the study of solutions and properties of differential equations, including questions related to stability [9, 10] or integrability [11–14].The so-called first integral method for finding exact solutions of nonlinear partial differential equations has also been successfully applied to a wide range of equations modeling various phenomena in physics (see [15] and the references therein). In this paper, we obtain a complete set of linearly independent first integrals for all linear equations of maximal symmetry of order n such that 2 ≤ n ≤ 8. These first integrals are obtained for the most general form of the equations in which they depend on an arbitrary function through an application of Nœther’s theorem. We also give the Hamiltonian formulation for the corresponding class of scalar equations and show amongst others that their general solution can also be obtained by a simple superposition formula from those of a scalar second-order source equation.


Introduction
An important class of linear ordinary differential equations (LODEs) consists of those equations having a symmetry algebra of maximal dimension.This is partly due to the simple characterization of such a class of equations [1,2], according to which they are precisely the iterative equations, and equivalently they can be reduced by a point transformation to the canonical form  () = 0.
Despite the fact that linear equations are the simplest types of differential equations, they are far from being completely understood and yet they frequently appear in the study of all other types of equations and in particular as canonical or reduced form of nonlinear or partial differential equations with more complex structures.
Linear equations with maximal symmetry algebras have been studied in some recent papers [3][4][5][6], but in all such papers the emphasis has been placed more on the coefficient characterization and the elementary symmetry or the transformation properties of these equations.Noether's theorem [7,8] is a powerful tool that allows us to associate each variational symmetry of the equation with a conservation law, and, in the case of ordinary differential equations (ODEs), these conservation laws correspond to first integrals which in this instance are true constants of motion.The role of these first integrals is often crucial in the study of solutions and properties of differential equations, including questions related to stability [9,10] or integrability [11][12][13][14].The so-called first integral method for finding exact solutions of nonlinear partial differential equations has also been successfully applied to a wide range of equations modeling various phenomena in physics (see [15] and the references therein).
In this paper, we obtain a complete set of linearly independent first integrals for all linear equations of maximal symmetry of order  such that 2 ≤  ≤ 8.These first integrals are obtained for the most general form of the equations in which they depend on an arbitrary function through an application of Noether's theorem.We also give the Hamiltonian formulation for the corresponding class of scalar equations and show amongst others that their general solution can also be obtained by a simple superposition formula from those of a scalar second-order source equation.

Basic Properties of Linear Iterative Equations
Let  ̸ = 0 and  be two functions of the variable  and consider the differential operator Ψ = (/) + .Denote by Δ  [] = 0 a LODE of order  in the dependent variable  = ().Then this equation is said to be iterative if up to a normalizing factor one has Δ  [] = Ψ  [], where and where  is the identity operator.A LODE Δ  [] = 0 has a maximal symmetry algebra if and only if it is iterative [1] or, equivalently [1,2], if and only if it can be reduced by a point transformation to the canonical form  () = 0.By the transformation  =  exp((1/) ∫   0  −1 (V)V), where  −1 () is the coefficient of the term of order  − 1, any given LODE may be put into the form which is referred to as the normal form of the equation.
For linear iterative equations in the latter form,  linearly independent solutions can be found in the form for some functions  and V of the independent variable .For a given function  = (), we shall also use the notation   ,   , and so forth for /,  2 / 2 , and so forth, but where convenient we shall also denote / simply by   .It follows in particular from (3) that  and V are two linearly independent solutions of the second-order so-called source equation: In particular, we have Thanks to Abel's identity, the Wronskian w(, V) = det (  V   V  ) of two solutions  and V of a LODE in the form (2) is a constant and may be normalized to one as we shall do by convention.
By a result of Lie [16], all equations of order  ≥ 2 reducible by a point transformation to the canonical form  () = 0 have a symmetry algebra g  of maximal dimension 8 for  = 2 and  + 4 for  ≥ 3. Owing to the linear property, the  + 1 vector fields are symmetries of any iterative equation of the form (2). The three additional ones, obtained in [1], are also generators of the Lie symmetry algebra g  of (2) in the case of equations of maximal symmetry.These +4 symmetries form a basis of g  for  ≥ 3.
For  = 2, a basis of g  consists of the same  + 4 symmetry generators together with the two non-Cartan symmetries given by Recall that, in (6a)-(6f),  and V are assumed to be two linearly independent solutions of the source equation ( 4).The symmetry generators   in (6a) associated with each solution   are often called solution symmetries and generate the abelian Lie algebra A  , while   is referred to as the homogeneity symmetry.Denoting by k  =   the characteristic form of a vector field k with characteristic , it follows that the characteristic forms of the infinitesimal generators listed in (6a)-(6f) are given by In the actual case of LODEs, the characteristic  of k has the simpler expression  = k() − k()  .

Variational Symmetries and First Integrals
Suppose that the equation Δ  [] = 0 is the Euler-Lagrange equation of some variational problem corresponding to a Lagrangian .Let k =  (, )   +  (, )   (8) be a symmetry generator with characteristic function  of the equation and set where k [] is the th prolongation of the vector field k and  *  is the adjoint Fréchet derivative of the differential function  =  [𝑦].The vanishing of k under these two operators, represents some key concepts.Indeed, in the sense of [7], k is called a variational symmetry for Δ  [] = 0 if it satisfies (10) and is called a divergence symmetry if it satisfies (11).The latter condition is equivalent to S(k) = div  for a certain differential function  = [].Divergence symmetries are thus a more relaxed form of variational symmetries, and they also give rise to conservation laws and first integral as variational symmetries do by Noether's theorem.Divergence symmetries satisfy the same condition required on generalized symmetries to be variational symmetries, and, in a more modern language [17], these divergence symmetries are simply called variational symmetries.The characteristic form of a conservation law associated with each divergence symmetry has the form Δ  [] = div , for a certain differential function  = [].In the actual case of an ODE, we have is the total differential operator.Thus  is a first integral of the equation.

(13d)
It can be easily verified that each of these equations is selfadjoint and thus admits a Lagrangian formulation in terms of the Euler-Lagrange operator.A Lagrangian   for each of the equations Δ  [] = 0 of order  in (13a)-(13d) can be taken in the form   = Δ  []/2, but this th order Lagrangian can always be reduced to one of order /2.As already noted, the results of [3,6] can be used to generate iterative ODEs of any given order and thus we omit the listing of those iterative equations of odd orders  = 3, 5, 7 whose first integrals are also to be found.Although odd order equations are not known to possess a Lagrangian formulation, in order for a symmetry k of any such equation to give rise to a first integral it suffices that it be a divergence symmetry, that is, that it satisfies the divergence symmetry condition (11).
On the other hand, under the assumption that  and V are linearly independent solutions of (4), one can verify that for each even order  ≥ 4 the most general divergence symmetry of each of (13b)-(13d) has the form In other words, for even orders  ≥ 4,   =   is the only generator in (6a)-(6f) which is not a term of the divergence symmetry w, and in particular   itself is not a divergence symmetry.Similarly, for  = 2, let w be an arbitrary divergence symmetry vector in the divergence symmetry algebra of (13a).Thus we have where w 1 and w 2 are like in (14a)-(14c) but with  = 2 and where   for  = 1, 2, 3 are arbitrary scalars and  1 and  2 are the non-Cartan symmetries already introduced.Then in the divergence symmetry condition D(w) = 0 expressed as a polynomial in  and its derivatives, it turns out that the coefficients of   and     are 2 3 and −3( 1  +  2 V), respectively.This shows that in the expression of w one must have   = 0 for  = 1, 2, 3. Consequently, w = w 1 + w 2 in this case also, and one readily verifies that any such w is indeed a divergence symmetry.
Let us now denote by L  the Lie subalgebra of all divergence symmetries of g  for a given order .We have thus established the following result.Proposition 1.For every LODE of maximal symmetry of the most general form (13a)-(13d) and of even order , a basis B  of the Lie algebra L  of its divergence symmetries is given by In particular, L  has dimension  + 3.
For equations of odd orders, the most general divergence symmetry w can be sought in the form where w 1 and w 2 are like in (14a)-(14c) but with  = 3, 5, 7 and  1 is a scalar.Then in the divergence symmetry condition D(w) = 0 expressed as a polynomial in  and its derivatives, the coefficient of  (+1) is 2( 2 + 2V + V 2 ), and owing to the linear independence of  and V the vanishing of this coefficient implies that  =  =  = 0.This reduces the expression of w to w = w 1 + 1   , and it is readily verified that the latter vector is always a divergence symmetry.We have thus established the following result.
Proposition 2. For odd orders , a basis B  of L  is given by In particular, L  has dimension  + 1.
(c) If for each  and  we view F , = F , [, q] as a differential function of  and q, then F , is linear in  (and its derivatives) and polynomial in q.
First integrals are often required to be in a form in which they depend only on the parameters of the underlying differential equation and are not in particular expressed in terms of a particular solution of the equation.The first integrals we have found can indeed be expressed solely in terms of the dependent variable  and the coefficient q and their derivatives.For instance, in the case of equations of odd order , given that F , are symmetric functions of  and  and are required to be constant only on the solution space, the required type of first integrals can be obtained simply by letting  =  in the expression of F , .Denoting by F   the resulting first integrals for each odd order  of the equation yields the following expressions: (4) , + 112q (4) + 2   (4) − 2   (5) + 2 (6) . ( We notice that, for each , the first integral F   = F   [, q] viewed as a differential function of  and q is quadratic homogeneous in  and polynomial in q. In [18], first integrals of the trivial equation  () = 0 were explicitly computed for  = 2, 3, . . ., 6, and they are directly expressed in terms of the linearly independent ones thus obtained.The method used in that paper for finding the first integrals  associated with each symmetry k of the equation is direct computation done by solving the boundary value problem However, the calculations done in [18] are only a special case of those done in this paper for equations of maximal symmetry and low order  ≤ 8. Indeed, we have considered equations of maximal symmetry in their most general form (13a)-(13d) and not in the simpler canonical form  () = 0. Of course the first integrals we have found here from a variational principle all satisfy (33a) and (33b).It follows from Liouville's formula that if  1 , . . .,   are linearly independent solutions of an equation of the form (2), then  linearly independent first integrals of the equations are given by the Wronskians However, this direct calculation does not relate the first integral to any variational symmetry of the equation and thus cannot exploit the various properties of such symmetries.Moreover, it requires the knowledge of the complete set of linear independent solutions of the equation itself, while those we found involve only two linearly independent solutions of the second-order source equation.In addition, these two symbolic solutions are not required to be known explicitly but are only assumed to satisfy the second-order equation.In fact, we have ultimately expressed the first integrals found solely in terms of q,  and their derivatives.

Hamiltonian Formulation
In order to gain more insight into the properties of LODEs of maximal symmetry, we shall move one step forward in this section by finding a Hamiltonian formulation of some of these equations and investigate some of its properties.Indeed, in the case of scalar equations, the study in the previous sections was largely facilitated thanks to formula (3) which provides the general solution of any iterative LODE through a very simple superposition principle in terms of two linearly independent solutions  and V of the second-order source equation (4).Thus one of our ultimate goals in this undertaking is to obtain an extension of such a formula to Hamiltonian systems associated with equations of maximal symmetry.
For second-order equations, the transformation into a Hamiltonian system is usually achieved by the common method via the Legendre transformation by defining the conjugate momenta as  = /, where  is the Lagrangian.However, for higher-order equations, one can make use of the Jacobi-Ostrogradski generalized momenta [19,20].
Indeed, consider in this more general approach a given th order Euler-Lagrange equation () = 0, where  is the usual Euler-Lagrange operator (see [7,Page 250]) applied to a Lagrangian [] = (,  () ).In the latter equality,  can be a multi-index of order /2 and  a multivariable but which in our own considerations will be limited to a single dependent variable of the independent variable .Then the generalized momenta are defined by the expressions As for the canonical coordinates, they are given by   =  (−1) , for  = 1, . . ., . (36) The corresponding Hamiltonian function then takes the form and in the new coordinates (, (  ), (  )) the Hamilton-Cartan equations are given as usual by For each LODE Δ  [] = 0, it is well known [7] as already noted that when the equation is of the Euler-Lagrange type, a Lagrangian  is found by setting Moreover, this th order Lagrangian can always be reduced to one of order /2 by eliminating some null Lagrangian terms from it.Denoting as usual by   the reduced Lagrangian corresponding  listed in (13a)-(13d), it is found that for the first two equations.In case  = 2, the canonical variables are (, ) with  =  and  = /  = −  , and, by the Legendre transformation, the Hamiltonian in this case has the expression with corresponding Hamiltonian system where we have Ḟ = /, for any function  = ().It turns out that (42) is linear as was to be expected given that  2 is quadratic, but also (42) coincides with the standard representation of the second-order source equation ( 4) as a first-order system.Thus two linearly independent solutions of (42) are (, −  ) and (V, −V  ), with the usual notations.
For  = 4, the canonical coordinates ( 1 ,  2 ,  1 ,  2 ) are obtained more explicitly from (35) and (36) as and the quadratic Hamiltonian  4 has the expression whereas the corresponding linear Hamiltonian system takes the form A question which is worth considering at this point is whether solutions to the Hamiltonian system (45) can also be obtained through a simple superposition formula from the solutions of the second-order system (42), as in the case of scalar equations of maximal symmetry.By construction of (45), this question is already positively answered by formulas (43) which provide an explicit expression for the unknown variables   and   directly in terms of the solutions of the scalar source equation ( 4).Indeed, if  = () is any given function of , then (43) shows that  = {,   , −10q  −  (3) ,   } is a solution to (45) if and only if  is a solution to the corresponding fourth-order scalar equation (13b).Therefore, a complete set of linearly independent solutions for (45) is given by   = {  , (  )  , −10q (  )  −  (3)   , (  )  } ,   =  3− V  ,  = 0, . . ., 3, where as usual  and V are two linearly independent solutions of (4), using also the fact that q = w(  , V  ) by (5).Note that, for any choice of  in the expression of , all equations in (45) are always satisfied, except for the third one which reduces exactly to (13b).More generally, it follows from (35) and (36) that the general solution to every Hamiltonian system derived from an equation of maximal symmetry as those given in (13a)-(13d) can be found by a mere superposition formula from the solutions of the second-order source equation (4).It will therefore be useful to obtain a characterization of the equivalence class under point transformations of these nontrivial families of Hamiltonian systems depending on the arbitrary function q, as well as its symmetry properties.
Restricting our attention for now to the fourth-order system (45), the existence of the superposition formula (46) indicates that this system of equations can be reduced in order, and we know this is possible in particular if the system has a two-parameter abelian symmetry subgroup.As is well known for Hamiltonian systems, symmetries are essentially in one-one correspondence with first integrals  of the system, the determining equation of which is given by   + {, } = 0 ⇐⇒ (47a) In (47a) and (47b), {, } is the associated Poisson bracket for the canonical coordinates and k is the Hamiltonian vector