Partial differential equations with differential constraints

Transcription

1 J. Differentil Equtions Prtil differentil equtions with differentil constrints Olg Krupková,b, Deprtment of Algebr nd Geometry, Fculty of Science, Plcký University, Tomkov 40, Olomouc, Czech Republic b Deprtment of Mthemtics, L Trobe University, Bundoor, Victori 3086, Austrli Received 6 October 2004; revised 2 Mrch 2005 Avilble online 18 April 2005 Abstrct A geometric setting for constrined exterior differentil systems on fibered mnifolds with n-dimensionl bses is proposed. Constrints given s submnifolds of et bundles loclly defined by systems of first-order prtil differentil equtions re shown to crry nturl geometric structure, clled the cnonicl distribution. Systems of second-order prtil differentil equtions subected to differentil constrints re modeled s exterior differentil systems defined on constrint submnifolds. As n importnt prticulr cse, Lgrngin systems subected to first-order differentil constrints re considered. Different kinds of constrints re introduced nd investigted Lgrngin constrints, constrints dpted to the fibered structure, constrints rising from codistribution, semi-holonomic constrints, holonomic constrints Elsevier Inc. All rights reserved. MSC: 35G20; 58A15; 58J60 Keywords: Fibered mnifold; Dynmicl form; Exterior differentil system; Constrint submnifold; Non-holonomic constrint; Cnonicl distribution of constrint; Constrined systems of PDEs; Constrined Lgrngin systems Deprtment of Algebr nd Geometry, Fculty of Science, Plcký University, Tomkov 40, Olomouc, Czech Republic. Fx: E-mil ddresses: krupkov@inf.upol.cz, O.Krupkov@ltrobe.edu.u /$ - see front mtter 2005 Elsevier Inc. All rights reserved. doi: /.de

2 O. Krupková / J. Differentil Equtions Introduction Within the clssicl clculus of vritions nd optiml control theory, equtions subected to different kinds of constrints re investigted, providing mthemticl models for motion of vrious systems ppering in mechnics nd engineering. Recently, nmely constrints given by systems of ordinry differentil equtions hve been intensively studied with the help of methods of differentil geometry nd globl nlysis, nd generl theory of non-holonomic systems in fibered mnifolds ws founded. This concerns geometric version of Chetev equtions [6] nd its generliztion to constrints given by higher-order ODEs, geometric model for constrined ODEs s differentil systems defined directly on constrint submnifolds, theory covering non-lgrngin systems s well s higher-order ODEs with higher-order differentil constrints, study of symmetries of constrined Lgrngin systems, Hmiltonin constrined systems, nd mny other questions see e.g. [5,8,13,22,23,25,29 34,36]. All the bove-mentioned results, however, hve been chieved for systems of ordinry differentil equtions; prtil differentil equtions, except of pioneer work [3], hve not yet been studied. In this pper we propose generliztion of the theory of non-holonomic systems to second-order prtil differentil equtions subected to constrints given by first-order PDEs. Our tsk is to trnsfer to this cse min ides from [22,25]. The exposition consists of the following four prts: In Section 2 we present geometric setting for systems of second-order prtil differentil equtions E σ x i, γ ν, γν x p, 2 γ ν x p x q = 0, 1σm, 1.1 for mppings x i γ ν x i, 1i n, 1νm, between smooth mnifolds. Eqs. 1.1 re modeled by dynmicl form nd its Lepge clss on et prolongtion of fibered mnifold π : Y X, where dim X = n nd dim Y = n + m, nd solutions re interpreted s integrl sections of corresponding exterior differentil system generted by n-forms. This pproch reltes the globl theory of differentil equtions to the clculus of vritions in fibered mnifolds [21]: it enbles, on one hnd, esily to consider vritionl equtions s specil cse, nd on the other hnd, to enlrge nd generlize to the non-vritionl cse some methods which hve been developed to investigte exclusively vritionl equtions. In Section 3 we study systems of first-order PDEs which hve the mening of differentil constrints in fibered mnifolds, i.e., which re fibered submnifolds of J 1 Y Y. In this pper we focus on significnt clss of constrints, which we cll regulr constrints chrcterized by rnk condition 3.3. As key-result it is shown tht every regulr constrint is endowed with nturl geometric structure, nmely, subbundle of the tngent bundle, which we cll the cnonicl distribution. This subdistribution of the Crtn distribution hs n nlogy in non-holonomic mechnics, where it plys role of generlized virtul displcements. Thus, we cn sy tht regulr constrints comply with generlized D Alembert principle.

3 356 O. Krupková / J. Differentil Equtions Section 4 dels with constrined PDEs. First of ll, we ssocite with unconstrined equtions new equtions, defined on the constrint submnifold. The geometric model for unconstrined PDEs, together with the cnonicl distribution of the constrint gives the constrined equtions represented by n exterior differentil system on the constrint submnifold. In prticulr, we re interested in constrined vritionl equtions, nd we find constrined Euler Lgrnge opertor. While in the unconstrined clculus of vritions on fibered mnifolds Lgrngin is differentil form which cn be loclly represented by function, L, it turns out tht constrined Lgrngin is differentil form which cnnot be represented by single function. Next, we study constrined PDEs s locl deformtions of unconstrined PDEs, nd we obtin equtions which generlize to the cse of PDEs Chetev equtions, known from non-holonomic mechnics. We lso show tht generlized Chetev equtions nd constrined equtions re equivlent. Section 5 is devoted to detil study of different kinds of constrints, which re covered by our setting. It turns out tht for prtil differentil equtions one hs more interesting constrints thn for ODEs. In prticulr, there pper constrints which we cll Lgrngin, nd π-dpted. Besides, one hs, similrly s in mechnics, constrints defined by distribution on Y, semi-holonomic, nd holonomic constrints. We study properties of these constrints nd their reltions. 2. Dynmicl forms in et bundles 2.1. Fibered mnifolds nd their prolongtions Throughout this pper, we ssume ll mnifolds nd mps be smooth, nd use stndrd nottions: T nd J r denotes the tngent nd the r-et prolongtion functor, respectively, d the exterior derivtive, the pull-bck, i ξ the contrction by vector field ξ, etc. The summtion convention is used unless otherwise explicitly stted. Let us briefly recll min concepts from the theory of fibered mnifolds nd the corresponding clculus. For more detils we refer to [19,21,35] see lso [4,14]. We consider fibered mnifold π : Y X with bse X of dimension n, nd totl spce Y, dim Y = m + n, nd its et prolongtions π r : J r Y X; for simplicity of nottions, we lso write J 0 Y = Y nd π 0 = π. There re nturlly induced fibered mnifolds π r,s : J r Y J s Y, where r>s0. In this pper we shll minly work with the first nd second et prolongtion of π, i.e., with fibered mnifolds π 1, π 2. Locl fibered coordintes on Y re denoted by x i,y σ, where 1i n, 1σm, nd the ssocited coordintes on J 1 Y nd J 2 Y by x i,y σ,y σ nd xi,y σ,y σ,yσ k, where 1 k n, respectively. In formuls, we use summtion over ll vlues of indices not only over non-decresing sequences. In clcultions we use on J 1 Y resp. J 2 Y, either cnonicl bsis of one forms, dx i,dy σ,dy σ resp. dxi,dy σ,dy σ,dyσ k, or bsis dpted to the contct structure, dx i, ω σ,dy σ, resp. dxi, ω σ, ω σ,dyσ k, where ω σ = dy σ y σ k dxk, ω σ = dyσ yσ k dxk. 2.1

4 O. Krupková / J. Differentil Equtions Next, we denote ω 0 = dx 1 dx n, ω 1 = i /x ω 0, ω 1 2 = i /x 2 ω 1,..., ω 1... n = i /x n ω 1... n By section γ of π we men mpping γ : U Y, defined on n open subset U of X, such tht π γ = id U. In fibered coordintes, components of section γ of π tke the form x i, γ σ, where the γ σ s re functions of the x i s. Components of the first et prolongtion J 1 γ which is section of π 1 tke the form x i, γ σ, γ σ /x. Similrly, components of the second et prolongtion J 2 γ of γ become x i, γ σ, γ σ /x, 2 γ σ /x x k. A section of π r is clled holonomic if it is the r-et prolongtion of section of π. A vector field ξ on J r Y, r 0, is clled π r -proectble if there exists vector field ξ 0 on X such tht T π r.ξ = ξ 0 π r, nd π r -verticl if it proects onto zero vector field on X, i.e., T π r.ξ = 0. Quite similrly one cn define π r,s -proectble or π r,s -verticl vector field on J r Y, where r>s. A differentil k-form η on J r Y is clled π r -horizontl resp. π r,s -horizontl ifi ξ η = 0 for every π r -verticl resp. π r,s -verticl vector field ξ on J r Y. Note tht π r -horizontl forms re those which in fibered coordintes contin wedge products of the bse differentils dx i only with components dependent upon ll the fibered coordintes. Similrly, π r,0 -horizontl forms contin wedge products of the totl spce differentils dx i s nd dy σ s only, etc. To every k-form η on J r Y one cn ssign unique horizontl k-form on J r+1 Y, denoted by hη nd clled the horizontl prt of η. The mpping h is defined to be n R-liner wedge product preserving mpping such tht for every function f on J r Y, hf = f π r+1,r, nd hdx i = dx i, hdy σ = y σ k dxk, hdy σ = yσ k dxk, etc. 2.3 In prticulr, hdf = d i fdx i, where d i is the ith totl derivtive opertor which for first-order function f tkes the form d i d dx i = x i + yσ i y σ + yσ ki yk σ. 2.4 By definition of h, for ny form η of degree k>n, hη = 0. A k-form η on J r Yr1 is clled contct if for every section γ of π, J r γ η = 0. A contct k-form is clled 1-contct resp. q-contct, q 2 if for every verticl vector field ξ, the k 1-form i ξ η is horizontl resp. q 1-contct. Theorem 2.1 Krupk [19]. Every k-form η on J r Y hs cnonicl decomposition π r+1,r η = hη + p 1η + p 2 η + +p k η, 2.5

5 358 O. Krupková / J. Differentil Equtions where hη is unique horizontl form, nd p q η, q = 1, 2,...,k, re unique q-contct forms. The forms hη nd p q η, q = 1, 2,...,k, bove re clled the horizontl prt of η, nd the q-contct prt of η, respectively Differentil equtions modeled by dynmicl forms Definition 2.2. By second-order dynmicl form on fibered mnifold π : Y X we understnd differentil n + 1-form on J 2 Y which is 1-contct, nd horizontl with respect to the proection onto Y. In fibered coordintes one gets E = E σ ω σ ω 0, 2.6 where E σ re functions depending upon x i,y ρ,y ρ p,y ρ pq. Asection γ of π defined on n open set U X is clled pth of E if E J 2 γ = The bove eqution, clled eqution for pths of dynmicl form E, tkes in fibered coordintes the form of system of m second-order prtil differentil equtions, or, more explicitly, E σ J 2 γ = 0, 1σm, 2.8 E σ x i, γ ρ, γρ x p, 2 γ ρ x p x q = 0, 1σm, 2.9 where m = dim Y dim X is the fiber dimension. Note tht equtions for pths of dynmicl form on fibered mnifold with the bse dimension n nd fiber dimension m cn be regrded s globl chrcteriztion of locl differentil equtions 2.9 for grphs of mppings R n R m. Dynmicl forms represent quite wide clss of systems of differentil equtions: in prticulr, they contin ll second-order vritionl PDEs. Definition 2.3. Let E be dynmicl form on J 2 Y. We sy tht n + 1-form α on J r Y is relted to E if p 1 α = E. Tking into ccount Theorem 2.1, we cn see tht α is relted to E if nd only if π r+1,rα = E + F, 2.10

6 O. Krupková / J. Differentil Equtions where F is n t lest 2-contct form. The fmily of ll to E relted n + 1-forms, defined possibly on open subsets of J r Y, will be clled the Lepge clss of E of order r, nd denoted by [α] r. We cn see tht every second-order dynmicl form hs relted Lepge clsses of order r for every r 2. Lepge clsses re used for geometricl description of equtions 2.7 i.e., 2.9 by mens of exterior differentil systems s follows: Proposition 2.4. Let be dynmicl form on J 2 Y, [α] r its Lepge clss of order r. For α [α] r consider the idel H α in the exterior lgebr on J r Y, generted by the system of n-forms i ξ α, where ξ runs over ll π r -verticl vector fields on J r Y The following conditions re equivlent: 1 A section γ of π is pth of E, i.e., E J 2 γ = 0. 2 For every α [α] r, J r γ is n integrl section of the idel H α, i.e., J r γ i ξ α = 0, for every π r -verticl vector field ξ on J r Y For every α [α] r, J r+1 γ i ξ ˆα = 0, for every π r+1 -verticl vector field ξ on J r+1 Y, 2.13 where ˆα is the t most 2-contct prt of α. Proof. Suppose 1. Then E σ J 2 γ = 0, 1σm. This mens tht for every π 2 -verticl vector field ζ on J 2 Y, ζ = ζ σ y σ + ζσ i yi σ + ζ σ i y σ i, 2.14 we hve J 2 γ i ζ E = J 2 γ E σ ζ σ ω 0 = E σ ζ σ J 2 γ ω 0 = If ξ is verticl vector field on J r Y, denote by ˆξ vector field on J r+1 Y which proects onto ξ. Since Eq depends only upon the totl spce components of vector fields on J 2 Y, we cn see tht lso for every π r -verticl vector field ξ on J r Y where r>2, J r+1 γ iˆξ π r+1,2e =

7 360 O. Krupková / J. Differentil Equtions Hence, for every α [α] r, nd every verticl ξ, J r γ i ξ α = J r+1 γ iˆξ π r+1,r α = J r+1 γ iˆξ p 1α = J r+1 γ iˆξ π r+1,2e = Conversely, suppose tht γ stisfies Eqs Tking ny α [α] r, nd using tht possibly up to proection E = p 1 α, we get from 2.17 by similr rguments s bove tht E J 2 γ = 0. Assertions 2 nd 3 re equivlent, s seen immeditely from Definition 2.5. We shll cll the idels H α nd Hˆα introduced in Proposition 2.4 Hmiltonin idel nd principl Hmiltonin idel of the n + 1-form α, respectively, nd the form ˆα the principl prt of α. Proposition 2.4 sys tht ll Hmiltonin idels nd principl Hmiltonin idels ssocited with E hve the sme holonomic integrl sections, nd these coincide with prolonged pths of E i.e., with solutions of Eqs. 2.7, respectively, 2.9. Remrk 2.6. The terminology for H α reflects tht one used in the clculus of vritions. If Eqs. 2.9 i.e., 2.7 re vritionl i.e., re Euler Lgrnge equtions, then relted Eqs re clled Hmilton Eqs. see [10,20,24], lso [7,9], etc Equtions polynomil in the second derivtives We shll study second-order PDEs which dmit first-order Lepge clss. In view of the bove considertions this mens tht equtions of this kind re described by mens of exterior differentil systems on J 1 Y. Proposition 2.7. Let E = E σ ω σ ω 0 be dynmicl form on J 2 Y. E hs Lepge clss of order 1 if nd only if the functions E σ,1σm, re polynomils of degree n in the vribles y ν i, i.e., E σ =A σ +B i 1 σν 1 y ν 1 i 1 +B i 1 2 i 2 σν 1 ν 2 y ν 1 i 1 y ν 2 2 i 2 + +B i 1... n i n σν 1...ν n y ν 1 i 1...y ν n n i n, 2.18 nd the coefficients B i 1... k i k σν 1...ν k indices i 1,...,i k. where 2k n re completely ntisymmetric in the Proof. In bsis dpted to the contct structure, every n + 1-form α on J 1 Y tkes the form polynomil in dy ν, i.e., α = β 0 +β ν 1 dy ν 1 +β 2 ν 1 ν 2 dy ν 1 dy ν 2 2 +β 2 3 ν 1 ν 2 ν 3 dy ν 1 dy ν 2 2 dy ν β... n ν 1...ν n dy ν 1... dy ν n n +β... n+1 ν 1...ν n+1 dy ν 1 dy ν n+1 n+1, 2.19

8 O. Krupková / J. Differentil Equtions where the β s re p-forms n + 1p 0 expressed by mens of wedge products of the dx i s nd ω ρ s only. Substituting dy ν = ω ν + yν i dxi, we obtin the lift π 2,1α of α expressed s sum of contct prts, nd we cn see tht ll components of π 2,1 α re polynomils in the vribles yν i. In prticulr, this concerns the first term, i.e., the 1-contct prt E = p 1 α, which is by ssumption dynmicl form. Tking into ccount tht the term ω ν 1 ω ν n+1 n+1 gives no contribution to E indeed, y ν 1 i 1 dx i 1 y ν 2 2 i 2 dx i 2 y ν n+1 n+1 i n+1 dx i n+1 = 0, we conclude tht the components of E re polynomils of degree t most n. Finlly, the ntisymmetry condition for the B s ppers, since B i 1... p i p σν 1...ν p rise s components t ω σ y ν 1 k 1 dx k 1 y ν 2 2 k 2 dx k 2 y ν p p k p dx k p ω i1...i p, which re completely ntisymmetric in the indices i 1...i p. Conversely, ssume tht fiber chrt components E σ,1σm, ofe re polynomils chrcterized by the proposition. Put α 0 = A σ ω σ ω 0 +B i 1 σν 1 ω σ dy ν 1 ω i B i 1 2 i 2 σν 1 ν 2 ω σ dy ν 1 dy ν 2 2 ω i1 i ! B i 1 2 i 2 3 i 3 σν 1 ν 2 ν 3 ω σ dy ν 1 dy ν 2 2 dy ν 3 3 ω i1 i 2 i n! B i 1... n i n σν 1...ν n ω σ dy ν 1 dy ν n n ω i1...i n Then α 0 is locl form on J 1 Y such tht p 1 α 0 = E, i.e., it genertes first-order Lepge clss of E. In ccordnce with [11], we sy tht dynmicl form E on J 2 Y is J 1 Y -pertinent if it possesses first-order Lepge clss, i.e., its components E σ tke the form described by Proposition 2.7. In wht follows, we denote first-order Lepge clss [α] 1 simply by [α], nd we write α 1 α 2 for α 1, α 2 [α] The n + 1-form α 0 givenby2.20 is locl first-order form relted with E, which is miniml in the sense tht it does not contin ny free terms. All the first-order relted n + 1-forms re then chrcterized s follows: Corollry 2.8. The first-order Lepge clss [α] of J 1 Y -pertinent dynmicl form E consists of ll locl forms α = α 0 + F, 2.22 where α 0 is given by 2.20, nd F is n t lest 2-contct form defined on n open subset of J 1 Y. The clss [α] contins subclss of forms belonging to the idel generted by the 1-contct forms ω σ,1σm; in prticulr, one cn even consider invrint representtives such tht F is π 1,0 -horizontl contins no dy ν.

9 362 O. Krupková / J. Differentil Equtions For dynmicl forms whose components re ffine in the second derivtives i.e., for qusiliner second-order PDE the sitution further simplifies: Corollry 2.9. Every dynmicl form E on J 2 Y with components ffine in the second derivtives, i.e., such tht E σ = A σ + B i σν y ν i, 2.23 is J 1 Y -pertinent, nd its first-order Lepge clss [α] consists of the following forms: α = A σ ω σ ω 0 + Bσν i ω σ dy ν ω i + F = E σ E σ yi ν yi ν ω σ ω 0 + E σ yi ν ω σ dy ν ω i + F, 2.24 where F is t lest 2-contct, nd, i denotes symmetriztion in the indicted indices Vritionl equtions Among equtions we hve considered up to now, there is n importnt fmily of vritionl equtions, hving mny specific properties. We briefly recll without proofs bsic concepts from the clculus of vritions on fibered mnifolds in order to put vritionl equtions into the bove generl scheme. The exposition follows [15,16,19,21], where more results nd proofs cn be found. A horizontl n-form λ on J 1 Y where n = dim X is clled first-order Lgrngin. A form ρ such tht hρ = λ, nd p 1 dρ is π 1,0 -horizontl is clled Lepgen equivlent of λ [15]. Lepgen equivlents of first-order Lgrngin λ = Lω 0 tke the form ρ = Θ λ + μ = Lω 0 + L y σ ω σ ω + μ, 2.25 where Θ λ is the Poincré Crtn form, nd μ is n rbitrry t lest 2-contct form. Fmily 2.25 of Lepgen equivlents of λ contins the following n-form: ρ λ = L ω 0 + n k=1 1 k! 2 k L y σ 1 y σ ω σ 1 ω σ k ω k 1 k, 2.26 k clled Krupk form see [17,2]. If ρ is Lepgen equivlent of λ then the ction functions of ρ nd λ re the sme, nd the pths of the dynmicl form E λ = p 1 dρ 2.27

10 O. Krupková / J. Differentil Equtions re extremls of the Lgrngin λ. E λ is clled the Euler Lgrnge form of λ, its components Euler Lgrnge expressions, nd equtions for pths of E re clled Euler Lgrnge equtions. It holds E λ = ε σ Lω σ ω 0, where ε σ L = L y σ d dx L y σ, 1σm Since λ is first-order Lgrngin, the Euler Lgrnge expressions 2.28 re ffine in the second derivtives. Keeping nottions of 2.18 we hve where E σ = A σ + B i σνy ν i = A σ + B i σν y ν i, 2.29 E σ = ε σ L, A σ = ε L σ L y σ d dx L y σ, 2 L Bσν i = BσνL i yi σ, 2.30 yν nd B σν i = 2 1 Bσν i + Bσν i. Above, d /dx is the cut totl derivtive d dx = d dx yν i yi ν = x + yν y ν Euler Lgrnge equtions tke one of the following equivlent intrinsic forms: J 1 γ i ξ dρ = 0 for every verticl vector field ξ on J 1 Y. E λ J 2 γ = The first eqution comes from the first vrition formul for the Lgrngin λ, the second one reflects the fct tht the Euler Lgrnge form E λ is dynmicl form. By definition of E λ, formul 2.29, nd Corollry 2.9 we immeditely get: Proposition Let λ be first-order Lgrngin. Then its Euler Lgrnge form E λ hs Lepge clss defined on J 1 Y. Moreover, in the first-order Lepge clss of E λ there re the following distinguished representtives: dθ λ dρ λ α 0λ α λ, 2.33

11 364 O. Krupková / J. Differentil Equtions where α 0λ = ε σ L ωσ ω 0 + B i σν L ω σ dy ν ω i, α λ = ε σ L ωσ ω 0 + B i σν L ω σ dy ν ω i Euler Lgrnge equtions 2.32 then red J 1 γ i ξ α = 0 for every verticl vector field ξ on J 1 Y, 2.35 where α is ny element belonging to the first-order Lepge clss of E λ, nd they re equtions for holonomic integrl sections of the idel H α Remrk It is known how to recognize whether dynmicl form E coincides t lest loclly with the Euler Lgrnge form of Lgrngin see [12] for secondorder ODEs, [1,18] for PDEs of ny order. Necessry nd sufficient conditions for vritionlity of second-order dynmicl forms tke the following form of conditions on the left-hnd sides of the corresponding equtions: E σ y ν i E ν y σ i = 0, E σ y ν + E ν y σ 2d i E ν y σ i = 0, E σ y ν E ν y σ + d E ν i yi σ d i d E ν y σ i = A locl Lgrngin then cn be computed using the Tonti Vinberg formul 1 L = y σ E σ x i,uy ν,uy ν,uyν k du Next, it is known tht dynmicl form E is loclly vritionl if nd only if the Lepge clss [α] 2 of E contins closed representtive i.e., there exists α [α] 2 such tht dα = 0 [18,22,11]. 3. Constrint structure in J 1 Y Definition 3.1. By non-holonomic constrint in J 1 Y we shll men fibered submnifold Q of π 1,0 : J 1 Y Y, codim Q = κ, where 1κmn 1. This mens tht in ny fibered chrt constrint Q cn be expressed by equtions f α x i,y σ,y σ = 0, 1ακ, 3.1

12 O. Krupková / J. Differentil Equtions such tht f α rnk y σ = κ, where α lbels rows nd σ, lbel columns. 3.2 If moreover f α rnk y σ = k, where α, lbel rows nd σ lbels columns, 3.3 for some k, 1k m 1, we sy tht Q is regulr non-holonomic constrint of cornk κ,k. Remrk 3.2. Notice tht condition 3.3 is invrint. Indeed, with obvious nottions we hve F α σ = f α ȳ σ = f α y ν l y ν l ȳ σ = f α y ν l y ν ȳ σ x x l = F αl ν Bν σ A l, i.e., in mtrix nottion, F 1 F 2. F κ AF 1 A 0 0 = AF 2. B = 0 A 0 AF κ 0 A Since the mtrices A, B re regulr, we get s desired. f α rnk ȳ σ = rnk F 1 F 2. = rnk F κ F 1 F 2. F κ F 1 F 2. F κ B. = rnk f α y σ, Let V, ψ be fibered chrt on Y, V 1, ψ 1 the ssocited chrt on J 1 Y, U V 1 n open set. A regulr constrint Q in J 1 Y of cornk κ,k nturlly gives rise to the following distributions, defined on U: 1 D U, nnihilted by the 1-forms df α,1ακ. The rnk condition 3.2 gurntees tht D U hs constnt cornk equl to κ on U, i.e., its rnk is n + m + nm κ.

13 366 O. Krupková / J. Differentil Equtions C U, nnihilted by the following 1-forms, α = f α dx + 1 n f α y σ ω σ, 1ακ, 1 n. 3.4 These 1-forms re not independent, however, due to the rnk condition 3.3, there exist functions cα,1k, 1ακ, 1 n, onu, such tht the k m-mtrix M = Mσ, where M σ = 1 f α n c α y σ, 3.5 hs mximl rnk equl to k. This mens tht =cα α =cα f α dx + 1 f α n c α y σ ω σ =cα f α dx +Mσ ωσ, 1k, 3.6 re independent t ech point of U. Hence, the distribution C U = nnih{, 1 k}, 3.7 hs constnt cornk k, i.e., rnk C U = n + m + nm k. 1-forms nnihilting the distribution C U will be clled cnonicl constrint 1-forms of the constrint Q. 3 C U, nnihilted by k + κ independent 1-forms,df α,1k, 1ακ. Immeditely from the bove constructions we cn see tht the following ssertions hold: Proposition 3.3. Q U is n integrl submnifold of D U. Hence, for every x Q, the forms df α x, 1ακ, nnihilte the tngent spce T x Q to the mnifold Q t x, i.e., long Q, D = nnih{df α, 1ακ} =TQ. Corollry 3.4. Let Q be constrint of codimension κ in J 1 Y, nd let f α = 0 nd f α = 0, where 1ακ, be two sets of equtions of Q on n open set U V1 J 1 Y. Then there re functions γ α β γ on U such tht t ech point of U, α β is regulr mtrix, nd df α = γ α β df β. In prticulr, t ech point x Q U, f α y σ = γ α f β β y σ. 3.8

14 O. Krupková / J. Differentil Equtions Proposition 3.5. C U is subdistribution of both C U nd D U. At the points of Q U, the distributions C U nd C U D coincide, nd define distribution of cornk k on Q U. Now, we shll show tht the bove locl distributions on the constrint Q unite into globl distribution on Q. Theorem 3.6. Let Q be regulr constrint in J 1 Y of cornk κ,k, let ι : Q J 1 Y be the cnonicl embedding of the submnifold Q into J 1 Y. If, 1 k, re independent locl cnonicl constrint 1-forms, put Then is distribution of cornk k on Q. φ = ι = M σ ι ι ω σ, 1 k. 3.9 C = nnih{φ, 1 k} 3.10 Proof. Tking into ccount Propositions 3.3 nd 3.5, it is sufficient to show tht if α defined on U 1 nd α defined on U 2 such tht U 1 U 2 Q = re constrint 1-forms of Q nnihilting the distribution C U 1 nd C U 2, respectively, then on U 1 U 2 Q, φ α = c α βl φβl 3.11 for some functions c α βl, mening tht C U1 = C U2 t the points of U 1 U 2 Q. Denote x i,y σ,y σ nd x i, ȳ σ, ȳ σ ssocited fibered coordintes on U 1 nd U 2, respectively, nd ssume tht the constrint Q is given by the equtions f α x i,y σ,y σ = 0onU 1, nd f α x i, ȳ σ, ȳ σ = 0onU 2, where 1ακ. Wehve α = f α dx + 1 n f α y σ ω σ, α = f α d x + 1 n f α ȳ σ ω σ Now, by trnsformtion rules nd by 3.8, we get n φ α = ι f α ȳ σ = ι γ α x β ω σ x l f β y ν l =ι f α yl ν ω ν y ν l ȳ σ ȳ =ι σ y ρ ωρ γ α f β β yl ν x x l y ν ȳ σ ȳ σ y ρ ωρ = c α βl ι βl = c α βl nφβl, 3.13 proving our ssertion.

15 368 O. Krupková / J. Differentil Equtions Definition 3.7. The distribution C on Q defined in Theorem 3.6 will be clled cnonicl distribution. 1-forms belonging to the nnihiltor, C 0,ofC, will be clled cnonicl constrint 1-forms. The idel in the exterior lgebr of differentil forms on Q generted by C 0 will be clled cnonicl constrint idel, nd denoted by IC 0. Elements of IC 0 will then be clled cnonicl constrint forms. Note tht, by definition, C is the chrcteristic distribution of the idel IC 0. Let us find vector fields belonging to the cnonicl distribution. Theorem 3.8. The cnonicl distribution C on Q is loclly spnned by the following vector fields: c x i x i + k Fi ι y m k+, =1 1i n, c y s y s + k =1 G s ι, 1s m k, ym k+, zj 1J nm κ, 3.14 where x i,y σ,z J,f α, 1i n, 1σm, 1J nm κ, 1ακ, denote fibered coordintes dpted to the submnifold ι : Q J 1 Y, the functions G s represent t ech point fundmentl system of solutions of the system of independent homogeneous lgebric equtions for m unknowns Ξ σ,1σm, M σ Ξσ = 0, 1 k, 3.15 nd, for every i = 1, 2,...,n, the F i re solutions of the equtions Mσ F i σ = Mσ yσ i f α cαi, 1 k, 3.16 where y σ re considered s functions of zj,f β corresponding to the choice of ll the prmeters equl to zero. Proof. The rnk condition 3.2 gurntees tht in neighborhood of every point in Q one cn find coordintes x i,y σ,z J,f α, where 1i n, 1σm, nd 1J nm κ, 1ακ. Consider the distribution C U on U J 1 Y such tht U Q =. For vector field ξ on U denote ξ = ξ i x i + Ξσ y σ + Ξ J z J + Ξ α f α. 3.17

16 O. Krupková / J. Differentil Equtions The condition i ξ = 0 for ll, gives us the following system of equtions for the components of ξ: i.e., c α f α ξ + M σ Ξ σ y σ l ξl = 0, 1 k, 3.18 M σ Ξσ = M σ yσ l c αl f α ξ l, 1 k, 3.19 where y x ν re functions of the dpted coordintes i,y σ,z J,f β. By ssumption, rnk M = k. This mens tht one cn express k of the functions Ξ σ, e.g. without loss of generlity Ξ m k+, where 1 k, by mens of prmeters ξ i, Ξ s,1in, 1s m k. Hence, the generl solution of i ξ = 0, 1 k, is ξ = ξ i x i + m k s=1 Ξ s k y s + Ξ m k+ ξ 1,...,ξ n, Ξ 1,...,Ξ m k y m k+ =1 + Ξ J z J + Ξ α f α, 3.20 where ξ i, Ξ s, Ξ J nd Ξ α re rbitrry functions nd Ξ m k+ re solutions of Eqs Hence, one cn tke independent vector fields on U spnning the distribution C U s follows: x i + k =1 F i y m k+, 1i n, y s + k =1 G s, 1s m k, ym k+ z J,, f α 1J nm κ, 1ακ, 3.21 where G s,1sm k, is fundmentl system of solutions of 3.15 i.e., 3.19 with ξ 1 = = ξ n = 0, nd F i,1i n, re solutions of 3.19 for Ξ1 = = Ξ m k = 0 here the subscript i corresponds to the choice ξ i = 1, ξ = 0 for = i. Since C = C U TQ, we finlly get { k C = spn x i + =1 F i y m k+, y s + k =1 Ḡ s y m k+, } z J, 3.22

17 370 O. Krupková / J. Differentil Equtions where, s bove, 1i n, 1s m k, 1J nm κ, nd G s ι. F i = F i ι, Ḡ s = The cnonicl distribution C on Q is subbundle of the tngent bundle TQ Q. In generl, however, it need not be completely integrble. We shll study conditions for the complete integrbility of C in Section 5. Remrk 3.9 Nottions dpted to the constrint structure. i The following conventions concerning nottion of indices will be used, nd summtion over repeted indices will be understood if not otherwise explicitly stted: 1i,, l n, 1α, β, γκ, 1J nm κ, 1σ, ν, ρm, 1,b,ck, 1p, r, s m k ii Tking into ccount tht the mtrix 3.5 in 3.6 hs mximl rnk, k, one cn express k of the contct 1-forms ω σ by mens of the constrint forms,1m, nd the remining ω ν s. Without loss of generlity we my suppose tht this concerns the forms ω m k+, where 1 k. In n dpted bsis x i,y σ,z J,f α nd in the nottions of the bove theorem it holds ω m k+ = μ b b Ms b ωs cα b f α dx = μ b b + G s ωs + F + G s ys ym k+ dx, 3.24 where μ b is n pproprite regulr mtrix. Here nd in wht follows, yσ re considered s functions of the dpted coordintes x i,y σ,z J,f α. Similrly, the rnk condition 3.2 gurntees tht one cn express the forms dz α by mens of df β,dx i,dy σ,dz J. Thus, we hve on J 1 Y the following bses of 1-forms, dpted to the constrint structure: dx i,dy s,,dz J,df A, or dx i, ω s,,dz J,df A ; 3.25 Consequently, with obvious nottions we my write ω m k+ ι ω m k+ = φ + Ḡ s ωs, 3.26 where ω s = ι ω s, nd φ = ι μ b b = μ b ιφb. We cn see tht, on Q, insted of cnonicl bsis dx i,dy σ,dz J, or bsis dx i, ω σ,dz J dpted to the induced contct structure, it is worth to work with bses dpted to the constrint structure,

18 O. Krupková / J. Differentil Equtions where the cnonicl constrint 1-forms pper: dx i,dy s, φ,dz J, dx i, ω s, φ,dz J iii Keeping the bove nottions we cn express the functions G s nd F in 3.21 nd 3.14 s follows: ppering G s = μ b Mb s, F = ym k+ G s ys μ b cb α f α We lso put y σ ι = gσ With this nottion, Ḡ s = μ b Mb s ι, F = gm k+ Ḡ s gs iv The vector fields c /x i nd c /y s on Q defined by 3.14 will be clled constrint prtil derivtive opertors. Weput d c dx i = c x i + gs i d c dx i = c x i + gs i c y s + zj i c y s, z J = d c dx i + zj i z J, 3.31 nd cll the bove opertors the ith cut constrint nd constrint totl derivtive opertor, respectively. We note tht the opertors d c /dx i ct on functions on Q, giving rise to functions on Q, the lift of Q in J 2 Y defined s the mnifold of ll points Jx 2γ J 2 Y such tht Jx 1γ Q. v The exterior derivtive of function f on Q cn be expressed s follows: f df = x + f y σ gσ dx + f y s ωs + f = x + f y s gs + f y m k+ F + Ḡ s gs dx f + y s + f Ḡ s y m k+ ω s + f y m k+ ωm k+ + f f y m k+ φ + f z J dzj z J dzj = d c f dx dx + cf y s ω s f + y m k+ φ + f z J dzj, 3.32

19 372 O. Krupková / J. Differentil Equtions since by 3.30 g m k+ = F + Ḡ s gs vi Let us compute the explicit expression for d φ which will often be used lter d φ = d ω m k+ dḡ s ωs Ḡ s d ωs = dg m k+ = nd denote Ḡ s d c gs dx i Ḡ s c g s y r Ḡ g s s Ḡ s dx dḡ s ωs + Ḡ s dgs dx d c g m k+ dx i dx i dx + d cḡ r dx z J g g s m k+ z J g ym k+b cg m k+ y m k+b m k+ y r ω r dx dz J dx cḡ s y r φ b dx ω r ω s Ḡ s z J dzj ω s Ḡ s y m k+b φb ω s, 3.34 g s m k+ CJ = Ḡ s z J g z J = F z J Ḡ s z J gs, Ci s = z J C J yi s Constrined systems 4.1. Constrined PDEs Let Q be regulr constrint in J 1 Y, IC 0 the ssocited cnonicl constrint idel. Since for every q-contct form η on J 1 Y ι η is q-contct form on Q, wehvethe following equivlence reltion on n + 1-forms on Q: α 1 α 2 if α 1 α 2 = F + φ, 4.1 where F is n t lest 2-contct n + 1-form on Q, nd φ is constrint n + 1-form. We denote by [[α]] the clss of α. If [α] is Lepge clss on J 1 Y then obviously for ny of its two elements, α 1 α 2 ι α 1 ι α

20 O. Krupková / J. Differentil Equtions This enbles us to ssocite with given system of second-order PDEs, polynomil in the second derivtives, system of equtions defined on the constrint Q. Recll tht the equtions we consider re chrcterized by dynmicl form with components left-hnd sides of the equtions given by Proposition 2.7 formul Definition 4.1. Let E be J 1 Y -pertinent dynmicl form on Y, [α] its Lepge clss on J 1 Y. By the constrined system ssocited with E nd the constrint Q we shll men the equivlence clss [[ι α]]. A generl element of the clss [[ι α]] is of the form ᾱ = ι α + F + φ, 4.3 where α [α] is ny n+1-form relted with E, F is t lest 2-contct, nd φ IC 0. In prticulr, we hve distinguished representtive ᾱ 0 = ι α 0 [[ι α]] cf. 2.20, s well s those ᾱ ᾱ 0 which belong to the idel generted by the forms ω σ = ι ω σ. Proposition 4.2. In dpted fibered coordintes x i,y σ,z J on Q, where ᾱ 0 A s ω s ω 0 +B i 1 sj 1 ω s dz J 1 ω i B i 1i 2 sj 1 J 2 ω s dz J 1 dz J 2 ω i1 i n! B i 1...i n sj 1...J n ω s dz J 1 dz J n ω i1...i n, 4.4 A s = Ā s + Ā m k+ Ḡ s + B i 1 sν 1 + B i 1 2 i 2 sν 1 ν 2 + B + + B i 1... n i n sν 1...ν n + B i 1 2 i 2 m k+ν 1 ν 2 Ḡ s + B i 1 m k+ ν 1 Ḡ s d c gν 1 dx i 1 i 1... n i n m k+ν 1...ν n Ḡ s d c gν 1 dx i 1 d c gν 2 2 dx i 2 d c gν 1 dx i 1 d c gν n n dx i n, B i 1 sj 1 = B i 1 sν 1 + B + 2 B i 1 2 i 2 sν 1 ν 2 i 1 m k+ ν 1 Ḡ s + B + +n B i 1... n i n sν 1...ν n g ν 1 z J 1 i 1 2 i 2 m k+ν 1 ν 2 Ḡ s + B g ν 1 z J 1 i 1... n i n m k+ν 1...ν n Ḡ s d c gν 2 2 dx i 2 g ν 1 z J 1 d c gν 2 2 dx i 2 d c gν n n dx i n,

21 374 O. Krupková / J. Differentil Equtions B i 1i 2 sj 1 J 2 = B i 1 2 i 2 sν 1 ν gν 1 z J 1 + B nn 1 2 g ν 2 2 z J 2 i 1 2 i 2 m k+ν 1 ν 2 Ḡ s d c gν 3 3. B i 1...i n sj 1...J n = B i 1... n i n sν 1...ν n + B nd Ā σ = A σ ι, B i 1 σν 1 = B i 1 σν 1 ι, etc. B i 1... n i n sν 1...ν n d c g ν n n dx i 3 dx i, n g ν 1 z J 1 + B i 1... n i n m k+ν 1...ν n Ḡ s g ν 2 2 z J 2 i 1... n i n m k+ν 1...ν n Ḡ s g ν n n z J 1 z J, 4.5 n g ν 1 Proof. By 2.20 nd in the nottions of Remrk 3.9 we hve ᾱ 0 = ι α 0 = Ā σ ω σ ω 0 + B i 1 σν 1 ω σ dg ν 1 ω i B i 1 2 i 2 σν 1 ν 2 ω σ dg ν 1 dg ν 2 2 ω i1 i n! B i 1... n i n σν 1...ν n ω σ dg ν 1 dg ν n n ω i1...i n Ā s + Ā m k+ Ḡ s ω s ω B i 1 sν 1 d c g ν 1 dx l 1 + B B i 1 2 i 2 sν 1 ν n! d c g ν 1 dx l 1 i 1 m k+ ν 1 Ḡ s + B dxl 1 + g ν 1 z J dzj 1 1 B i 1... n i n sν 1...ν n dx l 1 + gν 1 z J 1 dzj 1 ω s i 1 2 i 2 m k+ν 1 ν 2 Ḡ s + B d c g ν 2 2 dx l 2 d c g ν 1 dx l 1 ω s i 1... n i n m k+ν 1...ν n Ḡ s d c g ν n n dx l n dx l 1 + gν 1 z J dzj 1 ω i1 1 dx l 2 + gν 2 2 z J dzj 2 ω i1 i 2 2 ω s dxl n + g ν n n z J dzj n ω i1...i n, 4.6 n from which formuls 4.4, 4.5 esily follow. Corollry 4.3. If E σ re ffine in the second derivtives i.e., represent qusiliner second-order PDEs then ᾱ 0 A s ω s ω 0 + B i 1 sj 1 ω s dz J 1 ω i1, 4.7

22 O. Krupková / J. Differentil Equtions where A s = Ā s + Ā m k+ Ḡ s + B i 1 sν 1 B i 1 sj 1 = B i 1 sν 1 + B + B i 1 m k+ ν 1 Ḡ s i 1 m k+ ν 1 Ḡ s d c gν 1 dx i 1, g ν 1 z J Recll tht unconstrined equtions were PDEs for sections γ : W Y, W X, of the fibered mnifold π : Y X. Solutions of constrined equtions hve to obey the constrint condition J 1 γw Q, 4.9 i.e., hve to stisfy the system of κ first-order PDE defining the constrint Q, f κ x i,y σ,y σ J 1 γ = Now, in ccordnce with the understnding of unconstrined equtions s equtions for holonomic integrl sections of ny Hmiltonin idel H α relted with Lepge clss [α], we cn consider constrined equtions s equtions for holonomic integrl sections of n pproprite idel in the exterior lgebr on Q: Definition 4.4. Let Q be constrint in J 1 Y with the cnonicl distribution C, E dynmicl form on J 2 Y, nd [[ι α]] its corresponding constrined system. For every ᾱ [[ι α]] consider the idel Hᾱ on Q, generted by the system of n-forms i ξ ᾱ, where ξ runs over ll verticl vector fields on Q belonging to C We shll cll Hᾱ constrint Hmiltonin idel. Sections γ : W Y of π such tht J 1 γw Q, which re integrl sections of Hᾱ, i.e., stisfy J 1 γ i ξ ᾱ = 0, for every verticl vector field ξ C, 4.12 will be clled constrined pths of E. Eqs will be clled constrined equtions ssocited with E nd the constrint Q. In dpted fibered coordintes, Eqs represent system of m k second-order PDE for the components of γ, polynomil in the second derivtives: A s + B i 1 sj 1 z J 1 i 1 + B i 1i 2 sj 1 J 2 z J 1 i 1 z J 2 i 2 + +B i 1...i n sj 1...J n z J 1 i 1 z J n i n J 2 γ =

23 376 O. Krupková / J. Differentil Equtions In fct, due to the following proposition, in equtions 4.13 only m k unknown functions γ 1 x i,...γ m k x i pper. Therefore we shll lso refer to them s to reduced equtions for the constrined system [[ι α]]. Proposition 4.5. A section γ of π stisfies the constrint condition 4.9 i.e., 4.10 if nd only if J 1 γ is n integrl section of the cnonicl distribution C. This mens tht for every, J 1 γ φ = In coordintes, γ m k+ x = g m k+ J 1 γ Proof. Let γ be section of π stisfying 4.9. This mens tht δ = J 1 γ is holonomic section of the fibered mnifold Q X, mening tht δ is n integrl section of the induced contct distribution on Q. However, this distribution is nnihilted by the 1- forms ι ω σ = ω σ. Now, from 3.26 we cn see tht δ is n integrl section of C. The converse is trivil. We cn conclude tht constrined pths cn be loclly obtined by solving the system of simultneous kn first-order PDE 4.15 nd m k second-order PDE Notice tht complete integrbility of the distribution C is not so essentil, since we re looking for integrl sections which re loclly n-dimensionl submnifolds of Q, not for integrl mnifolds of C. In fct, in nlogy with non-holonomic mechnics ordinry differentil equtions one cn expect tht nmely the situtions where C is not completely integrble will be of interest in the theory nd pplictions of PDEs with differentil constrints Constrined Lgrngin systems If the unconstrined equtions re equtions for extremls of first-order Lgrngin λ, i.e., if E = E λ, we hve in the Lepge clss [α] of E λ distinguished representtives, which we cn use for construction of the corresponding constrined system see Proposition In the unconstrined clculus of vritions one usully tkes the form dθ λ see e.g. [10,9], however, in mny situtions the form dρ λ my be more useful [17,2,11], or one cn even utilize generl Lepgen n + 1-form dρ [7,20,24,26]. As we hve seen bove, in the constrined sitution, the constrined Lgrngin system is the equivlence clss [[ι dθ λ ]], nd for study of constrined equtions ny of its representtives is pproprite. Of course, the work with the most simple ones, ᾱ 0 or ᾱ 0λ, or with the most simple closed one, dθ λ, cn be most convenient.

24 O. Krupková / J. Differentil Equtions Definition 4.6. Let [[ι dθ λ ]] be constrined Lgrngin system. Ech of the forms ι dθ λ + φ, where φ IC 0, will be clled constrined Poincré Crtn n + 1- form of λ. Similrly, ech of the forms ι dρ λ + φ, where φ IC 0, will be clled constrined Krupk n + 1-form of λ. Pths of constrined system will be clled constrined extremls, nd Eqs where ᾱ is ny element of the clss [[ι dθ λ ]] will be clled constrined Euler Lgrnge equtions of λ. For λ = Lω 0 denote L = L ι, L = L y m k+ ι, 4.16 where the bove re functions of dpted fibered coordintes, x i,y σ,z J,onQ, nd Θ ι λ = L ω 0 + L y s ω s ω = L ω 0 + L z J z J y s ω s ω In keeping with nottions introduced in Remrk 3.9 we cn esily find the following reltion: Proposition 4.7. Proof. We hve ι Θ λ = Θ ι λ + L i C is ω s ω + L φ ω ι L Θ λ = Lω 0 + ι ω s ω + y s y s L y m k+ ι ω m k+ ω L = Lω 0 + ι + L Ḡ s ω s ω + L φ ω On the other hnd, Θ ι λ = L ω 0 + L z J z J y s y s ω s ω L = L ω 0 + ι + L Ḡ s ω s ω L i C is ωs ω, 4.20

25 378 O. Krupková / J. Differentil Equtions since from ι dl = d L one gets L L g σ z J = yi σ ι i L g r z J = yi r ι i z J + L g m k+ i yi m k+ ι z J L g = ι + L i r i Ḡ r z J L i C Ji y r i Compring 4.19 nd 4.20 we obtin the desired formul. For convenience of nottions let us introduce the C-modified Euler Lgrnge opertor nd cut C-modified Euler Lgrnge opertor, respectively: μ s = c y s d c dx i yi s cg r y s μ s = c y s d c dx i yi s cg r y s y r y r = c y s d c z J dx i yi s = c y s d c z J dx i yi s z J cg r y s z J cg r y s z J y r z J y r z J, z J Theorem 4.8. Let λ be Lgrngin in J 1 Y, Q J 1 Y regulr constrint. Let γ : W Y be section of the fibered mnifold π : Y X such tht J 1 γw Q. In dpted fibered coordintes, the constrined Euler Lgrnge equtions tke one of the following equivlent forms: 1 By mens of L, A s + BsJ i zj i J 2 γ = 0, 4.23 where A s, B i sj re given by 4.8, where cf Ā σ = ε σ L ι, B σν i 2 L = yi σ ι yν 2 By mens of L nd L, μ s L L μ s g m k+ Cs i d c L dx i J 2 γ = 0, 4.25

26 O. Krupková / J. Differentil Equtions mening tht the functions A s, B i sj B i sj re equivlently expressed s follows: A s = μ s L L μ s g m k+ Cs i d c L dx i, L z K = z J z K yi s + L i Ḡ z J s δ i C is Cs i L z J = L z K z J z K yi s + L g m k+ z K z J z K yi s Cs i L z J Proof. The first prt of the theorem is cler. Let us prove the second one. By Proposition 4.7 nd with nottions of Remrk 3.9 we obtin: ι dθ λ dθ ι λ + d L i C is ω s ω L i C is dgs ω 0 + L d φ ω dθ ι λ Cs i d c L dx i ω s ω 0 d L Cs i dx i + Cr i c gi r y s L Cs i z J L Ḡ s i z J + m k+ cg y s d Ḡ s dx Ḡ r c g r y s ω s ω 0 ω s dz J ω i However, μ s g m k+ = cg m k+ y s d c dx i z J y s i g m k+ z J cgi r y s z J y r i g m k+ z J = cg m k+ y s = cg m k+ y s d c Ḡ dx i s δ i Ci s cgi r y s d cḡ s dx + d c Ci s dx i cg r y s Ḡ r + cgi r y s Ḡ r δ i Ci r Ci r, 4.28 nd μ s g m k+ μ s gm k+ = z J g m k+ z K z K y s i = Ḡ z J s δ i Ci s zi J z J i

27 380 O. Krupková / J. Differentil Equtions Substituting into 4.27 we get ι dθ λ dθ ι λ + L z J L μ s gm k+ + Cs i d c L dx i g m k+ z K z K y s i ω s ω 0 Cs i L z J ω s dz J ω i Finlly, expressing dθ ι λ we obtin dθ ι λ c L y s d c L z J dx i z J yi s L z J z J y r L z K z J z K yi s ω s dz J ω i = μ s L ω s ω 0 z J c g r y s ω s ω 0 L z K z K yi s ω s dz J ω i Formuls 4.30 nd 4.31 give us the representtive ᾱ 0 dθ ι λ the components of which determine the corresponding constrined equtions, ᾱ 0 = A s ω s ω 0 + BsJ i ωs dz J ω i = μ s L L μ s gm k+ + Cs i z J L z K z K yi s L z J d c L dx i g m k+ z K ω s ω 0 z K y s i +C i s L z J ω s dz J ω i, 4.32 s desired. Remrk 4.9. Proposition 4.7 nd Theorem 4.8 show tht for generl constrints the n- form Θ ι λ is not Lepgen form for the constrined equtions. This mens tht ι λ hs not the mening of constrined Lgrngin. A proper Lepgen form is, however, ι Θ λ or ι ρ, where ρ is ny Lepgen equivlent of λ, since dι Θ λ = ι dθ λ gives rise to the constrined Euler Lgrnge equtions. In this wy, the role of constrined Lgrngin is plyed by the locl n-form λ C = Lω 0 + L φ ω. 4.33

28 O. Krupková / J. Differentil Equtions Consequently, constrined Lgrngin system typiclly cnnot be loclly determined by single function defined on the constrint, but is determined rther by 1 + nk constrint Lgrnge functions, L, L on Q. Definition The opertor defined by 4.25, i.e., E C s L, L = μ s L L μ s g m k+ will be clled the constrint Euler Lgrnge opertor. Cs i d c L dx i 4.34 We cn define the concept of constrint-horizontl form on Q s form nnihilted by verticl vector fields belonging to the cnonicl distribution C cf. [27]. Then λ C is constrint-horizontl, nd E C is mp cting on constrint-horizontl n-forms on Q, ssigning them clsses of dynmicl forms on Q J 2 Y Q is nturl prolongtion of Q. Indeed, E C λ C is determined up to dynmicl form Φ IC 0 ; in coordintes, 4.3. Chetev equtions E C λ C = E C s L, L ω s ω 0 + Φ φ ω We hve introduced differentil equtions with constrints s geometric obects defined directly on constrint mnifolds. Another but equivlent model for constrined equtions rises from their understnding s deformtions of the originl unconstrined equtions, defined on J 1 Y, in neighborhood of the constrint. We dopt this ide from [22,23] where it hs been proposed for the cse of second nd higher-order ODEs. Let Q J 1 Y be regulr constrint. To point x Q consider n pproprite open set U J 1 Y open in J 1 Y where Q is given by equtions f α = 0, nd the corresponding distribution C U defined on U. Recll tht by 3.7 C U is nnihilted by k linerly independent 1-forms defined on U, = cα f α dx + Mσ ωσ, where Mσ = 1 f α n c α y σ Denote by I U the idel on U generted by Let E be J 1 Y -pertinent dynmicl form on J 2 Y, [α] its Lepge clss. Recll tht E is chrcterized by Proposition 2.7. If Φ I U is dynmicl form, put E Φ = E π 2,1Φ E Φ is J 1 Y -pertinent dynmicl form on π 1 2,1 U, hence hs Lepge clss [α Φ] defined on U. Moreover, we cn esily see tht α 1 α 2 α 1Φ α 2Φ.

29 382 O. Krupková / J. Differentil Equtions Definition We shll cll E Φ deformtion of E induced by the constrint Q. Similrly, the Lepge clss [α Φ ] will be clled deformtion of [α] induced by Q. Equtions for pths of E Φ will be clled deformed equtions. A corresponding dynmicl form Φ will be clled energy-momentum form of the constrint Q. Note tht by definition, Φ = η, where η re horizontl n-forms defined on U; in fibered coordintes, η = h ω 0. With help of 4.36 we write Φ= η =h Mσ ωσ ω 0 = 1 n h cα f α y σ ω σ f α ω 0 =λ α y σ ω σ ω 0, 4.38 nd cll the functions λ α = 1 n h c α 4.39 Lgrnge multipliers. Hence, energy-momentum forms of the constrint Q red Φ = Φ σ ω σ f α ω 0, where Φ σ = λ α y σ, 4.40 nd we cn see tht they re determined by the constrint up to Lgrnge multipliers. Obviously, the concept of energy-momentum form of the constrint does not depend upon choice of locl genertors of the distribution C U. Indeed, if ψ re other independent 1-forms nnihilting C U, it holds ψ = A b b for regulr mtrix A b on U, nd we get Φ = η = A b η ψ b = κ ψ. Remrk The definition of energy-momentum form of the constrint Q gives locl n + 1-form on every pproprite open set U. However, one cn obtin globl form Φ with help of prtition of unity subordinte to cover {U ι } of Q. Moreover, s n immedite consequence of Corollry 3.4 nd Proposition 3.5 it turns our tht ny two energy-momentum forms long the constrint Q coincide up to Lgrnge multipliers. We shll be interested in constrined pths of the deformed equtions, i.e., those pths tht pss in the constrint mnifold J 1 γw Q U. Immeditely from the definitions we get: Proposition The following conditions re equivlent: 1 A section γ : W Y of π is constrined pth of E Φ. 2 For ny α Φ [α Φ ], J 1 γ is n integrl section of the Hmiltonin idel H αφ, nd J 1 γw Q U.

30 O. Krupková / J. Differentil Equtions γ stisfies the following system of second-order PDE: A σ + B i 1 σν 1 y ν 1 i 1 + B i 1 2 i 2 σν 1 ν 2 y ν 1 i 1 y ν 2 2 i 2 + +B i 1... n i n σν 1...ν n y ν 1 i 1...y ν f α n n i n = λ α y σ, 4.41 together with the equtions of the constrint, f α = 0. If, in prticulr E = E λ i.e., the unconstrined equtions re Euler Lgrnge equtions of Lgrngin λ = Lω 0, then the corresponding deformed equtions 4.41 tke the form L y σ d dx L y σ f α = λ α y σ Remrk Eqs were obtined lso in [3]. These equtions generlize to vritionl prtil differentil equtions the Chetev equtions, proposed by Chetev in 1930 to describe motion of mechnicl Lgrngin systems subected to constrints involving time, positions nd velocities of prticles [6] so-clled non-holonomic mechnics. The right-hnd sides of Chetev s equtions in mechnics re interpreted s components of force, clled constrint or Chetev force; it is determined up to Lgrnge multipliers, which hve to be evluted with help of deformed equtions. As we cn see from 4.41, for prtil differentil equtions the mening of Lgrnge multipliers nd deformed equtions is nlogous. Let us clrify the reltion between the deformed nd reduced equtions. Theorem For every U nd every dynmicl form Φ IU, the constrined system ssocited with E Φ coincides with the constrined system ssocited with E, i.e., [[ι α Φ ]]=[[ι α]] For sections γ : W Y of π such tht J 1 γw Q, deformed equtions nd reduced equtions re equivlent. Proof. By definition of [α Φ ], every element of the clss is of the form where F is n t lest 2-contct form on U. Hence α Φ = α Φ + F, 4.44 ι α Φ = ι α + ι Φ + ι F ι α, 4.45

31 384 O. Krupková / J. Differentil Equtions since ι Φ is constrint form on Q, nd ι F is n t lest 2-contct form on Q. This mens tht ι α Φ [[ι α]]. Conversely, for every fixed Φ, ι α ι α ι Φ = ι α Φ, 4.46 i.e., ι α [[ι α Φ ]]. The second prt of Theorem 4.15 is direct consequence of the first prt. 5. Prticulr cses of regulr constrints in J 1 Y In this section we introduce some prticulr cses of constrints, such s Lgrngin constrints, π-dpted constrints, constrints defined by distribution on Y, semiholonomic constrints, nd holonomic constrints Lgrngin constrints Let Q be regulr constrint in J 1 Y, C = nnih{ φ } its cnonicl distribution, I the constrint idel. For n open subset U J 1 Y where Q is given by equtions f α = 0 consider the relted constrint distribution C U = nnih{ } nd the constrint idel I U on U. Definition 5.1. A constrint Q is clled Lgrngin in U if for ll, the forms p 1 d re horizontl with respect to the proection onto Y. Q is clled Lgrngin constrint if it is Lgrngin in n open neighborhood of the submnifold Q. We note tht the definition of Lgrngin constrint does not depend upon choice of forms nnihilting the distribution C U. Indeed, if ψ is nother system of independent 1-forms nnihilting C U, one hs ψ = A b b, where A b is regulr mtrix on U. Hence p 1 dψ = p 1 A b db + p 1 da b b = A b p 1d b + hda b b 5.1 which is π 1,0 -horizontl, since ll re π 1,0 -horizontl, s cn be seen from their definition. Next, note tht the definition of Lgrngin constrint mens tht for ll nd i, the 1-contct prt of d ω i = dc αi f α ω 0 + M σ ωσ ω i is n Euler Lgrnge form. This mens, however, tht the n-forms re Lepgen, nd i = c αi f α ω 0 + M σ ωσ ω i = c αi f α ω 0 + M σ δ i ωσ ω 5.2 re locl Lgrngins for the constrint Q. Λ i = c αi f α ω 0 5.3