Variational problems of some second order Lagrangians given by Pfaff forms

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1 Vritionl problems of some second order Lgrngins given by Pfff forms P. Popescu M. Popescu Abstrct. In this pper we study the dynmics of some second order Lgrngins tht come from Pfff forms i.e. differentil forms on tngent bundles. In the non-singulr cse minly considered in the pper the generlized Euler-Lgrnge eqution is third order differentil eqution. We prove tht the solutions of the differentil equtions of motion of chrge in field nd the Euler equtions of rigid body cn be obtined s prticulr solutions of suitble Pfff forms with non-negtive second vritions long their solutions. A non-stndrd Hmiltonin pproch is lso considered in the non-singulr cse using energy functions ssocited with suitble semi-sprys. M.S.C. 00: 70G5 70H03 70H06 70H07 70H30 70H50 53C80. Key words: Pfff form; Lgrngin; Euler-Lgrnge eqution; vrition; semi-spry. Introduction The Euler-Lgrnge eqution of first order Lgrngin is one of the most known nd widely used vritionl eqution in mthemtics mechnics nd physics. Its solutions re the criticl curves of the ction defined by the Lgrngins on curves; in the cse when the Lgrngin comes from Riemnnin non-riemnnin or Finslerin metric these solutions re known s geodesics since they loclly minimise the distnce. The second vrition decides if the solution is n extreme (see [ Ch. Sect.]). The locl expression of the first order Euler-Lgrnge eqution contins the second derivtives nd in the cse of hyperregulr Lgrngin its solutions re integrl curves of globl second order differentil eqution. In this pper we consider n other type of dynmics where the locl expression of generlized Euler-Lgrnge eqution contins the third derivtives nd in the regulr cse the solutions re integrl curves of globl third order differentil eqution. The generlized Euler-Lgrnge eqution is obtined by vritionl method on n ction of second order Lgrngin defined by Pfff form (i.e. differentil -form on the tngent spce of mnifold). Blkn Journl of Geometry nd Its Applictions Vol.7 No. 0 pp c Blkn Society of Geometers Geometry Blkn Press 0.

2 Vritionl problems of some second order Lgrngins 83 According to [3 Sect. 6.3] some specil regulrity conditions cn be considered in this cse nd they re studied in detil in our pper. The second order Lgrngin is liner in the second order velocities (ccelertions) s in [8]. The Euler-Lgrnge eqution of higher order Lgrngin ws firstly described by Ostrogrdski nd studied for exmple in the monogrphs [3] or in [5]. The second order Lgrngin considered in our pper comes from Pfff form nd hs null Hessin. Besides its Euler-Lgrnge eqution we re interested lso in the second derivtive of the vrition deduced in clssicl vritionl wy. We prove tht the solutions of the differentil equtions of motion of chrge in field (formuls (7) in [ Section 7]) nd the Euler equtions of rigid body [6] cn be obtined s prticulr solutions of suitble Pfff forms with some non-negtive second vritions. A dul Hmiltonin pproch to higher order Lgrngins prticulrly on second order ws first used lso by Ostrogrdski (see for exmple [3 5]). But techniclly this pproch cn not be used in the prticulr cse when the Lgrngin is liner in the second order velocities (or ccelertions); the second order moment cn not be relted to the ccelertions since the Legendre trnsformtion is degenerted on fibers. We use here n unusul Hmiltonin pproch s in [8 0] in the cse of second nd higher order Lgrngin liner in the second or higher order velocities. The line in [8] uses generl Hmiltonin dulity tht is extended in [9] to the higher order cse. More exctly we consider energy functions of the Pfff form ech ssocited with suitble semi-spry; then we obtin the locl solutions of the generlized Euler-Lgrnge eqution in some prticulr but relevnt cses given by the integrl curves of the second order differentil equtions coming from the semisprys (see Propositions 3.3 nd 3.). Actions on curves given by Pfff forms A Pfff form is differentible -form ω X (R T M): (.) ω = ω 0 ω i dx i ω i dy i. We define the ction I 0 of ω on curve γ : I = [ b] T M by the formul I 0 (γ) = (ω 0 ω i dx i ω i d x i ). If ω i = ω i 0 ω 0 = L : R T M R then it is esy to see tht I 0 is the ction I for the Lgrngin L. Denoting L (t x i y ()i y ()i ) = ω 0 (t x i y ()i ) ω i (t x i y ()i )y ()i ω i (t x i y ()i )y ()i we obtin tht I 0 is in fct the vrition of second order Lgrngin ffine in the second order velocities (ccelertions) s studied in [8]. Let us consider two points x y M nd γ 0 = (x i 0(t)) curve joining x nd y i.e. x i 0(0) = x nd x i 0() = y. Let us consider llowed vritions of γ 0 s vritions by curves joining x nd y hving the locl form γ ε = (x i ε(t)) where x i ε(t) = x i 0(t)εh i (t) nd stisfying the conditions (.) h i () = h i (b) = 0 dhi dhi () = (b) = 0.

3 8 P. Popescu M. Popescu A Pfff form cn be relted to second order dynmicl form considered in [3]. According to [3 Section ] first order dynmicl form on the bundle Y = R M M is one contct nd horizontl two form ν on J (Y ) hving the locl form ν = ν i (t x j y j )dx i ν i (t x j y j )dy i. Obviously first order dynmicl form is equivlent to give Pfff form with ω 0 = 0. An dvntge to use Pfff forms is hving the Lgrngin forms (i.e. ω i = ω i = 0) in the sme setting. An other motivtion to study Pfff forms is given by the possibility to use their ctions on curves through some second order Lgrngins tht verticl Hessins re null. One hve: d dε I 0 (γ ε ) ε=0 = ( ω 0 x i hi ω 0 y i dh i ω i dh i ) ( ω j x i hi ω j dh i y i )dxj 0 ( ω j x i hi ω j y i dh i )d x j 0. The cse of non-singulr Pfff forms A Pfff form ω given loclly by (.) is regulr if the verticl -form (.3) ( ω j y i ω i y j )dy i dy j ω i d h i. is regulr i.e. the mtrix ( ωj y i ωi y j ) ij is non-singulr. If the verticl -form.3 is not vnishing i.e. its mtrix is only non-null we sy tht the Pfff form is non-singulr. Let us consider now for singulr curve γ vrition tht stisfies the conditions (.). We hve: d dε I 0 (γ ε ) ε=0 (.) = ω 0 x i ω j x i dx j 0 ω j x i d x j 0 d (ω 0 y i ω j y i dx j 0 ω i ω j y i d x j 0 ) d ω i = 0. If the Pfff form ω is non-singulr then the eqution is of third order. regulr Pfff form one cn prove the following result. For Proposition.. If the Pfff form ω is regulr then the solutions of the generlized Euler-Lgrnge eqution (.) re the sme with the solutions of third order eqution given by globl second order semi-spry S : T M T 3 M. We omit the proof since it is not relevnt for the rest of the pper. Some importnt clsses of Pfff forms re when ω 0 = 0 (for exmple the cse of time independent Lgrngins L = L(x i y i )) nd when ω 0 = ω i = 0 (for exmple this is the cse of L = L(y i ));

4 Vritionl problems of some second order Lgrngins 85 If ω = ω j (y i )dy j then there re constnts c i so tht the eqution (.) hs the form ( ω j y i ω j y i )d x j 0 = c i. Exmple. Let us consider coordintes (x y) on R nd (x y X Y ) on R = T R. Let ω = Y dx XdY. The equtions (.) hve the form: d ( d ) ( ) y d dy = 0 tht simply implies d3 y d 3 t = 0 nd d3 x d 3 t = 0. The generl solution is: x(t) = C C t C 3 t y(t) = C C 5 t C 6 t. Exmple. In R s in Exmple. bove let ω = ydx xdy Y dx XdY. The equtions (.) hve the form dy d3 y dx d 3 = 0 t d3 x d 3 t = 0. The generl solution is x(t) = c cos t c 3 sin t c 5 x(t) = c cos t c sin t c 6 thus the integrl curves re ellipses nd stright lines. If t < t < t 3 re given then for every three distinct points A α (x α y α ) R α = 3 there is unique integrl curve in the fmily tht contins the three points i.e. t (x(t) y(t)) x(t α ) = x α y(t α ) = y α α = 3. Notice tht this feture chrcterizes the dynmics generted by third order differentil eqution while in generl n integrl curve is determined by three distinct points. Let us notice tht for second order differentil eqution n integrl curve is determined in generl by two distinct points. Let us consider now the cse dim M =. In this cse since the only sqewsymmetric mtrix of first order is the null mtrix the eqution (.) is lwys of second order for every Pfff form ω = ω 0 ωdx ωdy. In the cse when the locl functions ω 0 ω nd ω do not depend on y the Euler eqution hs the form (.5) ω d x 0 x ω x ( ) dx0 ω dx 0 xt ω 0 x ω t ω t = 0. According to [ Section.] stndrd Lgrngin hs the form (.6) L(t x y) = P (x t)y Q(x t)y R(x t). Its Euler-Lgrnge eqution is P x P x (x ) P t x (Q t R x ) = 0 where subscripts x t denote prtil derivtives nd x = dx x = d x. In [ Proposition..] one prove tht there is stndrd Lgrngin description (.6) for second order eqution x (t x) (x ) b(t x)x c(t x) = 0 iff b x = t ; then P = exp( x (t s)ds) nd R = x(qt (t s) c(t s)p (t s))ds where Q = Q(x t) is n rbitrry function. The following result cn be proved by strightforwrd verifiction.

5 86 P. Popescu M. Popescu Proposition.. The eqution (.5) dmits stndrd Lgrngin description. We clculte now the second derivtive of I (γ ε ) (the second vrition). Tking into ccount of the conditions (.) we obtin: = d dε I 0 (γ ε ) ε=0 ( ω 0 x i x j hi h j d ( ω 0 x i y j ω 0 y i x j )hi h j ω 0 dh i dh j y i y j ) ( ω k x i x j dx k 0 hi h j d (( ω k x i y j ω k y i x j )dxk 0 )hi h j ( d (ω j x i ω i x j )hi h j ( ω j y i ω i dh j y j )dhi ) ( ω k x i x j d x k 0 hi h j d (( ω k x i y j ω k y i x j )d x k 0 )hi h j ω k dx k 0 dh i dh j y i y j ) ω k d x k 0 dh i dh j y i y j ) ( d ( ω j x i ω i x j )hi h j ( ω j x i ω i dh j x j )dhi d ( ω j y i ω i dh j y j )dhi ). We prticulrize below this long formul in some importnt prticulr cses.. Some prticulr cses If ω = ω j (y i )dy j then there re the constnts c i such tht the eqution (.) becomes (.7) ( ωj y i ω ) i d x j 0 y j = c i. (.8) In this cse the second vrition is d dε I 0 (γ ε ) ε=0 = ( ω k y i y j ω i y k y j If m = dim M then we obtin the qudrtic form (( ω k y i y j ω i y k y j ω j y k y i ω j x k y k y i )d 0 dh i dh j. ) d x k ) 0. ij=m Let us consider now two exmples. Even the equtions of motion used in the following exmples hve the second order their integrl curves re obtined from some suitble equtions of Pfff forms. First we consider system tht hs the form dy = c cy by 3 (.9) dy = c y 3 cy dy 3 = c 3 by y where the coefficients re constnts. Notice tht the equtions of motion of chrge in field (formuls (7) in [ Section 7]) hve this form.

6 Vritionl problems of some second order Lgrngins 87 Proposition.3. There is Pfff form ω = ω j (y i )dy j on R 3 such tht the solutions of the system (.9) re lso solutions of the generlized Euler-Lgrnge eqution (.7). Proof. One cn check tht we cn consider the Pfff form ω = ω dy ω dy ω 3 dy 3 where (.0) (.) ω (y j ) = c 3 y by y y y 3 ω (y j ) = c y 3 y y cy y 3 ω 3 (y j ) = c y cy y 3 by y 3. One cn find other Pfff forms with the property sked by Proposition.3 looking for ω i = B ij y j C ijky i y j C ijk = C ikj (ll constnts). Let us investigte the second derivtive of the vrition. The mtrix (.8) hs in this cse the form dig( dy c cc 3 bc b dy bc cc 3 c c dy3 cc 3 bc c ). Let us tke s in [ Section 7] ( b c) = (0 0 ) (the z-xis) nd (c c c 3 ) = (0 c c 3 ) (in the Y Z plne) we obtin z = c 3 thus the bove mtrix becomes dig( c 3 c 3 0). It follows tht long the solutions ( d dε I (γ ε ) ε=0 ) hs nonpositive or non-negtive sign ccording to c 3 ; we cn find suitble c 3 ccording to z(t) = c 3t αt β. The second exmple is constructed using the Euler equtions of rigid body s follows. Let us consider system of second order equtions hving the form (.) x = β y z y = β z x z = β 3 x y. According to [6] the Euler equtions of rigid body hve the bove form (.) where (.3) β = I I 3 I β = I 3 I I β 3 = I I I 3. Proposition.. There is Pfff form ω = ω j (y i )dy j on R 3 such tht the solutions of the system (.) re solutions of the generlized Euler-Lgrnge eqution (.7) too. Proof. One cn check tht we cn consider the Pfff form with ω (y j ) = β 3 y ( y ) β y ( y 3) ω (y j ) = β y ( y 3) β 3 y ( y ) ω 3 (y j ) = β y 3 ( y ) β y 3 ( y ). One cn find other Pfff forms with the property sked by Proposition.3 looking for ω = A y (y ) B y (y 3 ) ω = A y (y 3 ) B y (y ) ω 3 = A 3 y 3 (y ) B 3 y 3 (y ) where A i nd B i re constnts nd (A B 3 ) = β (A 3 B ) = β nd (A B ) = β 3.

7 88 P. Popescu M. Popescu In the explicit cse of the Euler eqution of the rigid body when (.3) holds one cn tke ω (y y y 3 ) = I I 3 y ( y ) I I y ( y 3) δ (I 3 I ) ( 6 y ) 3 ω (y y y 3 ) = I I y ( y 3) I I 3 y ( y ) δ (I I 3) ( 6 y ) 3 ω 3 (y y y 3 ) = I 3 I y 3 ( y ) I 3 I y 3 ( y ) δ 3(I I ) ( 6 y 3 ) 3. Let us investigte the second derivtive of the vrition. The mtrix (.8) hs the form I I I 3 y y y 3 I I I 3 y 3 b y 3b I I I 3 y y y 3 y b I I I 3 y b 3 I I I 3 y b I I I 3 y b 3 I I I 3 y y y 3 3 I I I 3 I I I 3 I I I 3 where = (I I 3 ) ( I δ I I 3 δ I I 3 ) = (I 3 I ) ( δ I I 3 δ I I 3 I ) 3 = (I I ) ( δ 3 I I δ 3 I I I 3 ) b = I 3 y I 3 I y I 3 y I 3 I y b = I 3 y I I y I 3 3 y 3 I I 3 y 3 b 3 = I 3 y I I y I 3 3 y 3 I I 3 y 3. It is esy to see tht for δ i lrge enough the bove mtrix comes from positively or negtively definite qudrtic form ccording to the sign of y y y 3. Proposition.5. Let us consider the system (.) with the coefficients (.3) coming from the Euler eqution of the rigid body in n bounded domin U where y y y 3 0. Then there is Pfff form ω = ω j (y i )dy j defined for (y i ) U such tht the solutions of the system (.) re extreml solutions for the generlized Euler-Lgrnge eqution (.7) i.e. the second vrition hs constnt sign long these solutions. 3 A Hmiltonin description of non-singulr Pfff forms In this section we study the solutions of the generlized Euler-Lgrnge equtions for non-singulr Pfff forms considering n energy function ssocited with Pfff form nd semi-spry. A section S : R T M R T M of the ffine bundle R T M π R T M is clled (first order) semi-spry on T M. It cn be regrded s well s (time dependent) vector field Γ 0 on the mnifold T M since T M T T M. Let ω be Pfff form s in (.) nd S : R T M R T M (t x i y i ) S (t x i y i S i (t x j y j )) be semi-spry. We consider the energy: E S : T T M R E(x i y i q i p i ) = (q i ω i )y i (p i ω i )S i ω 0. Proposition 3.. The energy E S is globl function on T T M. y i Proof. One hve (q i ω i )y i (p i ω i )S i ω 0 = q i y i y i yi x p i i y i (ω i x ω i i ) (p i ω i )S i (p i ω i ) x i x i x y i y j ω j 0 = (q i ω i )y i (p i ω i )S i ω 0.

8 Vritionl problems of some second order Lgrngins 89 Let (3.) X ES = E S q i x i E S p i y i E S x i E S q i y i be the Hmiltonin vector field of the Hmiltonin E S ccording to the cnonicl symplectic form on T T M nd let M T T M be the submnifold defined by p i ω i = 0. Proposition 3.. If n integrl curve of X ES is tngent to the submnifold M then the curve projects to n integrl curve of the generlized Euler-Lgrnge eqution of the Pfff form ω. Proof. Let t (x i (t) y i (t) q i (t) p i (t)) be n integrl curve of X ES. By (3.): dx i = dy i yi = Si (3.) dq i = ω j x i yj ω j x i Sj (p j ω j ) Sj x i ω 0 x i dpi = ω j y i yj q i ω i ω j y i Sj (p j ω j ) Sj y i ω 0 y i. Since p i = ω i dxi d ω i = y i nd d x i = S i one hve = d (ω j y i dx j ω i ω j y i d x j ω 0 y i ) ω j x i dx j ω j x i d x j ω 0 x i. Then the Euler-Lgrnge eqution (.) holds. Using the definition of the submnifold M then its tngent subspce is generted by the locl frme of vectors { x i ω j x i p j y i ω j y i p j Thus the vector X ES is tngent to M in certin point of T T M hving s locl coordintes ((x i y i q i p i = ω i (x j y j )) iff ( (3.3) S i ωj y i ω ) i y j y i ω j x i ω i y j yi q j ω j ω 0 y j = 0. Exmple 3. Let us consider the setting of Exmple. nd ω = (XY )dxy dy. We use below the nottions x = x y = x X = y Y = y. Let {S i (x j y j )} i= be some rel functions considered s the components of semi-spry S on R. Then E S (x i y i q i p i ) = (q i ω i )y i (p i ω i )S i nd the differentil system tht gives the integrl curves of X ES hs the form dx i = yi dy i = Si q j }. p i dq i =0 dp i = q i ω j y i Sj (p j ω j ) Sj y i. We hve q i (t) = qi 0 (= const.) long the solution. The condition (3.3) gives S = q0 nd S = q0. Tking semi-spry on R with constnt coefficients we obtin the equtions dxi = y i yi = c i. It follows tht d3 x i = 0. Thus ll the solutions of (.) re obtined in this cse for suitble semi-spry on R with constnt coefficients. The bove exmple cn be extended s follows.

9 90 P. Popescu M. Popescu Proposition 3.3. Let us suppose tht there re some coordintes such tht the locl coefficients of regulr Pfff form ω depend only on (y i ). Then there re some locl semi-sprys whose locl coefficients depend only on (y i ) such tht:. An integrl curve Γ of X ES tht intersects M is included in M.. An integrl curve Γ projects on M on curve γ tht is solution of the generlized Euler-Lgrnge eqution of the Pfff form ω. 3. The curve γ is obtined s n integrl curve of semi-spry whose locl coefficients depend only on (y i ). Proof. If the locl coefficients of ω depend only on (y i ) then the second eqution (3.) implies q i (t) = qi 0 (= const.) nd the comptibility condition (3.3) becomes ( S i ωj y i ω ) i y j ω i y j yi qj 0 ω j ω 0 y j = 0. Since the Pfff form is regulr it follows tht the mtrix ( ωj y i ) ω i y j is non-singulr nd lso the resulting solution (S i ) hs ech component depending only on (y i ). Exmple. Consider setting of Exmple. nd the nottions x = x y = x X = y Y = y. Let {S i (x j y j )} i= be some rel functions considered s the components of semi-spry S on R. Then the differentil system tht gives the integrl curves of X ES reds dx i = yi dy i = Si dq i = ω j x i yj dp i = q i ω i ω j y i Sj (p j ω j ) Sj y i. Considering S (x i y i ) = x c nd S (x i y i ) = x c then tking into ccount Exmple. the integrl curves of ll semi-sprys S hving this form give ll the solutions of the generlized Euler-Lgrnge eqution (.) of ω. The bove exmple cn be extended s follows. Proposition 3.. Let us suppose tht there re some coordintes such tht the locl form of regulr Pfff form is ω = ω i (x j )dx i ω i (y j )dy i nd lso ω i x ωj j x = 0. i Then there re some locl semi-sprys S such tht:. An integrl curve Γ of X ES tht intersects M is included in M.. An integrl curve Γ projects on M on curve γ tht is solution of the generlized Euler-Lgrnge eqution of the Pfff form ω. Proof. Let us consider semi-spry given by the condition S j g ij = ω i c i where c i re constnts. Let t γ (x i (t) y i (t)) be n integrl curve of the semi-spry S nd denote by Γ the curve t Γ (x i (t) y i (t) q i (t) = ω i (x j (j))c i p i (t) = ω i (y j (t))). Then Γ M is n integrl curve of the Hmiltonin vector field X ES (3.) s it cn be checked by strightforwrd verifiction. The ssertion. follows from the construction of Γ tking into ccount the uniqueness of the integrl curve of X ES pssing through given point of M. The ssertion. follows using Proposition 3..

10 Vritionl problems of some second order Lgrngins 9 References [] C.S. Actrinei A pth integrl leding to higher order Lgrngins J. Phys. A: Mth. Theor. 0 (007) F99-F933. [] J.L. Cieśliński T. Nikiciuk A direct pproch to the construction of stndrd nd non-stndrd Lgrngins for dissiptive-like dynmicl systems with vrible coefficients J. Phys. A: Mth. Theor. 3 (00) [3] O. Krupkov The geometry of ordinry vritionl equtions Lect. Notes in Mth. Vol. 678 Springer-Verlg Berlin 997. [] L.D. Lndu E.M. Lifshitz The Clssicl Theory of Fields Cours of Theorethicl Physics vol. th revised English Edition Butterworth & Heinemnn Publ. 99. [5] M. Léon P. RodriguesGenerlized Clssicl Mechnics nd Field Theory North Hollnd 985. [6] J. Mrsden T. Rtiu Introduction to Mechnics nd Symmetry Second Edition Springer-Verlg New York Inc [7] R. Y. Mtsyuk The Vritionl Principle for the Uniform Accelertion nd Qusi-Spin in Two Dimensionl Spce-Time Symmetry Integrbility nd Geometry: Methods nd Applictions SIGMA (008) 06. [8] M. Popescu Totlly singulr Lgrngins nd ffine Hmiltonins Blkn Journl of Geometry nd its Applictions (009) [9] P. Popescu M. Popescu Affine Hmiltonins in Higher Order Geometry Int. J. Theor. Phys. 6 0 (007) [0] M. Mrcel P. Popescu Totlly singulr Lgrngins nd ffine Hmiltonins of higher order Blkn Journl of Geometry nd its Applictions 6 (0) -3. [] H. Rund The Hmilton-Jcobi theory in the clculus of vritions. Its role in mthemtics nd physics D. Vn Nostrnd Compny Ltd. London 966. [] C. Udriste A. Pite Optimiztion problems vi second order Lgrngins Blkn Journl of Geometry nd its Applictions 6 (0) [3] E.T. Whittker Tretise on the Anlyticl Dynmics of Prticles nd Rigid Bodies; with n Introduction to the Problem of Three Bodies Second edition University Press Cmbridge 97. Author s ddress: Pul Popescu nd Mrcel Popescu Deprtment of Applied Mthemtics University of Criov Fculty of Exct Sciences 3 A.I.Cuz st. Criov Romni. E-mil: pul p popescu@yhoo.com mrcelcpopescu@yhoo.com