Reduction of systems of first-order differential equations via Lambda-symmetries

Reduction of systems of first-or d er differen tial equations via Λ -symmetries Giampaolo Cicogna ∗ Dipartimen to di Fisica “E.F ermi” dell’Univ ersit` a di Pisa and Istituto Nazionale di Fisica Nucleare, Sez. di Pisa Largo B. Pon tecorv o 3 , Ed. B-C, I- 5 6127, Pisa, Italy Abstract The notion of λ -symmetries, originally int ro duced by C. Muriel and J .L. Romero, is extended to the case of sys tems of fir st-order ODE’s (and of dynam- ical systems in particular). It is shown that the existence of a symmetry of this t yp e pro duces a r eduction o f the differential equations, re s tricting the pres ence of the v aria ble s inv olved in the problem. The r esults are co mpa red with the case of standard (i.e. exa ct) Lie- po in t s ymmetries and are also illustra ted by some examples. P A CS : 02.20 .Sv; 02.3 0.Hq Keywor ds : λ -sy mmetries; r eduction pro cedures; dynamical sys tems 1 In tro duction It is a well known prop erty that if a n ordinar y differential equation (ODE ) admits a L ie p oint-symmetry , then the order of the equation can b e low ere d by one (see e.g. [1]). The notion of λ -symmetry has b e en intro duced in 2001 by Muriel and Romero [2, 3] with the main purp ose of obtaining this reduction even in the absence of standard Lie symmetries. The idea consists in intro ducing a suitable mo dification of the pr olongatio n r ules o f the vector field in such a way that the lowering procedur e still works, even if λ -symmetrie s are not symmetries in the prop er sense, as they do not map in gener al s olutions into solutions. λ -Symmetries are rela ted to symmetries of integral exp onential type [1, 2, 3], to hidden and to some clas s es o f p otential s ymmetries (see [3, 4, 5] and ref- erences ther ein). The mea ning o f λ -pro longation has b een clarified (together with a p ossible genera lization of the pr o cedure) b y means of classic al theo ry ∗ Email: cicogna@df.unipi.it 1 of characteristics o f vector fields [6]. λ -Symmetr ies hav e b een extended to par- tial differential equa tions [7, 8] (and called in that co nt ext µ -symmetries), and also interpreted in terms of a deformed L ie der iv ative in a more geometrica l ap- proach [9]. A nontrivial relationship with nonlo ca l symmetries has b een recently po int ed o ut [10]; an interpretation in terms of appropria tely defined changes of reference fr ames has b een also prop ose d [11]. F or the implications of λ - and µ -symmetries in No ether-type cons e rv ation rules se e [12, 1 3]. In the case of first-or der ODE’s, Lie symmetries cannot low e r the order of the equations, but they provide a sort of “reduction” o f the complexity of the system, or – mor e precisely – a reduction of the num b er of the inv olved v ariables (see [1], Theorem 2.66 ). In this pap er, we will r estrict precisely to the case of systems of first-or der ODE’s (with us ua l regularity a nd nondegeneracy assumptions: see e.g. [1, 14]), and of dynamica l sy stems (DS) in particula r, where the applica tion of λ -symmetries requires so me attention and whe r e they exhibit some relev ant p eculiarities. W e shall prove that some forms o f reduction are a llow ed also in these cases . An applicatio n of λ -s ymmetries to sy stems of ODE’s has been a lr eady con- sidered in a particula r case [1 5]; in the present pap er we wan t to examine mor e general situations. 2 Systems of ODE’s Let us recall first of all that in the ca se o f a single dep endent v ariable u = u ( t ) (w e shall always deno te by t ∈ R the indep enden t v aria ble, acc o rding to its natural interpretation as the time v a riable in the case of DS), the first-orde r λ -prolonga tion X (1) λ of a vector field X X = τ ( t, u ) ∂ ∂ t + ϕ ( t, u ) ∂ ∂ u (1) is defined as X (1) λ = X (1) + λ Q ∂ ∂ ˙ u (2) where X (1) is the standar d pro lo ngation [1, 14], ˙ u = d u/ d t , λ = λ ( t, u, ˙ u ) is an arbitrar y C ∞ function, and Q = ϕ − τ ˙ u . Considering systems o f equatio ns , and then q > 1 dep e nden t v ar ia bles u a = u a ( t ), the natura l extens io n o f definition (2) is X (1) Λ = X (1) + (Λ Q ) a ∂ ∂ ˙ u a (3) where the sum ov e r a = 1 , . . . , q is understo o d, with X = τ ∂ ∂ t + ϕ a ∂ ∂ u a , Q a = ϕ a − τ ˙ u a (4) and where now Λ is a q × q matr ix of C ∞ functions dep ending on t, u a , ˙ u a . The case Λ = λ I is the one co nsidered, in the context of DS and a lso for sy stems of ODE’s of any order, by Muriel and Romer o [15]. 2 Given a system of q first- o rder ODE’s (we sha ll assume for simplicity that the num b er o f the equations is the same as the num b er of dep endent v ar iables u a ( t )) F a ( t, u b , ˙ u b ) = 0 a, b = 1 , . . . , q (5) we s ha ll sa y that this system is Λ- symmetric under a vector field X if ther e is a matrix Λ such that X (1) Λ F a   F a =0 = 0 . (6) It is clea r from (3) that Λ is not uniquely defined: indeed, fo r a ny matrix R such that RQ = Q then also Λ ′ = Λ R satisfies the ab ov e condition. This ar bitr ariness in the definition o f Λ, far fr om b eing distur bing, may b e useful in pra ctice, as it allows the choice of the more conv enient matr ix Λ in v iew of the given pro ble m. W e shall say that the system (5 ) is Λ- invariant under X if ther e is a matrix Λ s uch that X (1) Λ F a = 0 . (7) It is not to o r estrictive to a ssume that the system of ODE’s we ar e g oing to consider can be put into a Λ-inv ariant form. Indeed, extending to Λ-symmetrie s a well k nown result [1, 6], it can b e s hown that if a sys tem is Λ- symmetric, then there exists a q × q matrix A = A ( t, u, ˙ u ) such that X (1) Λ F a = A ab F b . It is no w eno ug h to pr ov e, a pplying standar d arguments (cf. e.g. [1 6, 17, 18]), the existence of s ome s ome q × q in vertible ma trix S , p oss ibly dep e nding on t, u, ˙ u , such that X (1) Λ S + S A = 0 and the (locally ) equiv alen t system G a ≡ S ab F b = 0 turns out to be Λ- inv ariant. W e will co nsider in the following, unless otherwis e stated, only Λ-inv ariant sys- tems. The matrix Λ plays the role o f an additiona l “unknown” in the determining equations which are deduced from the Λ- inv ariance condition τ ∂ F a ∂ t + ϕ b ∂ F a ∂ u b +  ϕ (1) b + (Λ Q ) b  ∂ F a ∂ ˙ u b = 0 (8) where ϕ (1) b is the co efficient of the standa r d first-o r der pr olongation, and which clearly s trongly dep end on the explicit form of the functions F a . F o r instance, in the case where F a = ˙ u a − f a ( t, u ), i.e . the case o f dynamical systems (see Section 3 ), these equations are { ϕ , f } a = ∂ ∂ t ( ϕ a − τ f a ) − ∂ τ ∂ u b f a f b + Λ ab Q b , where { ϕ , f } a = ϕ b ∂ f a ∂ u b − f b ∂ ϕ a ∂ u b , which clearly b ecome { ϕ , f } a = (Λ ϕ ) a (9) 3 in the case of autonomous systems and time-indep endent vector fields X with τ ≡ 0 (see [1 9]). Let us now intro duce “symmetry- adapted” co ordina tes w a (sometimes a lso called ca nonical c o ordinates) characterized by the prop erty of b eing inv ariant under the action of the v ecto r field X : X w a = τ ∂ w a ∂ t + ϕ b ∂ w a ∂ u b = 0 ; (10) they are obtained through the asso cia ted characteristic equa tions d t τ = d u a ϕ a . In this wa y we intro duce exactly q new v a r iables w a ; one at least of these, sa y w q , will dep end explicitly on t , and we will choo se this as the new indep endent v a riable a nd ca ll it η . In pa r ticular, if τ ≡ 0, we ca n cho ose η = t . As ( q + 1 )- th v a riable, w hich will b e called z , we will take the co ordina te “alo ng the a ction of X ”, i.e. such that X z = 1. Summar izing, the new se t of v ar iables is η , w α ( η ) , z ( η ) , α = 1 , . . . , q − 1 (11) (among these, w α and w q ≡ η a re in v a riant under X ) and clear ly do not dep end on Λ . W e have now to write the given vector field X a nd its first Λ -prolong ation (3) in terms of these co or dinates. W e get first X = X (1) = ∂ /∂ z , and we then find that eq. (3) takes the form X (1) Λ = ∂ ∂ z + M α ∂ ∂ w ′ α + M q ∂ ∂ z ′ (12) where w ′ α = d w α / d η , z ′ = d z / d η and (here and in the fo llowing the s um will b e alwa ys unders to o d over the repea ted indices α = 1 , . . . , q − 1 and a = 1 , . . . , q ; D t is the total deriv ative) M α = ( D t η ) − 2  D t η ∂ w α ∂ u a − D t w α ∂ η ∂ u a  (Λ Q ) a (13) M q = ( D t η ) − 2  D t η ∂ z ∂ u a − D t z ∂ η ∂ u a  (Λ Q ) a . (14) In particular , if τ ≡ 0 , and then with η = t , we hav e more simply ∂ w ′ α ∂ ˙ u a = ∂ w α ∂ u a , ∂ z ′ ∂ ˙ u a = ∂ z ∂ u a and M α = ∂ w α ∂ u a (Λ Q ) a , M q = ∂ z ∂ u a (Λ Q ) a . (15) 4 The above expres sions (13 ,1 4) can b e obtained either by direct calculation ex- pressing by the c ha in rule the op erator s ∂ /∂ ˙ u a in terms of ∂ / ∂ w ′ α , ∂ / ∂ z ′ , or – more elegantly – starting from the algebr aic rela tion [ X (1) Λ , D t ] = − D t ( τ ) D t + (Λ Q ) a ∂ ∂ u a (16) which can b e ea sily proved a nd genera lizes to Λ -symmetries other similar known ident ities [2, 3]. F rom this, one directly gets indeed X (1) Λ ( w ′ α ) = X (1) Λ  D t w α D t η  = X (1) Λ ( D t w α )( D t η ) − ( D t w α ) X (1) Λ ( D t η ) ( D t η ) 2 = = ( D t η ) − 2  D t η ∂ w α ∂ u a − D t w α ∂ η ∂ u a  (Λ Q ) a = M α (17) thanks to X (1) Λ w α = X w α = 0, X (1) Λ η = X η = 0; similar ly fo r X (1) Λ ( z ′ ). It ca n be interesting to p oint out that eq. (17 ) puts in clear evidence the difference with res p ect to exact symmetries: indeed, starting fr om the q ( X - inv a riant) v ar iables w α , η o ne o btains q − 1 first-o rder differential quantities w ′ α which a re inv ariant under X (1) , but in genera l not under X (1) Λ . In turn, the given system of differential equatio ns will take the form (we will use the e · to denote the e x pressions in the new v aria bles) e F a ( η , w α , w ′ α , z , z ′ ) = 0 (18) and the condition of its Λ -inv ariance under X now be comes ∂ e F a ∂ z + M α ∂ e F a ∂ w ′ α + M q ∂ e F a ∂ z ′ = 0 . (19) This a llows us to state the following first form of r eduction: Theorem 1 . If t he system (5) is Λ -invariant u nder a ve ctor field X , then, onc e writt en in t he symmetry-adapte d c o or dinates η , w α , w ′ α , z , z ′ , it turns out to dep end on only 2 q quantities (inste ad of 2 q + 1 ): i.e. on the q variables w α , η and on other q first-or der differ ential Λ -invariant qu ant ities ζ a = ζ a ( η , z , w α , w ′ α , z ′ ) which ar e obtaine d fr om the char acteristic e quations d z = d w ′ α M α = d z ′ M q (20) c oming fr om c ondition (19). Examples 1 and 2 will illustrate this result. 5 3 The case of Dynamical Systems Let us now consider the particular ly imp or tant case o f the dynamical systems, i.e. the sy s tems o f first-o rder O DE’s which a re written “in explicit form”: ˙ u a = f a ( t, u ) . (21) Clearly , o nce symmetry-a dapted co ordinates ar e in tro duced, the system b e- comes “auto ma tically” a function of the 2 q qua n tities w α , η , ζ a , a s granted b y Theorem 1. But it ca n b e preferable or mo re convenien t (e.g. in view o f the physical int erpretation in terms of “evolution” pr oblem, or a lso if the explicit expr ession of the ζ a is not kno w n 1 ) to adopt a different p oint of view, i.e. to preserve the form of the system as an explicit DS, i.e. to rewrite it as follows w ′ α = e f α ( η , w , z ) (22) z ′ = e f q ( η , w , z ) . (23) and to lo ok fo r the dep endence on z of the r.h.s. This p oint of v iew will b e elucidated by Examples 3 and 4. Recalling the express ion (12 ) of the first Λ-pr olongatio n of X , we then easily deduce in this case: Theorem 2. If a D S is Λ -invariant under X , the dep endenc e on z of the r.h.s. of e q.s (22,23) is given by ∂ e f α ∂ z = M α ; ∂ e f q ∂ z = M q . Then, if for some α one has M α = 0 , the c orr esp onding e f α do es n ot dep end on z . If M α = 0 for al l α = 1 , . . . , q − 1 , then only e f q dep ends on z and the system splits into a system for the q − 1 variables w α = w α ( η ) and the last e quation (23) which is an ODE for the variables z and η . It is useful to compar e the situa tion cov er ed by Theo rems 1 and 2 with the case o f exact symmetry: the difference is that in the case of exac t sy mmetr y a ll terms of eq. (18) are indep endent of z ; the same is tr ue for all the terms at the r.h.s. o f (2 2,23): then, in this cas e , the last equation for z and η turns o ut to be a quadr ature, as is w ell known [1]. Clearly , if Λ = 0 i.e. if X is a n exact symmetry , then M α = M q = 0. Conv ers ely , it can b e shown that if M α = M q = 0 then the symmetr y X is exact. This is par ticula rly clear in the c a se τ ≡ 0 (and then η = t ): indeed, in this case the conditions M α = M q = 0 can b e written (see (15)) J ab (Λ Q ) b = 0 where J is the (in vertible !) Jaco bian matrix o f the transformatio n fro m u a to 1 If one is int erested to kno w “a priori” the expressions of the q differential Λ-inv ariant quan tities ζ a = ζ a ( η, z , w α , w ′ α , z ′ ), one has to express M α , M q in terms of w α , η i n order to solve (20). 6 w α , z . Then Λ Q = 0 , which is the same as Λ = 0 (recall tha t Λ is no t uniquely defined). Notice in particular that the term (Λ Q ) a app earing in the expressions (13 ,14), when written in the new co or dinates, b ecomes ( e Λ e Q ) a = e Λ aq indeed e Q ≡ (0 , 0 , . . . , 1). This s hows that o nly the last column of e Λ is relev a nt. Finally , let us recall the following r esult: Theorem 3. (Muriel-Romero [1 5]) If Λ = λ I , then M α = 0 for al l α = 1 , . . . , q − 1 , and the c onclusion of the last p art of The or em 2 holds. Indeed, from X w α = X η = 0, X z = 1 and the definition of Q , one easily deduces Q a ∂ w α ∂ u a = − τ D t w α ; Q a ∂ η ∂ u a = − τ D t η and Q a ∂ z ∂ u a = − τ D t z + 1 hence, in the cas e (Λ Q ) a = λQ a considered in [15], one gets M α = 0 ; M q = ( D t η ) − 2 λ . Notice a lso that (16) b ecomes in this case [ X (1) λ , D t ] = − D t ( τ ) D t + λQ a ∂ ∂ u a = − D t ( τ ) D t + λX − λ τ D t . 4 Examples Example 1 . This is a very simple example, which can provide a clear illustra- tion of Theorem 1. Consider a n y sys tem F a ( t, u 1 , u 2 , ˙ u 1 , ˙ u 2 ) = 0 ( a = 1 , 2) of t wo first-or der ODE’s for the v ar ia bles u 1 = u 1 ( t ) , u 2 = u 2 ( t ) and co nsider the vector field X = ∂ ∂ u 2 . It is e asily seen that if one chooses Λ =  0 1 0 1  then, with our notation, w 1 = u 1 , η = t and z = u 2 ; eq. (13,1 4) give M 1 = M 2 = 1 and therefor e fro m (20) ζ 1 = ˙ w 1 − z = ˙ u 1 − u 2 , ζ 2 = ˙ z − z = ˙ u 2 − u 2 . Then, Λ - inv ariance under X gives that the F a depe nd only on the q uantit ies t, ˙ u 1 − u 2 , ˙ u 2 − u 2 , in a g reement w ith Theor em 1. Extens ion to mor e than 2 v a riables u a is immediate. 7 Example 2 . Consider a sys tem of ODE’s for the tw o v aria bles u 1 = u 1 ( t ) , u 2 = u 2 ( t ) of the fo r m h ( s 1 , s 2 ) ( ˙ u 1 − u 1 u 2 ) + a ( t )( u 2 1 + u 2 2 ) u 1 + b 2 ( t )( u 2 1 + u 2 2 ) u 2 = 0 h ( s 1 , s 2 ) ( ˙ u 2 + u 2 1 ) + a ( t )( u 2 1 + u 2 2 ) u 2 − b 2 ( t )( u 2 1 + u 2 2 ) u 1 = 0 where h is a function of s 1 = u 1 ˙ u 1 + u 2 ˙ u 2 , s 2 = u 1 ˙ u 2 − ˙ u 1 u 2 + u 3 1 + u 1 u 2 2 and where a ( t ) , b ( t ) are arbitra ry functions o f t ; it is clear ly not symmetric under the rotation op erator X = u 2 ∂ ∂ u 1 − u 1 ∂ ∂ u 2 (unless h ≡ 0), howev er it turns out to b e Λ- symmetric (but not Λ-inv a riant) under rotatio ns if Λ = λ I with λ = u 2 . Indeed, e.g., one has X (1) Λ h = 0, X (1) Λ ( ˙ u 1 − u 1 u 2 ) = ( ˙ u 2 + u 2 1 ), etc. Introducing symmetry- adapted co ordinates, which are w 1 = r = ( u 2 1 + u 2 2 ) 1 / 2 , η = t, z = θ , with obvious notations, the system bec o mes ˙ r h ( r ˙ r , r 2 ( ˙ θ + r cos θ )) + a ( t ) r 3 = 0 ( ˙ θ + r cos θ ) h ( r ˙ r , r 2 ( ˙ θ + r cos θ )) − b 2 ( t ) r 2 = 0 which turns to be Λ-inv a riant under X = ∂ /∂ θ with λ = r sin θ . As exp ected, thanks to Theorem 1, this s y stem contains only the fo ur quantities r, t a nd ζ 1 = ˙ r , ζ 2 = ˙ θ + r cos θ . If, e.g., h = s 2 , the system can b e als o put in the explicit fo rm o f a DS: ˙ r = ± ( a ( t ) /b ( t )) r ˙ θ = ± b ( t ) − r co s θ and – as a consequence – acco rding to Theorems 2 and 3, o ne (and only one) of the ab ove equa tio ns do es not c o nt ain z (here: θ ). Then the s y stem can b e easily solved. Example 3 . Consider a ny DS for u a = u a ( t ) , a = 1 , 2 , 3 , of the form ˙ u 1 = h 1 ( t, w 1 , w 2 ) + ( a − 3 b ) u 2 u 3 + b u 3 3 + h 2 ( t, w 1 , w 2 ) u 3 + h 3 ( t, w 1 , w 2 ) u 2 3 ˙ u 2 = h 2 ( t, w 1 , w 2 ) + 2 u 3 h 3 ( t, w 1 , w 2 ) + au 2 3 ˙ u 3 = cu 3 + 2 h 3 ( t, w 1 , w 2 ) where a, b , c ar e consta nts and h a are functions of t, w 1 = 2 u 2 − u 2 3 , w 2 = 3 u 1 − 3 u 2 u 3 + u 3 3 . Systems of this fo r m a re Λ-inv ariant under the vector field X = u 2 ∂ ∂ u 1 + u 3 ∂ ∂ u 2 + ∂ ∂ u 3 with Λ =   0 0 ( a − 3 b ) u 2 0 0 (2 a − c ) u 3 0 0 c   . The X -inv ariant quan tities are just w 1 , w 2 , together with η = t . The co efficients M α , M q (see eq.s (13,14 )), with z = u 3 , a r e M 1 = 4 ( a − c ) z , M 3 = c , 8 M 2 = 3( a − 3 b − c ) u 2 − 6 ( a − c ) u 2 3 = 3 ( a − 3 b − c )( w 1 + z 2 ) / 2 − 6( a − c ) z 2 . The characteristic equatio ns (20) can then b e easily so lved to obta in the three first-order differential Λ-inv ar iant qua n tities ζ 1 = ˙ w 1 − 2 ( a − c ) z 2 , ζ 2 = ˙ w 2 − (3 / 2)( a − 3 b − c ) w 1 z − (3 / 2)( a + b − c ) z 3 , ζ 3 = ˙ z − c z . Direct ca lc ulation shows that this system b ecomes ζ a − g a ( t, w 1 , w 2 ) = 0 where g 1 = 2 h 1 , g 2 = 3 h 1 − 3 h 3 w 1 , g 3 = 2 h 3 , and then contains only the quantities t, w 1 , w 2 , ζ a , in agreement with Theorem 1. If instea d one prefers to write the system as an explicit DS, then it is ˙ w 1 = 2 ( a − c ) z 2 + g 1 ( t, w 1 , w 2 ) ˙ w 2 = 3 2 ( a − 3 b − c ) w 1 z + 3 2 ( − a − b + c ) z 3 + g 2 ( t, w 1 , w 2 ) ˙ z = cz + g 3 ( t, w 1 , w 2 ) . Now, if a = c , the firs t equation do es not contain z ; if a = c a nd b = 0, only the third equatio n co nt ains z , in ag reement with Theorem 2. If a = c = 0, then only the second equation con tains z and “plays the role” of the q − th equation in o ur notatio n. The c ase a = b = c = 0 is of course the ca se of exac t symmetr y Λ = 0. Example 4 . This is a n example with non-autonomo us DS and vector field X with τ 6 = 0 and ther efore η 6 = t . Consider the DS for u a = u a ( t ) , a = 1 , 2 , 3 , ˙ u 1 = t + h 1 ( s, w 1 , w 2 ) exp( − λ 1 t ) ˙ u 2 = 1 + h 2 ( s, w 1 , w 2 ) exp( − λ 2 t ) ˙ u 3 = u 2 + h 2 exp( − λ 2 t ) 1 − exp( u 2  λ 2 − λ 3 )  λ 3 − λ 2 + h 3 exp( − λ 3 u 2 ) where h a are no nv anishing functions of s = u 2 − t, w 1 = u 1 − t 2 / 2 , w 2 = u 3 − u 2 2 / 2. This system is Λ- inv a riant under X = ∂ ∂ t + t ∂ ∂ u 1 + ∂ ∂ u 2 + u 2 ∂ ∂ u 3 with Λ = diag onal( λ 1 , λ 2 , λ 3 ). W e can choose as in v a riants under X just w 1 , w 2 and η = u 2 − t , with z = u 2 . It is now more useful to rewr ite the sys tem in these co ordinates preser ving its form of explicit DS, we ge t then w ′ 1 = ( h 1 /h 2 ) exp (( λ 2 − λ 1 )( z − η )) w ′ 2 =  1 − exp( λ 2 − λ 3 ) z λ 3 − λ 2 − z  + ( h 3 /h 2 ) exp  z ( λ 2 − λ 3 ) − λ 2 η )  z ′ = (1 / h 2 ) exp  λ 2 ( z − η )  . W e see that if λ 1 = λ 2 , or λ 2 = λ 3 (notice that the case λ 2 = λ 3 is well defined), one of the ab ov e equatio ns do es not contain z in ag reement with Theor em 2; whereas if λ 1 = λ 2 = λ 3 only one equatio n contains z , as stated by Theorem 3 ; the same happ ens also if λ 1 = λ 2 = 0, according to Theorem 2. 9 5 Concluding remarks An interesting prop erty which relates inv aria nce with λ -symmetries is the fol- lowing. Co nsider the cas e of an a utonomous DS ˙ u a = f a ( u ) which is Λ-inv a riant under a vector field X o f th e form X = ϕ a ( u ) ∂ /∂ u a . If w = w ( u ) is a ny in v a riant under X , i.e. ϕ a ∂ w/ ∂ u a , then its Lie deriva tive along f a , i.e. D ( f ) t w ≡ f a ∂ w ∂ u a satisfies X  D ( f ) t w  =  ϕ a ∂ ∂ u a  f b ∂ ∂ u b  w = Λ ab ϕ b ∂ w ∂ u a hence X  D ( f ) t w  = 0 if Λ = λ I , having used the co mm uta tion rule (9) and the inv ariance prop erty of w . On the o ther ha nd, eq. (17) gives directly , for X of the a bove form, X (1) Λ ( D t w ) = (Λ ϕ ) a ∂ w ∂ u a . The strong difference is that the latter result is purely algebr aic, being a con- sequence of the relation (16), and expre s ses a pr op erty o f the vector field X which holds indep endently of the presence of any DS (i.e., of an y c ho ice of the functions f a ). The for mer result, instead, states that the time evolution under the dynamics describ ed by the DS ˙ u a = f a of a quantit y w ( u ) which is inv aria n t under a vector field X pr eserves this inv aria nce even if X is not a (sta ndard) symmetry of the DS; it is enough to require that X is a λ -symmetry o f the DS. It can b e no tice d that the present statement, concerning Lie der iv atives, can be suitably extended to the case of several vector fields X for the g iven DS (s e e [20], Prop. 2.1). Several other aspects of λ -symmetrie s (and of all their generaliza tio ns as well) could b e further inv estiga ted. Apa rt fro m their g eometrical interpretation (se e the papers quoted in the Introduction), their action on changes of co or dina tes should b e b etter understo o d, as well a s their general role in finding so lutions of differen tia l equations which do not admit standard symmetries: see e.g . [2, 3, 5, 6] a nd the r eferences therein; compare a lso, for insta nce, with [21], for what concerns the problem of finding int egrating factors for ODE’s and its relationship with symmetry pro per ties. It can b e observed, fina lly , that a n y ODE ∆( t, u, ˙ u, ¨ u, . . . ) = 0 of ar bitrary order > 1 can b e tra nsformed into a system of first-o rder ODE’s, and ther efore our results could b e applied also to this ca se. T his is true in pr inciple: the only nearly obvious remar k is that o ne has to consider no long er vector fields of the form X = τ ( ∂ /∂ t ) + ϕ ( ∂ /∂ u ) inv olving o nly the tw o v ar iables t and u , but also extended vector fields X = τ ( ∂ /∂ t ) + ϕ (0) ( ∂ /∂ u ) + ϕ (1) ( ∂ /∂ ˙ u ) + ϕ (2) ( ∂ /∂ ¨ u ) + . . . . It is “conceptually” differen t to look for vector fields of the former or of the latter form; on the other hand, the “co ncrete effect” o f the ex istence of a symmetry is different in the tw o c o ntexts (i.e., low ering the order in the case of the ODE’s, 10 and resp ectively reducing the pr e sence of the inv olved v aria bles in the cas e of first-order systems, as shown). This holds in particular for λ - and Λ-symmetr ie s, where also the prolongation r ules of the vector fields are markedly differ e nt in the t wo case s . T o emphasize this different ro le of Λ-symmetries in the context of first-o rder systems, it should b e p er haps more a ppropriate to call them ρ - symmetries (where ρ sta nds for “ reducing”, in contrast with λ , which co uld stand fo r “ low ering ”). Ac kno wledgmen ts It is a pleasur e to thank Giusepp e Gaeta and Diego Ca ta lano F erra ioli for de- tailed discussio ns and useful co mmen ts. 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