a new approach to the converse of noether s theorem the concepts of vector fields and forms along a map are used to establish a condition characterising symmetries of the hamiltonian system associated with a regular lagrangian this condition does not mention any second order differential equation field but is expressed in terms of the geometry of the second order tangent bundle this result is also generalised to the case of lagrangian functions depending on higher order derivatives

internal deformation of lie algebroids and symplectic realizations given a lie algebroid and a bundle over its base which is endowed with a localizable poisson structure and a flat connection we construct an extended bundle whose dual is endowed with an almost poisson structure that is a quadratic poisson structure when a certain compatibility property is satisfied this new formalism on lie algebroids describes systems with internal degrees of freedom

almost hermitian geometries that characterize hamilton jacobi distributions the author considers horizontal lagrangian sub bundles of a cotangent bundle t n that contain the dynamical vector field of a hamiltonian system it is shown that if the hamiltonian is quadratic on the fibres of t n then all such sub bundles can be characterized as solutions to a system of algebro partial differential equations that are defined solely in terms of vertical quantities this system of equations has a universal solution the universal solution is applied to the study of the geometry of the lorentz force law it is shown that the dynamical flow of the lorentz force law on t n is a geodesic vector field for a metric connection on t n with torsion the torsion of this connection is given by an extension of cartan s first structure equation to non linear connections

on the differential geometry of the euler lagrange equations and the inverse problem of lagrangian dynamics the conditions for a system of second order differential equations to be derivable from a lagrangian the conditions of self adjointness in the terminology of santilli 1978 and others are related in the time independent case to the differential geometry of the tangent bundle of configuration space these conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second order differential equations necessary and sufficient conditions for such a vector field to be derivable from a lagrangian may be stated as the existence of a two form with certain properties it is interesting that it is a deduction not an assumption that this two form is closed and thus defines a symplectic structure some other differential geometric properties of euler lagrange second order differential equations are described

generation of 1 f noise from a broken symmetry model for the arbitrary absolute pitch of musical melodies a model is presented to generate power spectrum noise with intensity proportional to 1 f as a function of frequency f the model arises from a broken symmetry variable which corresponds to absolute pitch where fluctuations occur in an attempt to restore that symmetry influenced by interactions in the creation of musical melodies

the inverse problem for six dimensional codimension two nilradical lie algebras ado s theorem asserts that every real lie algebra g of dimension n has a faithful representation as a subalgebra of gl p r for some p the theorem offers no practical information about the size of p in relation to n and in principle p may be very large compared to n this article is concerned with finding representations for a certain class of six dimensional lie algebras specifically real indecomposable algebras that have a codimension two nilradical these algebras were classified by turkowski and comprise of 40 cases some of which contain up to four parameters linear representations are found for each algebra in these classes more precisely a matrix lie group is given whose lie algebra corresponds to each algebra in turkowski s list and can be found by differentiating and evaluating at the identity element of the group in addition a basis for the right invariant vector fields that are dual to the maurer cartan forms are given thereby providing an effective realization of lie s third theorem t

levy flights in inhomogeneous environments and 1 f noise abstract complex dynamical systems which are governed by anomalous diffusion often can be described by langevin equations driven by levy stable noise in this article we generalize nonlinear stochastic differential equations driven by gaussian noise and generating signals with 1 f power spectral density by replacing the gaussian noise with a more general levy stable noise the equations with the gaussian noise arise as a special case when the index of stability 2 we expect that this generalization may be useful for describing 1 f fluctuations in the systems subjected to levy stable noise

symmetries of nonlinear differential equations and linearisation a non linear ordinary differential equation is linearisable if it possesses sl 3 r symmetry the conditions under which the abelian two dimensional subalgebras of sl 3 r are sufficient for linearisation are established

dynamic scaling approach to study time series fluctuations we propose an approach for properly analyzing stochastic time series by mapping the dynamics of time series fluctuations onto a suitable nonequilibrium surface growth problem in this framework the fluctuation sampling time interval plays the role of time variable whereas the physical time is treated as the analog of spatial variable in this way we found that the fluctuations of many real world time series satisfy the analog of the family viscek dynamic scaling ansatz this finding permits us to use the powerful tools of kinetic roughening theory to classify model and forecast the fluctuations of real world time series

the lax mizohata theorem for kirchhoff type equations abstract in this paper we prove the necessity of the hyperbolicity to the cauchy problem for the kirchhoff type equations to be locally in a neighbourhood of the given solution and local in time well posed the method of proof is based on the gauge invariance of the kirchhoff type equations which allows to use high frequency components of solutions and on some ideas from the lyapunov stability theory

the helmholtz conditions revisited a new approach to the inverse problem of lagrangian dynamics deals with the general problem of finding a multiplier matrix that can give to a prescribed system of second order ordinary equations the structure of euler lagrange equations the approach is based on a generalisation of previous studies on linear systems the main result concerns a set of necessary and sufficient conditions for the existence of a multiplier which contains an infinite set of algebraic equations the coefficients of which can be used to derive necessary conditions involving only the given right hand sides of the differential equations an outline is given of interesting points for future studies and an example is presented for which all multipliers are explicitly constructed

extended tangent bundle formalism for time dependent lagrangian systems the lagrangian formalism for time dependent systems is developed using vector fields defined in the extended tangent bundle t q r the definition of the time independent extended lagrangian function l associated to a time dependent lagrangian l is given and then the techniques of symplectic mechanics are used to prove some properties of the equations determining the euler lagrange vector field the relations between the time independent and the time dependent helmholtz conditions are analyzed from a geometric perspective and finally the noether theorem for the extended lagrangian l is considered

lagrangian systems with constraints a geometric approach to the method of lagrange multipliers a geometric approach to the method of lagrange multipliers is presented using the framework of the tangent bundle geometry the nonholonomic systems with constraint functions linear in the velocities are studied in the first place and then and using this study of the nonholonomic mechanical systems as a previous result the holonomic systems are considered the lagrangian inverse problem is also analysed and finally the theory is illustrated with several examples

second order equation fields and the inverse problem of lagrangian dynamics the transformation properties of determined autonomous systems of second order ordinary differential equations identified as vector fields on the tangent bundle of the space of dependent variables are derived and studied the inverse problem of lagrangian dynamics is studied from this transformation viewpoint as well as the problem of alternative lagrangians in particular regular lagrangians which are analytic as functions of the first derivatives are considered finally the inverse problem for second order systems corresponding to the geodesic flow of a symmetric linear connection is investigated

variational and integrable connections the two dimensional symmetric connections whose geodesic equations are derivable from a lagrangian function are divided into five classes this classification is compared with that of douglas for more general classes of systems of ordinary differential equations three of the five classes of connections are further investigated and in most cases specific lagrangians are exhibited in particular those connections that are engendered by lagrangians homogeneous in velocities are characterized in terms of the ricci tensor of each connection finally several examples of variational connections that possess integrals of motion are given thereby extending the known class of completely integrable systems

lie algebras associated with scalar second order ordinary differential equations second order ordinary differential equations are classified according to their lie algebra of point symmetries the existence of these symmetries provides a way to solve the equations or to transform them to simpler forms canonical forms of generators for equations with three point symmetries are established it is further shown that an equation cannot have exactly r 4 5 6 7 point symmetries representative s of equivalence class es of equations possessing s 1 2 3 8 point symmetry generator s are then obtained

bi invariant and noninvariant metrics on lie groups a restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied specifically for solvable lie algebras of dimension up to and including six all algebras for which there is a compatible pseudo riemannian metric on the corresponding linear lie group are found of the 19 such metrics four are bi invariant the lie algebras are taken from tables compiled originally by mubarakzyanov izv vyssh uchebn zaved mat 4 104 116 1963 and morozov izv vyssh uchebn zaved mat 4 161 171 1958

the inverse problem of the calculus of variations for discrete systems fil barbero linan maria instituto de ciencias matematicas espana universidad politecnica de madrid espana

extended covariance for the lagrange equations of motion a geometric analysis the infinitesimal transformations which leave the lagrangian structure of the equations of motion unchanged are intrinsically characterized a new condition is given with which to obtain alternative lagrangians

modeling 1 f noise the noise of signals or currents consisting of a sequence of pulses elementary events or moving discrete objects particles is analyzed a simple analytically solvable model is investigated in detail both analytically and numerically it is shown that 1 f noise may result from the statistics of the pulses transit times with random increments of the time intervals between the pulses the model also serves as a basis for revealing parameter dependences of 1 f noise and allows one to make some generalizations as a result the intensity of 1 f noise is expressed through the distribution and characteristic functions of the time intervals between the subsequent transit times of the pulses the conclusion that 1 f noise may result from the clustering of the signal pulses elementary events or particles can be drawn from an analysis of the model systems

alternative lagrangians for spherically symmetric potentials a new geometrical account of the existence of alternative s equivalent lagrangians is given for a particle moving in three dimensional euclidean space under the influence of a central force the fact that the lagrangian for this familiar problem is not unique was pointed out by henneaux and shepley j math phys 23 2101 1982 the present analysis makes the reasons for the phenomenon more transparent than theirs

symmetries first integrals and the inverse problem of lagrangian mechanics deals with the following question given a symmetry vector field y of a system of second order ordinary differential equations and an associated constant of the motion f is it possible to find a lagrangian l for the system such that y becomes a noether symmetry with respect to l and f its implied noether constant for one degree of freedom systems the answer to this question is affirmative in addition attention is paid to the construction of a suitable constant of the motion f for given symmetry y and vice versa several examples are discussed

when nonlinear differential equations are equivalent to linear differential equations a necessary and sufficient condition is established for the existence of a 1 1 transformation of a system of nonlinear differential equations to a system of linear equations the obtained theorems enable one to construct such transformations from the invariance groups of differential equations the hodograph transformation the legendre transformation and lie s transformation of the monge ampsre equation are shown to be special cases noninvertible transformations are also considered examples include burgers equation a nonlinear diffusion equation and the liouville equation introduction in this work we study transformations mapping nonlinear differen tial equations to linear differential equations in a 1 1 manner based upon the group analysis of differential equations we obtain necessary and sufficient conditions for the existence of such transformations the established theorems not only allow us to determine the existence of the transformations but also enable us to actually construct these transformations from invariance groups of the nonlinear equations in the following analysis two types of transformations are considered 1 the invariance groups of differential equations and 2 the mappings which transform nonlinear differential equations to linear differential equations theorems will be proved based upon the following observations clearly if there exists a 1 1 mapping between any two differential equations it must inject properties of one equation into the other including their invariance properties for this reason the often ignored fact that any linear differential equation admits an invariance group related to the superpo sition principle becomes very significant the generator of this particular group depends upon an arbitrary solution of the linear equation it follows then that any nonlinear equation transformable to a linear equation by a 1 1 mappingmust admit an invariance group whose generator depends upon an arbitrary solution of some linear differential equation the idea of comparing invariance groups of differential equations in the search of mappings connecting the equations was first used by bluman in his study of burgers equation 1 and it was applied to the study of the mappings of one dimensional linear parabolic equations to the heat equation 2 in the first section of this paper some basic properties of various transformations are summarized the theorems are presented in the second section the examples in the third section include the hodograph transformation the legendre transformation lie s transformation of themonge ampre equation and the equation ux uxx uyy 0 in the last section we examine properties of noninvertible mappings of burgers equation the diffusion equation u 2ux uy 0 and the liouville equation u y e u respectively into linear equations

stochastic nonlinear differential equation generating 1 f noise starting from the simple point process model of 1 f noise we derive a stochastic nonlinear differential equation for the signal exhibiting 1 f noise in any desirably wide range of frequency a stochastic differential equation the general langevin equation with a multiplicative noise that gives 1 f noise is derived the solution of the equation exhibits the power law distribution the process with 1 f noise is demonstrated by the numerical solution of the derived equation with the appropriate restriction of the diffusion of the signal in some finite interval

nonlinear stochastic models of 1 f noise and power law distributions starting from the developed generalized point process model of 1 f noise b kaulakys et al phys rev e 71 2005 051105 we derive the nonlinear stochastic differential equations for the signal exhibiting 1 f b noise and 1 x l distribution density of the signal intensity with different values of b and l the processes with 1 f b are demonstrated by the numerical solution of the derived equations with the appropriate restriction of the diffusion of the signal in some finite interval the proposed consideration may be used for modeling and analysis of stochastic processes in different systems with the powerlaw distributions long range memory or with the elements of self organization r 2006 elsevier b v all rights reserved

the lax mizohata theorem for nonlinear gauge invariant equations we prove that ifthe cauchy problem for the nonlinear partial differential equation is well posed in a neighbourhood of a given solution then all the characteristics of the equation are real valued for the linear equations this is a statement of the lax mizohata theorem equations considered in the paper include gauge invariant localequations and systems and non local kirchhoff type equations

on the existence of local and global lagrangians for ordinary differential equations necessary and sufficient conditions for the existence of local and global lagrangians for ordinary differential equations of arbitrary order are described in terms of the geometry of higher order tangent bundles the results are applied to the study of gauge invariant differential equations and the second order differential equation defined by the 2 1 dimensional yang mills lagrangian with the chern simons term is discussed

time dependent lagrangian systems a geometric approach using semibasic forms a new presentation of the geometric formalism in r tq of the time dependent lagrangian systems is given the formalism is used to prove some properties of the equations determining the euler lagrange vector field xl for relating the theory with the time independent case and for studying the time dependent lagrangian inverse problem

a geometric approach to a generalized virial theorem the virial theorem introduced by clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics is studied by making use of symplectic formalism as an approach in the case of both the hamiltonian and lagrangian systems the possibility of establishing virial like theorems from one parameter groups of non strictly canonical transformations is analysed the case of systems with a position dependent mass is also discussed using the modern symplectic approach to quantum mechanics we arrive at the quantum virial theorem in full analogy with the classical case

a separability theorem for dynamical systems admitting alternative lagrangian descriptions by generalising earlier results the authors prove that whenever a lagrangian dynamical system can be associated with a 1 1 type tensor field which is left invariant by the dynamics satisfies the nijenhuis condition and is compatible with the tangent bundle structure the system necessarily decomposes into a collection of lower dimensional mutually non interacting lagrangian subsystems

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

a linear connection for higher order ordinary differential equations a linear connection is defined on the space of n jets of sections of where b is one dimensional this is a first step toward classifying n 1 th order time dependent ordinary differential equations the module of vector fields on splits into coordinate invariant components n 1 of them isomorphic and one component generated by the differential equation field the connection is shown to uniquely determine the differential equation field to which it is associated

geometric lagrangian approach to first order systems and applications the geometric theory of presymplectic systems is developed to study both the lagrangian and hamiltonian formulations of a system described by a lagrangian linear in the velocities the results are used to study some related problems for second order differential equations and regular lagrangians

origin of the lagrangian constraints and their relation with the hamiltonian formulation the theory of presymplectic systems is used for the study of mechanical systems described by singular lagrangians in order to clarify the geometric meaning of the euler lagrange equations for such systems the two different types of primary constraint functions arising in the lagrangian formulation are analyzed by means of the relation between the image under the vertical endomorphism of the kernel of the presymplectic form l and its vertical part the connection with the hamiltonian dirac theory is also studied and the theory is illustrated with several examples

the constraint algorithm for time dependent lagrangians the aim of this paper is to develop a constraint algorithm for time dependent lagrangian systems which permit us to solve the motion equations this algorithm extends the gotay and nester algorithm for autonomous lagrangians which is in fact a particular case to do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used

alternative lagrangians for central potentials two methods for the construction of alternative lagrangians to the standard lagrangian l t v where v is a central potential are presented the direct integration of the hessian matrix and the use of the noether symmetries of l the lagrangians obtained are of the so called nonlinear homogenous nlh type and the study is centered in those lagrangians not preserving the spherical symmetry of l finally the associated hamiltonian formalism is presented

global existence for nonlinear hyperbolic systems of kirchhoff type abstract in this paper we prove the global solvability of the cauchy problem with small and suitably smooth initial data for nonlinear hyperbolic systems with integro differential coefficients of kirchhoff type this extends to a wide class of nonlinear hyperbolic equations some results in 4 5 7 and 10 our method is similar to the technique employed by greenberg hu in 10 in the case of the classical kirchhoff equation

singular lagrangians affine in velocities the properties of lagrangians affine in velocities are analysed in a geometric way these systems are necessarily singular and exhibit in general gauge invariance the analysis of constraint functions and gauge symmetry leads us to a complete classification of such lagrangians

on the lie symmetries of the classical kepler problem the standard version of noether s theorem when applied to the classical kepler problem leads to the constants of energy and angular momentum but does not give the hidden symmetry known as the runge lenz vector lie s theory of differential equations is used to obtain all three constants of motion the transformations of solutions under the point transformations to which these constants correspond are studied the results are generalised to n dimensions

nonlinear garch model and 1 f noise abstract auto regressive conditionally heteroskedastic arch family models are still used by practitioners in business and economic policy making as a conditional volatility forecasting models furthermore arch models still are attracting an interest of the researchers in this contribution we consider the well known garch 1 1 process and its nonlinear modifications reminiscent of ngarch model we investigate the possibility to reproduce power law statistics probability density function and power spectral density using arch family models for this purpose we derive stochastic differential equations from the garch processes in consideration we find the obtained equations to be similar to a general class of stochastic differential equations known to reproduce power law statistics we show that linear garch 1 1 process has power law distribution but its power spectral density is brownian noise like however the nonlinear modifications exhibit both power law distribution and power spectral density of the 1 f form including 1 f noise

point process model of 1 f noise vs a sum of lorentzians we present a simple point process model of 1 f noise covering different values of the exponent the signal of the model consists of pulses or events the interpulse interevent interarrival recurrence or waiting times of the signal are described by the general langevin equation with the multiplicative noise and stochastically diffuse in some interval resulting in a power law distribution our model is free from the requirement of a wide distribution of relaxation times and from the power law forms of the pulses it contains only one relaxation rate and yields 1 f spectra in a wide range of frequencies we obtain explicit expressions for the power spectra and present numerical illustrations of the model further we analyze the relation of the point process model of 1 f noise with the bernamont surdin mcwhorter model representing the signals as a sum of the uncorrelated components we show that the point process model is complementary to the model based on the sum of signals with a wide range distribution of the relaxation times in contrast to the gaussian distribution of the signal intensity of the sum of the uncorrelated components the point process exhibits asymptotically a power law distribution of the signal intensity the developed multiplicative point process model of 1 f noise may be used for modeling and analysis of stochastic processes in different systems with the power law distribution of the intensity of pulsing signals

1 f noise from nonlinear stochastic differential equations we consider a class of nonlinear stochastic differential equations giving the power law behavior of the power spectral density in any desirably wide range of frequency such equations were obtained starting from the point process models of 1 fbeta noise in this article the power law behavior of spectrum is derived directly from the stochastic differential equations without using the point process models the analysis reveals that the power spectrum may be represented as a sum of the lorentzian spectra such a derivation provides additional justification of equations expands the class of equations generating 1 fbeta noise and provides further insights into the origin of 1 fbeta noise

tsallis distributions and 1 f noise from nonlinear stochastic differential equations probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems in this article we unite modeling of such distributions with the model of widespread 1 f noise we propose a class of nonlinear stochastic differential equations giving both the q exponential or q gaussian distributions of signal intensity revealing long range correlations and 1 f behavior of the power spectral density the superstatistical framework to get 1 f noise with q exponential and q gaussian distributions of the signal intensity is proposed as well

linearization of second order ordinary differential equations via cartan s equivalence method abstract we use cartan s equivalence method to study the differential invariants of a single second order ordinary differential equation relative to the pseudo group of point transformations as a result of the analysis a simple characterization is given of those second order equations which are linearizable by a point transformation

model selection for identifying power law scaling long range temporal and spatial correlations have been reported in a remarkable number of studies in particular power law scaling in neural activity raised considerable interest we here provide a straightforward algorithm not only to quantify power law scaling but to test it against alternatives using bayesian model comparison our algorithm builds on the well established detrended fluctuation analysis dfa after removing trends of a signal we determine its mean squared fluctuations in consecutive intervals in contrast to dfa we use the values per interval to approximate the distribution of these mean squared fluctuations this allows for estimating the corresponding log likelihood as a function of interval size without presuming the fluctuations to be normally distributed as is the case in conventional dfa we demonstrate the validity and robustness of our algorithm using a variety of simulated signals ranging from scale free fluctuations with known hurst exponents via more conventional dynamical systems resembling exponentially correlated fluctuations to a toy model of neural mass activity we also illustrate its use for encephalographic signals we further discuss confounding factors like the finite signal size our model comparison provides a proper means to identify power law scaling including the range over which it is present

infinitesimal symmetry transformations ii some one dimensional nonlinear systems the converse problem of similarity analysis is discussed for the infinitesimal symmetry transformations of ordinary second order differential equations which are nonlinear in x and may be linear or nonlinear in x a natural classification of the problem arises according to the highest order n of nonlinearity in x the completely general maximal lie algebra is obtained for the case n 3 in the case n 4 one has besides the system of differential equations for the infinitesimal generators an extra set of anholonomic constraints which operates as a symmetry breaking mechanism producing a strong reduction in the number of surviving parameters miscellaneous examples are given which illustrate some features of similarity analysis of nonlinear systems the infinitesimal point transformation symmetries of the van der pol oscillator are also briefly discussed

extended legendre transformation approach to the time dependent hamiltonian formalism the hamiltonian formalism for time dependent systems is developed using vector fields defined in the extended cotangent bundle t q r the extended hamiltonian h is obtained using the extended legendre transformation associated to an extended lagrangian l previously defined in t q r and then the techniques of symplectic mechanics are employed in t q r to study the properties of the extended hamiltonian system finally this approach is related with the multisymplectic formalism and other extended approaches

quasi coordinates from the point of view of lie algebroid structures a geometrical description of the lagrangian dynamics in quasi coordinates on the tangent bundle using the lie algebroid framework is given linear nonholonomic systems on lie algebroids are solved in local coordinates adapted to the constraints through lagrangian multipliers and gibbs appell generalized methods

on the inverse problem of the calculus of variations in field theory the inverse problem of the calculus of variations is investigated in the case of field theory uniqueness of the action principle is demonstrated for the vector laplace equation in a non decomposable riemannian space as well as for the harmonic map equation

cartan s method of equivalence and second order equation fields the theory underlying cartan s method of equivalence is outlined and applied to second order equation fields in particular the technique is used to obtain the canonical horizontal distribution associated to a second order equation field a construction which has been presented before but in a rather ad hoc manner

a geometrical version of the helmholtz conditions in time dependent lagrangian dynamics appropriate geometrical machinery for the study of time dependent lagrangian dynamics is developed it is applied to the inverse problem of the calculus of variations and a set of necessary and sufficient conditions for the existence of a lagrangian are given in terms of the existence of a 2 form with suitable properties which are exactly equivalent to the helmholtz conditions

killing tensors in spaces of constant curvature a killing tensor is one possible way of generalizing the notion of a killing vector on a riemannian or pseudo riemannian manifold it is explained how killing tensors may be identified with functions that are homogeneous polynomials in the fibers on the associated cotangent bundle as such killing tensors may be identified with first integrals of the hamiltonian geodesic flow which are homogeneous polynomials in the momenta again using this identification it is shown that in flat spaces the dimension of the vector space of killing tensors is maximal and that the killing tensors are generated by the killing vectors finally using riemann s model for the metric in spaces of constant curvature a comparison argument is used to show that similar results are valid in that more general context

a complete set of bianchi identities for tensor fields along the tangent bundle projection the various derivations defined along the tangent bundle projection tau in a series of papers by martinez carinena and sarlet 1992 are expressed as components of a single linear connection del on e the tangent bundle of the evolution space e r t m this connection is equivalent to a system of second order ordinary differential equations sode on m using the linear connection we calculate the torsion and curvature of e del the components of which are expressed in terms of the tensors along tau defined by martinez et al from these the full set of bianchi identities are calculated we also show that the generalized jacobi equation defined by several authors is precisely the horizontal component of the conventional jacobi equation along geodesics of e del finally we use this to show that if a jacobi field of the lift of a sode solution is a certain lift then it can be extended to a symmetry of the sode

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve