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Port-Hamiltonian formulation for Higher-order PDEs
5th IFAC Workshop on Lagrangian and Hamiltonian Methods 5th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Control 5th IFACLinear Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control 5th IFAC Workshop on Lagrangian and Hamiltonian Methods July 4-7,Linear 2015. Lyon, France for Non Control Available online at www.sciencedirect.com July 4-7, 2015. Lyon, France for Non Linear Control July 4-7, 2015. Lyon, France July 4-7, 2015. Lyon, France
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Port-Hamiltonian formulation Port-Hamiltonian formulation Port-Hamiltonian formulation Port-Hamiltonian formulation Higher-order PDEs Higher-order PDEs Higher-order PDEs Higher-order PDEs∗ ∗
for for for for
M. o M. Sch¨ Sch¨ oberl berl ∗∗ K. K. Schlacher Schlacher ∗∗ M. Sch¨ o berl ∗ K. Schlacher ∗ M. Sch¨ o berl K. Schlacher ∗ ∗ Johannes Kepler University Linz, Institute of Automatic Control Johannes Kepler University Linz, Institute of Automatic Control ∗ University Linz, Institute Automatic Systems ∗ Johannes Kepler Control Systems Technology, Austria Johannes KeplerControl University Linz,Technology, Institute of ofAustria Automatic Control Control Control Systems Technology, Austria (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected]) Control Systems Technology, Austria (e-mail: (e-mail: [email protected], [email protected], [email protected]) [email protected])
and and and and
Abstract: Abstract: In In this this paper paper we we consider consider partial partial differential differential equations equations in in aa port-Hamiltonian port-Hamiltonian setting. setting. Abstract: In we consider partial differential in setting. A of class, especially for control is to A crucial crucial property property of this this system class, especially for equations control purposes, purposes, is the the property property to be be Abstract: In this this paper paper wesystem consider partial differential equations in aa port-Hamiltonian port-Hamiltonian setting. A crucial property of this system class, especially for control purposes, is the property to be able to link a power balance relation to the structure of the equations. However, to derive able to link a powerofbalance relation to the structure the equations. to derivetothis this A crucial property this system class, especially forofcontrol purposes,However, is the property be able link balance relation to the equations. However, derive power relation one to into account effects energy flows via the powerto balance relation one has has to take take intostructure account of also the effects of of energy to flows via this the able to balance link a a power power balance relation to the the structure ofalso thethe equations. However, to derive this power balance relation one has to take into account also the effects of energy flows via the boundary. This can be handled in a straightforward manner when the Hamiltonian depends on boundary. This relation can be handled straightforward manner when the Hamiltonian depends on power balance one hasintoa take into account also the effects of energy flows via the boundary. This can be handled in a straightforward manner when the Hamiltonian depends on derivative variables of first order, e.g. by using integration by parts. If higher-order derivatives derivative variables order, by using integration parts. higher-order depends derivatives boundary. This can of be first handled in e.g. a straightforward mannerbywhen theIf Hamiltonian on derivative variables first order, using parts. If appear then by used due appear (higher-order (higher-order field theory) thenby integration by parts partsby cannot be used without withoutderivatives due care, care, derivative variables of offield firsttheory) order, e.g. e.g. byintegration using integration integration bycannot parts. be If higher-order higher-order derivatives appear (higher-order field theory) then integration by parts cannot be used without due thus we suggest an approach by using the so-called Cartan-form. In this paper we concentrate thus we(higher-order suggest an approach by using so-called by Cartan-form. In be thisused paper we concentrate appear field theory) thenthe integration parts cannot without due care, care, thus we approach using Cartan-form. this concentrate on theories in port-Hamiltonian framework and the derivation of on second-order second-order theories in a a by port-Hamiltonian framework and we weIn visualize the we derivation of aa thus we suggest suggest an an approach by using the the so-called so-called Cartan-form. Invisualize this paper paper we concentrate on second-order theories in a port-Hamiltonian framework and we visualize the derivation of power balance relation by using mechanical examples such as plates modeled as a first-order power balance relation mechanical examples suchand as we plates modeled a first-order on second-order theoriesby in using a port-Hamiltonian framework visualize the as derivation of aa power balance relation by using mechanical examples such as plates modeled as a first-order and as second-order field theory. and as balance second-order fieldbytheory. power relation using mechanical examples such as plates modeled as a first-order and as second-order field theory. and as second-order field theory. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Differential Differential geometric geometric methods, methods, Hamiltonian Hamiltonian Systems, Systems, Partial Partial differential differential equations equations Keywords: Differential geometric methods, Hamiltonian Systems, Partial differential Keywords: Differential geometric methods, Hamiltonian Systems, Partial differential equations equations 1. variational 1. INTRODUCTION INTRODUCTION variational calculus calculus whereas whereas in in the the Hamiltonian Hamiltonian picture picture 1. INTRODUCTION variational calculus whereas in the Hamiltonian the focus is on analyzing power flows. the focus is calculus on analyzing power 1. INTRODUCTION variational whereas in flows. the Hamiltonian picture picture the focus power the focus is is on on analyzing analyzing power flows. flows. Our This Our main main tool tool will will be be the the application application of of the the CartanCartanThis paper paper is is focusing focusing on on partial partial differential differential equations equations Our main tool will be the application of the Cartanform approach which is a standard concept in variational This paper is focusing on partial differential equations (PDEs) that allow for a Hamiltonian formulation. It form approach which is a standard concept in variational Our main tool will be the application of the Cartan(PDEs) thatis allow for on a Hamiltonian formulation. It form This paper focusing partial differential equations approach is concept calculus, see Saunders (1989); Giachetta et al. (PDEs) allow Hamiltonian It should noted that weaa have to account calculus, see e.g. e.g.which Saunders (1989); Giachetta et variational al. (1997). (1997). approach which is aa standard standard concept in in variational should be bethat noted thatfor have to take take formulation. account of of the the (PDEs) that allow forwe Hamiltonian formulation. It form calculus, see e.g. Saunders (1989); Giachetta et al. (1997). However, it is well known that the Cartan-form is should be noted that we have to take account of the fact that for PDEs different Hamiltonian formulation exHowever, it is well known that the Cartan-form is not not calculus, see e.g. Saunders (1989); Giachetta et al. (1997). fact thatbefornoted PDEsthat different Hamiltonian ex- However, should we have to take formulation account of the it is well known that the Cartan-form not unique when considering higher-order field theories, is which fact that for PDEs different Hamiltonian formulation exist. For instance, an approach based on a Stokes-Dirac unique when considering higher-order field theories, which However, it is well known that the Cartan-form is not ist. For approach based on formulation a Stokes-Dirac fact thatinstance, for PDEsan different Hamiltonian ex- unique when field which of is connected to in ist. For an on Stokes-Dirac structure can in der Maschke of course course is considering connected higher-order to the the problem problem in performing performing when considering higher-order field theories, theories, which structure can be be found found in van van based der Schaft Schaft and Maschke unique ist. For instance, instance, an approach approach based on a a and Stokes-Dirac of course is connected to the problem in performing repeated integration by parts of higher-order mixed partial structure can be found in van der Schaft and Maschke (2002); Maschke and van der Schaft (2005); Macchelli et al. repeated integration by parts of higher-order mixed partial of course is connected to the problem in performing (2002); Maschke van der Schaft et al. repeated structure can beand found in van der(2005); SchaftMacchelli and Maschke integration by higher-order mixed partial derivatives. This is nature but will (2002); Maschke van (2005); et (2004a,b) and alternative strategy basedMacchelli on jet-bundles jet-bundles derivatives. This problem problem is of ofof intrinsic nature but we we will integration by parts parts ofintrinsic higher-order mixed partial (2004a,b) and an anand alternative strategy based on (2002); Maschke and van der der Schaft Schaft (2005); Macchelli et al. al. repeated derivatives. This problem is of intrinsic nature but we will demonstrate that by choosing a coordinate chart adapted (2004a,b) and an alternative strategy based on jet-bundles has been considered in Ennsbrunner and Schlacher (2005); demonstrate that by choosing a coordinate chart adapted derivatives. This problem is of intrinsic nature but we will has been considered in Ennsbrunner Schlacher (2005); demonstrate (2004a,b) and an alternative strategyand based on jet-bundles that by aa coordinate chart adapted to we able power has been in Ennsbrunner Schlacher Schlacher (2008); and (2012, 2013b,(2005); 2014). demonstrate to the the boundary boundary manifold, we are are able to to derive derive the power thatmanifold, by choosing choosing coordinate chartthe adapted Schlacher (2008); Sch¨ Sch¨ berl and Siuka Siukaand (2012, 2013b, 2014). has been considered considered inooberl Ennsbrunner and Schlacher (2005); to the boundary manifold, we are able to derive the power ports in a unique fashion for second-order field theories. Schlacher (2008); Sch¨ o berl and Siuka (2012, 2013b, 2014). ports in a unique fashion for second-order field theories. to the boundary manifold, we are able to derive the power Schlacher (2008); Sch¨ oberlwill and Siuka (2012, 2013b, 2014). In ap- ports in a unique fashion for second-order field theories. In this this contribution contribution we we will extend extend the the jet-bundle jet-bundle apports in a unique fashion forfollows. second-order field theories. The In this contribution we will extend the jet-bundle approach by explicitly considering second-order field theoThe paper paper is is organized organized as as follows. In In the the second second section section proach explicitly considering second-order field theoIn this by contribution we will extend the jet-bundle ap- The paper is organized as follows. In the second section we introduce the geometric machinery and the notation. proach by explicitly considering second-order field theories, whereas in the cited references above only the firstwe introduce the geometric machinery and the notation. The paper is organized as follows. In the second section ries, whereas in the cited references above onlyfield the theofirst- we proach by explicitly considering second-order introduce the machinery the First-order will in Section 33 and ries, in the references only order case discussed. By the field First-order theory will be be analyzed analyzed in and Section and in in introducetheory the geometric geometric machinery and the notation. notation. orderwhereas case has has been discussed. By above the order order ofthe field we ries, whereas in been the cited cited references above only of theaa firstfirstFirst-order theory will be analyzed in Section 3 and in the fourth section the Mindlin plate example will order has discussed. the of field theory we what order derivative variables the fourth theory section will the be Mindlin plate example3 will be analyzed in Section and be in theorycase we indicate indicate what order of of By derivative variables (jetorder case has been been discussed. By the order order of aa (jetfield First-order the fourth section the Mindlin example be examined. Section is then devotedplate to second-order second-order theory theory we indicate order (jetvariables) the Hamiltonian -- the the order of examined. then devoted to theory fourthSection section55 is the Mindlin plate example will will be variables) enter thewhat Hamiltonian ordervariables of the the PDEs PDEs theory we enter indicate what order of of derivative derivative variables (jet- the examined. Section 5 is then devoted to second-order theory and subsequently the Kirchhoff plate example will be variables) enter the Hamiltonian the order of the PDEs can be higher of course. First-order theory has the signifand subsequently the Kirchhoff plate example will be examined. Section 5 is then devoted to second-order theory can be higher course. First-order theory variables) enterof the Hamiltonian - the orderhas of the the signifPDEs and subsequently the Kirchhoff plate example will be presented. The contribution closes with conclusions and can be higher of course. First-order theory has the significant advantage that based on the Hamiltonian and the presented. The contribution closes with conclusions and and subsequently the Kirchhoff plate example will be icant advantage that based on the Hamiltonian and the can be higher of course. First-order theory has the signif- presented. an icant that Hamiltonian and PDEs aa power can be derived in an outlook. outlook.The The contribution contribution closes closes with with conclusions conclusions and and PDEs advantage power balance balance can on be the derived in aa straightforstraightforicant advantage that based based on the Hamiltonian and the the presented. an outlook. PDEs a power balance can be derived in a straightforward manner, by using integration by parts together with an outlook. ward manner, using integration by parts with PDEs a powerbybalance can be derived in atogether straightfor2. ward manner, by using parts together with Stokes’ theorem. this is longer possible in 2. MATHEMATICAL MATHEMATICAL FRAMEWORK FRAMEWORK Stokes’ theorem. Unfortunately thisby is no no longer possible in ward manner, byUnfortunately using integration integration by parts together with 2. MATHEMATICAL FRAMEWORK Stokes’ theorem. Unfortunately this is no longer possible in the second-order scenario as integration by parts cannot 2. MATHEMATICAL FRAMEWORK the second-order scenario as integration by parts cannot Stokes’ theorem. Unfortunately this is no longer possible in the second-order scenario by parts this paper paper we we will will apply apply differential differential geometric geometric methods methods be without for partial In this be used used without cautiousness cautiousness for repeated repeated mixed partial In the second-order scenario as as integration integration by mixed parts cannot cannot In this paper we will apply differential geometric methods be used without cautiousness for repeated mixed partial and to keep the formulas short and readable we will derivatives. Based on Sch¨ o berl and Schlacher (2015) where andthis to paper keep the formulas and readable we methods will use use derivatives. Basedcautiousness on Sch¨ oberl and Schlachermixed (2015)partial where In we will applyshort differential geometric be used without for repeated and to keep the formulas short and readable we will derivatives. Based on Sch¨ o berl and Schlacher (2015) where tensor notation and especially Einsteins convention on we have analyzed this problem mainly from a Lagrangian tensor and especially Einsteins convention on we have analyzed problem mainly from a(2015) Lagrangian to notation keep the formulas short and readable we will use use derivatives. Based this on Sch¨ oberl and Schlacher where and notation and especially on we have analyzed problem from a sums we indicate the range rangeconvention of the the indices indices point of (the Hamiltonian perspective been sums where where we will will not indicate Einsteins the of point of view view (the this Hamiltonian perspective has been only only tensor tensor notation andnot especially Einsteins convention on we have analyzed this problem mainly mainly from has a Lagrangian Lagrangian we will not the the point of (the perspective been when they the In we touched) we work that is to when where they are are clear from the context. context. In the theof following we touched) we aim aim toHamiltonian work out out aa method method thathas is suitable suitable to sums sums where weclear will from not indicate indicate the range range offollowing the indices indices point of view view (theto Hamiltonian perspective has been only only when they are clear from the context. In the following we touched) we aim to work out a method that is suitable to will introduce the notation and mathematical objects that study second-order field theories in a Hamiltonian setting. will introduce notation andcontext. mathematical objects that study second-order in a Hamiltonian setting. they arethe clear from the In the following we touched) we aim to field worktheories out a method that is suitable to when and objects study second-order field theories in be the sequel we Sch¨ oothat berl The and the Hamiltonian approach can be used used in inthe thenotation sequel where where we closely closely follow follow Sch¨ berl The Lagrangian Lagrangian and the Hamiltonian approach setting. can be be will will introduce introduce the notation and mathematical mathematical objects that study second-order field theories in a a Hamiltonian Hamiltonian setting. will be used in the sequel where we closely follow Sch¨ o The Lagrangian and the Hamiltonian approach can be and Schlacher (2015), as the mathematical framework based on the same mathematical machinery, but of course and be Schlacher (2015), the we mathematical framework basedLagrangian on the sameand mathematical machinery, but ofcan course used in the sequelaswhere closely follow Sch¨ oberl berl The the Hamiltonian approach be will (2015), as mathematical framework based on but in Lagrangian and Hamiltonian setting for the of differs as in the theSchlacher Lagrangian and the the Hamiltonian setting for field field the interpretation interpretation of the the results resultsmachinery, differs significantly, significantly, as and and Schlacher (2015), as the the mathematical framework based on the the same same mathematical mathematical machinery, but of of course course in the Lagrangian and the Hamiltonian setting for the interpretation of the results differs significantly, as theories is similar. in the Lagrangian picture these concepts are used for theories is similar. and the Hamiltonian setting for field in the Lagrangian ofpicture these differs concepts are used for the Lagrangian field the interpretation the results significantly, as in theories is similar. in in the the Lagrangian Lagrangian picture picture these these concepts concepts are are used used for for theories is similar. 244 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 2405-8963 ©© IFAC (International Federation of Automatic Control) 244 Copyright ©2015, 2015 IFAC 244Control. Peer review under responsibility of International Federation of Automatic Copyright © 2015 IFAC 244 Copyright © 2015 IFAC 10.1016/j.ifacol.2015.10.247
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We use ∧ for the exterior product (wedge product), d is the exterior derivative and ⌋ denotes the natural contraction between tensor fields. Furthermore, C ∞ (·) denotes the set of smooth functions on the corresponding manifold. The Lie-derivative of a differential-form ω with respect to a vector field is denoted as Lv ω. The following relation (Cartan’s magic formula) Lv ω = v⌋dω + d(v⌋ω) (1) will be heavily used in the forthcoming sections. We will consider bundle structures in order to be able to separate dependent and independent coordinates. Let us consider the bundle X → D with coordinates (X i , xα ) for X and coordinates (X i ) for D where xα , α = 1, . . . , m are the dependent and X i , i = 1, . . . , r the independent coordinates on the domain D. A section of the bundle X → D is a map s : D → X , i.e. xα = sα (X i ). The k−th order partial derivatives of a section s are ∂k sα = ∂[J] sα = sα [J] 1 j (∂X ) 1 · · · (∂X r )jr where J is an ordered multi index with ji = k = #J. Hence J = j1 . . . jr and it should be noted that an index ji indicates that the derivative with respect to the independent variable X i is performed ji times. Consequently, a section s can be prolonged to the n-th jet which is denoted as j n (s), and the geometric object that contains this information is the n-th order jet-manifold J n (X ) with adapted coordinates (X i , xα [J] ) with #J = . 0, . . . , n where xα = xα [0...0] Example 1. Let us consider the case where r = 2 and m = 1, i.e. we have the coordinates (X 1 , X 2 , x1 ). Consequently, the first jet-manifold J 1 (X ) is equipped with the coordinates (X 1 , X 2 , x1[00] , x1[10] , x1[01] ) , x1 = x1[00] . The second jet-manifold J 2 (X ) possesses the additional coordinates (x1[20] , x1[11] , x1[02] ). A section s is an assignment (X 1 , X 2 ) → (X 1 , X 2 , s1 (X 1 , X 2 )) and the jet-manifold can be seen as a container for the partial derivatives of s and in classical notation e.g. ∂x1 ∂ 2 x1 ∂ 2 x1 1 1 , x = , x = x1[10] = [11] [02] ∂X 1 ∂X 1 ∂X 2 ∂(X 2 )2 1 1 1 2 is met, where the assignment x = s (X , X ) is implicitly assumed. Multi-indices J and I can be added component wise and the special index 1i denotes a multi index with zeros except the i−th entry is one. Hence, J + 1i increases ji by one. We will treat so-called energy densities in the sequel (a quantity that can be integrated), where we pay special attendance to densities of the form H = HΩ with H ∈ C ∞ (J n (X )) where Ω denotes the volume element on the manifold D, i.e. Ω = dX 1 ∧ . . . ∧ dX r with dim(D) = r. Additionally, we denote by H(s) = D (j n s)∗ (H) the integrated quantity, with the n-th jet-prolongation of the map x = s(X). Remark 1. (Notation) ∗ To simplify notation we will denote n ∗ (j s) (H) as s (H) whenever the order of the jet D D prolongation of s follows form the context. Furthermore, it should be noted that with H we denote a density (differential-form), with H the function part of the density, 245
245
and with H the integrated quantity that depends on a section s. The total derivative d[1i ] with respect to the independent variable X i reads as ∂ ∂ [J] d[1i ] = ∂i + xα (2) , ∂α[J] = [J+1i ] ∂α , ∂i = ∂X i ∂xα [J] and (d[1i ] f ) ◦ j n+1 (s) = ∂i (f ◦ j n (s)) is met by construction for all f ∈ C ∞ (J n (X )). The dual elements to the total derivatives are the so-called contact forms. In local coordinates we have α α i ω[J] = dxα (3) [J] − x[J+1i ] dX , #J ≥ 0. Example 2. For the same configuration as in example 1 the contact form including first-order jet-variables is 1 = dx1[00] − x1[10] dX 1 − x1[01] dX 2 ω[00] and for second-order jet-variables we have 1 = dx1[10] − x1[20] dX 1 − x1[11] dX 2 ω[10] 1 ω[01] = dx1[01] − x1[11] dX 1 − x1[02] dX 2 .
Furthermore, we introduce the horizontal projection hor α i as hor(dX i ) = dX i , hor(dxα [J] ) = x[J+1i ] dX which has α the property that hor(ω[J] ) = 0. Finally, we need so-called vertical vector fields and their jet-prolongation. A vertical vector field v ∈ V(X ) on the bundle X → D is of the form v = v α (X, x)∂α , hence it possesses only components into the fiber direction. Remark 2. In contrast to v = v α (X, x)∂α which is a vertical vector field on X , a vector field on X (a section of T (X )) is of the form w = wi (X, x)∂i + wα (X, x)∂α with ∂i = ∂/∂X i and ∂α = ∂/∂xα . Hence, vertical vector fields meet X˙ i = wi = 0. The n-th jet-prolongation of an element v ∈ V(X ) is given as j n (v) = v + d[J] (v α )∂α[J] , d[J] = (d[11 ] )j1 . . . (d[1r ] )jr (4) where 1 ≤ #J ≤ n, see e.g. Olver (1986); Saunders (1989). 3. FIRST-ORDER HAMILTONIAN DENSITY Let us consider a Hamiltonian of first-order defined on X → D. Hence, we have H = HΩ with H ∈ C ∞ (J 1 (X )). Depending on a chosen section s we are led to compute H(s) = D s∗ (H). Let us consider now an evolutionary vector-field v = v α ∂α , corresponding to the PDEs x˙ α = v α , v α ∈ C ∞ (J 2 (X )) (5) and we are interested in ∂t H(s) where s is a formal solution of the PDEs parameterized in t, i.e. ∂t s(X) = v(X, ∂[J] s(X)) , #J ≤ 2 where we have suppressed the explicit dependence of s with respect to the evolution parameter. We compute ˙ H = ∂t H(s) = s∗ (Lj 1 (v) H) (6) D
and using the explicit formula (4) for the first jetprolongation of v, we obtain Lj 1 (v) H = (v α ∂α H + (d[1i ] v α )∂α[1i ] H)Ω.
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Integration by parts of (6) gives s∗ v α (∂α − d[1i ] ∂α[1i ] )HΩ + d[1i ] (v α ∂α[1i ] H)Ω D
i with ω α = dxα − xα [1 ] dX such that i s∗ (H) = s∗ (C). D
D
and finally with Stokes’ theorem we have the decomposition ∂t H(s) = s∗ (v α (δα H)Ω) + s∗ (v α (∂α[1i ] H)Ωi ) (7)
Obviously hor(C) = H is met and it remains to choose the [1 ] functions ρα j properly, in order to be able to separate the domain and the boundary impact.
where Ωi = ∂i ⌋Ω, see also Schlacher (2008); Sch¨oberl and Siuka (2013b), where we have used the variational [1 ] derivative δα = ∂α − d[1i ] ∂α i . Thus, we have obtained a power balance relation when H = HΩ corresponds to an energy density. Obviously, in this power balance relation we have a contribution in the domain and one on/over the boundary. We can restate (7) to obtain H˙ = s∗ (v⌋δH) + s∗ (v⌋δ ∂ H) (8)
Hence, we consider the relation Lj 1 (v) C = j 1 (v)⌋dC + d(j 1 (v)⌋C)
D
∂D
D
∂D
with δH = (δα H)dxα ∧Ω and δ ∂ H = (∂αi H)dxα ∧Ωi . Hence choosing v = (J − R)(δH) + u⌋G (9) where J is a skew- symmetric map and R is positive semidefinite, we can express the impact on the domain as s∗ (−R(δH)⌋δH + y⌋u) = D s∗ (−(δα H)Rαβ (δβ H) + yξ uξ )Ω D
with y = G ∗ (δH) and the input map G together with its adjoint G ∗ . The boundary impact (depending on the boundary conditions of the PDEs) is determined by the expression v⌋δ ∂ H. Corollary 3. Given evolutionary PDEs of the form (5) with the generalized vector field v as in (9), the formal change of the first-order Hamiltonian H along solutions of (5) can be decomposed in a dissipative term in the domain −R(δH)⌋δH, in a power port in the domain y⌋u and in a power port on the boundary v⌋δ ∂ H. In the following we wish to derive this decomposition (7) without the explicit use of integration by parts. Based on this alternative strategy we are able to generalize the concept to second-order theories. 3.1 The Cartan-form approach There is also a different possibility to derive this decomposition which does not make explicit use of integration by parts. Using (1) we can rewrite the Lie-derivative Lj 1 (v) H as it appears in (6) in the form Lj 1 (v) H = j 1 (v)⌋dH + d(j 1 (v)⌋H). Hence, we have a natural decomposition into a domain and a boundary term, as by Stokes’ theorem d(j 1 (v)⌋H) contributes to the boundary. Unfortunately we cannot pick the domain impact directly as j 1 (v)⌋dH may depend on derivatives of the field v that induces further terms on the boundary. To solve this problem we try to reformulate the problem such that no prolongation of v is necessary - this can be performed by using the contact forms (3). Due to α the condition s∗ (ω[J] ) = 0 , #J = 0 we can modify the integrand using the Cartan-form C with j] α C = HΩ + ρ[1 (10) α ω ∧ Ωj 246
(11)
and instead of (6) we compute s∗ (Lj 1 (v) C). D
Using (11), j 1 (v)⌋dC will deliver the domain impact and from j 1 (v)⌋C the boundary impact can be extracted, if we [1 ] choose the functions ρα i such that hor(j 1 (v)⌋dC) = hor(v⌋dC) is met. In the latter expression the horizontal projection has to be used, since we have to take into account that the pullback with s cancels out contact parts. Hence, we compute hor(j 1 (v)⌋dC) and with (10) we obtain α [1i ] α i] α −ρ[1 (∂α Hv α +∂α[1i ] Hv[1 α v[1i ] )Ω−hor(dρα )∧v Ωi (12) i]
where we have used the fact that hor(ω α ) = 0. This expression (12) is independent of the jet-prolongation of α the vector field v[1 iff i] [1i ] i] ρ[1 α = ∂α H.
(13)
Using [1i ] [1i ] i] hor(dρ[1 α ) ∧ Ωi = d[1i ] (ρα )Ω = d[1i ] (∂α H)Ω
and (13) we can restate (12) as α i] hor(j 1 (v)⌋dC) = v α (∂α H−d[1i ] ρ[1 α )Ω = v (δα H)Ω. (14) From (14) we see that we have recovered the domain impact on D (compare with the first integrand in (7)). The boundary impact on ∂D follows from i] j 1 (v)⌋C = v α ρ[1 α Ωi and using (13) we have
(15)
j 1 (v)⌋C = v α (∂α[1i ] H)Ωi (16) corresponding to the second term in (7). Hence, we see that for first-order theories we can easily derive the boundary ports based on the knowledge of H by computing the [1 ] derivatives with respect to the jet variables, i.e. ∂α i H [1i ] with ∂α = ∂/∂xα [1i ] . This will be demonstrated by the following example. 4. EXAMPLE - THE MINDLIN PLATE The example of the Mindlin plate has been analyzed using a Stokes-Dirac Structure in Macchelli et al. (2005) and using variational calculus in Sch¨oberl and Siuka (2013a). We again investigate this example in the setting described in the previous chapter, mainly to demonstrate the derivation of the boundary ports and then also for the purpose of comparison with the Kirchhoff plate example. Let us consider a manifold X with the coordinates (X 1 , X 2 , x1 , . . . , x6 ). The spatial (independent) variables are X 1 and X 2 , whereas x1 , x2 and x3 are the plate
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deflection and the two rotation angles of a cross section. Additionally, x4 , x5 and x6 are the corresponding momenta, where we use the identification x4 = p1 , x5 = p2 and x6 = p3 . The Hamiltonian density reads as� H = HΩ, � H = T + V with 2T = (p1 )2 + α (p2 )2 + (p3 )2 and 1−ν 2 (x[01] + x3[10] )2 2 +(x2[10] )2 + 2νx3[01] x2[10] + (x3[01] )2
2V = (x1[10] − x2 )2 + (x1[01] − x3 )2 +
It should be noted that at each part of the boundary three ports appear - a pairing including the shear force and two pairings corresponding to bending/twisting moments. The collocated quantities are the velocity and the angular velocities. Remark 4. It can be easily verified that e.g. [10]
[10]
[10]
QX = ∂1 H , MX = −∂2 H , MXY = −∂3 H is met.
and corresponds to the energy in the system. For simplicity we set all the plate parameters to one apart from the Poisson’s ratio ν and a constant parameter α ∈ R+ . The variational derivatives that will be used in this setting read as [10] [01] δ1 = −d[10] ∂1 − d[01] ∂1 [10] [01] δ2 = ∂2 − d[10] ∂2 − d[01] ∂2 . (17) [10] [01] δ3 = ∂3 − d[10] ∂3 − d[01] ∂3 δi = ∂i , i = 4, 5, 6 Introducing the shearing forces QX = (x1[10] − x2 ) , QY = (x1[01] − x3 ) and the bending moments MX = −(x2[10] + νx3[01] ) , MY = −(x3[01] + νx2[10] ), 2 3 as well as MXY = − 1−ν 2 (x[01] + x[10] ), we can formulate the evolutionary PDEs as x˙ 1 = p1 , x˙ 2 = αp2 , x˙ 3 = αp3 and
p˙ 1 = −d[10] QX − d[01] QY p˙ 2 = −QX + d[10] MX + d[01] MXY
247
(18)
p˙ 3 = −QY + d[01] MY + d[10] MXY . Thus, the PDEs have the form x˙ = v = J (δH) (19) where J is the canonical skew-symmetric mapping such that 1 x˙ δ1 H 0 0 0 1 0 0 x˙ 2 0 0 0 0 1 0 δ2 H 3 x˙ 0 0 0 0 0 1 δ3 H = −1 0 0 0 0 0 δ H 4 p˙ 1 p˙ 0 −1 0 0 0 0 δ5 H 2 0 0 −1 0 0 0 δ6 H p˙ 3 with the variational derivatives δi , i = 1 . . . 6 as given in ∂ ∂ (17) where δi = ∂i = ∂x for i = 4 . . . 6. i = ∂p i−3 Hence, due to (19) and using (8) the domain impact vanishes. The boundary impact follows from the second � term in (8), namely ∂D s∗ (v⌋δ ∂ H). 4.1 The boundary impact
[1 ]
Consequently, since v⌋δ ∂ H = v α (∂α i H)Ωi , see (16), we [1 ] have to compute ∂α i H using the Hamiltonian H = T + V and since no jet variables concerning the momenta pi appear we derive the power balance as � � 1 � x˙ QX − x˙ 2 MX − x˙ 3 MXY dX 2 H˙ ∂D = ∂D � � 1 � x˙ QY − x˙ 2 MXY − x˙ 3 MY dX 1 . − ∂D
247
5. SECOND-ORDER HAMILTONIAN DENSITY In this section we will generalize the Cartan-form solution to the case of a second-order Hamiltonian. To derive the domain and the boundary impact with respect to a power balance relation we will proceed similarly as in the firstorder case. However, it will turn out that in the secondorder case the derivation of the boundary impact is much more involved, as it is no longer possible to obtain the boundary ports from a given Hamiltonian H without additional assumptions. 5.1 The domain impact We now consider a second-order Hamiltonian density with H = C ∞ (J 2 (X )) and we additively modify it with contact forms of the type α α i ω[I−1 = dxα [I−1i ] − x[I] dX , 1 ≤ #I ≤ 2 i] such that C = HΩ +
�
α ρ[αI−1i |I] ω[I−1 ∧ Ωi i]
is met. Remark 5. It should be noted there is no unique assignment of a contact form with respect to a jet-variable xα [I] (in contrast to the first-order case). Following Ennsbrunner (2006), we use a double multiindex for ρα in order to indicate which contact form corresponds to the jet-variable xα [I] . Furthermore, we take the sum over all possible contact forms. Similarly as in the first-order case, we compute � α dC = dH ∧ Ω + dρ[αI−1i |I] ∧ ω[I−1 ∧ Ωi i] � α + ρ[αI−1i |I] ∧ dω[I−1 ∧ Ωi i]
(using a second-order jet prolongation of v) and consequently we derive the following expression for hor(j 2 (v)⌋dC) � � � α [ I−1i |I] α α ∂α[0] Hv[0] + ∂α[I] Hv[I] − ρα v[I] Ω � α ) ∧ Ωi −hor( d(ρ[αI−1i |I] )v[I−1 i]
with the zero multi-index 0 = 0 . . . 0 and 1 ≤ #I ≤ 2. Remark 6. It should be noted that this latter expression simplifies to (12) in the first-order case. In order to cancel out the jet-prolongation of the vector field v, we have to choose �
ρ[αI−1i |I] = ∂α[I] H ,
#I = 2
(20)
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and for #I = 1, I = 1l we have to assign [ 0|1l ] [ 1l |1l +1i ] ρα = ∂α[1l ] H − d[1i ] (ρα ).
6. EXAMPLE - KIRCHHOFF PLATE (21)
2
Hence, hor(j (v)⌋dC) simplifies to α hor(j 2 (v)⌋dC) = ∂α[0] H − d[1i ] (ρ[α0|1i ] ) v[0] Ω
where we have used the fact that all expressions v[I] for #I �= 0 are canceled out. Finally, we obtain α hor(j 2 (v)⌋dC) = (δα H)v[0] Ω
with δα H =
2
(−1)#I d[I] ∂α[I] H
2V = (x1[20] )2 + (x1[02] )2 + 2νx1[20] x1[02] + 2(1 − ν)(x1[11] )2 (22)
#I=0
where we have set the physical parameters to one apart from the Poisson’s ratio ν. The PDEs 1read as δ1 H 0 1 0 1 x˙ = , J = −1 0 −1 0 δ1H p˙ 1
where we have used [ 1i |1k +1i ] ) −d[1i ] (ρ[α0|1i ] ) = −d[1i ] ∂α[1i ] H + d[1i ] d[1k ] (ρα together with (20) and (21).
with
From this calculation we see that, although the choice for pα is not unique (see (20)) we again derive the domain impact of the form 2 s∗ (v⌋δH) , δH = ( (−1)#I d[I] ∂α[I] H)dxα ∧ Ω D
Let us consider the case of a (rectangular) Kirchhoff plate with the two independent variables X 1 = X and X 2 = Y , such that we have two spatial variables and the time serves as an evolution parameter. Hence, X is equipped with the coordinates (X 1 , X 2 , x1 , p1 ) where x1 again denotes the deflection and x2 = p1 the linear momentum. The Hamiltonian is given as H = T + V with T = 21 (p1 )2 and
#I=0
with the only difference that now δ is a second order operator. Thus, we see again that if we choose v as in (9) (with a second-order Hamiltonian density) the same reasoning according to the dissipation and the domain port is appropriate. The boundary impact needs a more careful analysis - this is done in the forthcoming section. 5.2 The boundary impact We will exploit the degree of freedom in choosing ρ in such a way that we obtain the minimal number of boundary ports. To reduce the different possibilities for ρ (see again (20)) we parameterize the boundary of the manifold D which is denoted as ∂D as X r = const. which is always possible (at least locally) by a change of the independent coordinates. The boundary volume form in adapted coordinates can be introduced as Ωr = ∂r ⌋Ω. Hence, the boundary conditions are derived by means of [ 0|1r ] α α j 1 (v)⌋C = ρα Ωr v[0] + ρ[α1k |1k +1r ] v[1 (23) k]
(with a summation over k), which is the generalization of (15) to the second order case. In order to guarantee a minimal number of ports, we set
ρ[α1k |1k +1r ] = 0 , 1k �= 1r or equivalently in the relations (20) we have to choose [ 1 |1 +1 ] ρα r k r . This is always possible as for the multi-index I = 1k + 1r , #I = 2, 1k �= 1r there are always two possible [ I−1 |I] [ 1 |1 +1 ] choices for the functions ρα i namely ρα k k r or [ 1r |1k +1r ] . ρα Remark 7. It should be noted that the products of expressions with ρ and with v (and derivatives of v) are candidates for possible power ports. In a Lagrangian scenario the vector field v corresponds to a variation, and one can argue that using adapted coordinates only v[0] and v[0...01] = v[1r ] are independent, see Sch¨ oberl and Schlacher (2015). 248
δ1 =
∂ [1 ] [J] , δ1 = ∂1 − d[1i ] ∂1 i + d[J] ∂1 , #J = 2 ∂p1
which is x˙ 1 = p1 , p˙ 1 = −(x1[40] + x1[04] + 2x1[22] ). Remark 8. The PDEs can be derived using a Lagrangian L = T − V using the Euler-Lagrange operator, see e.g. Sch¨oberl and Schlacher (2015). The Hamiltonian includes second-order jet variables, and the application of the variational derivative implies then forth-order PDEs in our evolutionary setting. Since we use a skew-symmetric map J we can conclude that H˙ D in H˙ = H˙ D + H˙ ∂D is equal to zero and that only a boundary impact has to be considered. Remark 9. From the potential energy V we see, that we have one term involving mixed partial derivatives, namely (1 − ν)(x1[11] )2 . Calculation of the Lie-derivative with respect to an evolutionary field v results in 1 2(1 − ν)x1[11] v[11] dX 1 ∧ dX 2 . Now we have two possibilities to apply the integration 1 by parts regarding v[11] . Both possibilities lead to the same conditions on the domain (PDEs). In the following paragraph we will demonstrate by using the Cartan-form approach, that to obtain the minimal number of boundary (corresponding the mechanical meaningful ones) ports on the boundary X 2 = Y = const. we have to consider the integration by parts with respect to the variable X 1 first. 6.1 The boundary impact To obtain the boundary impact on the boundary X 2 = Y = const. we need to consider the relation (23) which in our example with 1r = [01] leads to [ 00|01] 1 v[00]
(ρ1
[ 10|11] 1 v[10]
+ ρ1
[ 01|02] 1 v[01] )Ωr
+ ρ1
(24)
since r = 2. The non-zero components for ρ1 of jet order #I = 2 are [01,02]
ρ1
[10,11] ρ1
[02]
[10,20]
= ∂1 H , ρ1 [01,11] ρ1
[11] ∂1 H
[20]
= ∂1 H
(25)
+ = where the remarkable as well as observation is that for the mixed spatial derivative we get two possible contact forms. For #I = 1 we obtain
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[00,01]
= ∂1 H − d[01] ρ1
[00,10]
= ∂1 H − d[10] ρ1
ρ1 ρ1
[01]
[01,02]
− d[10] ρ1
[01,11]
[10]
[10,20]
− d[01] ρ1
[10,11]
(26) .
1 is not admissible on the boundary of interest we Since v[10] [ 10|11]
can choose ρ1 follows.
[01,11]
= 0 and consequently ρ1
[11]
= ∂1 H
Hence, the boundary impact follows from (24) [ 01|02] 1 [ 00|01] 1 v[01] )Ωr , r = 2 v[00] + ρ1 (ρ1 ∂D
[ 10|11]
as we have set ρ1
= 0. [ 10|11]
1 Remark 10. The expression ρ1 v[10] on the boundary 2 X = Y = const. would be obtained by integration by parts with respect to X 2 , see Remark 9. [ 00|01]
We need to compute the coefficients ρ1 and using (25) and (26) we obtain [01,02]
ρ1
[ 01|02]
and ρ1
[02]
= ∂1 H = x1[02] + νx1[20]
as well as [01,11] [00,01] [01,02] ρ1 = −(x1[03] +(2−ν)x1[21] ), −d[10] ρ1 = −d[01] ρ1 (27) where we have used [01,11] [11] = ∂1 H = 2(1 − ν)x1[11] . ρ1 Finally, on the boundary X 2 = const. we have [ 00|01] [ 01|02] 1 1 ˙ ρ1 + v[01] ρ1 )dX 1 H∂D = − (v[00] = x˙ 1 FS + x˙ 1[01] MB dX 1
with the shear force FS = (x1[03] + (2 − ν)x1[21] ) appearing in the power port x˙ 1 FS and the bending moment MB = −(x1[02] + νx1[20] ) in x˙ 1[01] MB . The collocated quantities are 1 1 x˙ 1 = v[00] (angular velocity). (velocity) and x˙ 1[01] = v[01] In contrast to the Mindlin plate only two ports appear (the minimal number). From the equation (24) it can be deduced that a wrong choice for ρ1 would give again three ports, which is wrong from a mechanical point of view. Hence, it is vitally important to perform the integration by parts properly, which in our language amounts for a correct selection of the coefficients of the Cartan-form in accordance with the parameterization of the boundary. 7. CONCLUSION We have provided a strategy to compute the domain impact and the boundary ports for second-order field theories in a port-Hamiltonian fashion suitable for the computer algebra implementation. The main difficulty in the second-order case is the fact that the derivation of the boundary ports is not straightforward, as for mixed partial derivatives integration by parts cannot be uniquely performed. This is also reflected in the non uniqueness of the Cartan-form. However, taking into account the parameterization of the boundary manifold it is possible to uniquely determine the minimal number of ports. In future investigations we shall be focusing on generalizing the results of Sch¨ oberl and Siuka (2013b) regarding control concepts from the first-order to the second-order case. 249
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REFERENCES Ennsbrunner, H. (2006). Infinite-dimensional EulerLagrange and Port Hamiltonian Systems. University Linz, PhD-Thesis. Ennsbrunner, H. and Schlacher, K. (2005). On the geometrical representation and interconnection of infinite dimensional port controlled hamiltonian systems. In Proceedings IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 5263– 5268. Giachetta, G., Mangiarotti, L., and Sardanashvily, G. (1997). New Lagrangian and Hamiltonian Methods in Field Theory. World Scientific, Singapore. Macchelli, A., Melchiorri, C., and Bassi, L. (2005). Port based modelling and control of the mindlin plate. In Proceedings 44th IEEE Conference on Decision and Control (CDC), 5989–5994. Macchelli, A., van der Schaft, A., and Melchiorri, C. (2004a). Port hamiltonian formulation of infinite dimensional systems i. modeling. In Proceedings 43rd IEEE Conference on Decision and Control (CDC), 3762–3767. Macchelli, A., van der Schaft, A., and Melchiorri, C. (2004b). Port hamiltonian formulation of infinite dimensional systems ii. boundary control by interconnection. In Proceedings 43rd IEEE Conference on Decision and Control (CDC), 3768–3773. Maschke, B. and van der Schaft, A. (2005). Compositional Modelling of Distributed-Parameter Systems, 115–154. Advanced Topics in Control Systems Theory. Springer Lecture Notes in Control and Information Sciences, London. Olver, P. (1986). Applications of Lie Groups to Differential Equations. Springer-Verlag, New York. Saunders, D. (1989). The Geometry of Jet Bundles. Cambridge University Press, Cambridge. Schlacher, K. (2008). Mathematical modeling for nonlinear control: a hamiltonian approach. Mathematics and Computers in simulation, 97, 829–849. Sch¨oberl, M. and Schlacher, K. (2015). Lagrangian and port-hamiltonian formulation for distributed-parameter systems. In Proceedings, 8th Mathmod Conference, Vienna, Austria, accepted. Sch¨oberl, M. and Siuka, A. (2012). On the porthamiltonian representation of systems described by partial differential equations. In Proceedings 4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control, 1–6. Sch¨oberl, M. and Siuka, A. (2013a). Analysis and comparison of port-hamiltonian formulations for field theories - demonstrated by means of the mindlin plate. In Proceedings 12th European Control Conference (ECC), 548–553. Sch¨oberl, M. and Siuka, A. (2013b). On casimir functionals for infinite-dimensional port-hamiltonian control systems. IEEE Transactions on Automatic Control, 58, 1823–1828. Sch¨oberl, M. and Siuka, A. (2014). Jet bundle formulation of infinite-dimensional port-hamiltonian systems using differential operators. Automatica, 50, 607–613. van der Schaft, A. and Maschke, B. (2002). Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics, 42, 166–194.
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Port-Hamiltonian formulation Port-Hamiltonian formulation Port-Hamiltonian formulation Port-Hamiltonian formulation Higher-order PDEs Higher-order PDEs Higher-order PDEs Higher-order PDEs∗ ∗
for for for for
M. o M. Sch¨ Sch¨ oberl berl ∗∗ K. K. Schlacher Schlacher ∗∗ M. Sch¨ o berl ∗ K. Schlacher ∗ M. Sch¨ o berl K. Schlacher ∗ ∗ Johannes Kepler University Linz, Institute of Automatic Control Johannes Kepler University Linz, Institute of Automatic Control ∗ University Linz, Institute Automatic Systems ∗ Johannes Kepler Control Systems Technology, Austria Johannes KeplerControl University Linz,Technology, Institute of ofAustria Automatic Control Control Control Systems Technology, Austria (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected]) Control Systems Technology, Austria (e-mail: (e-mail: [email protected], [email protected], [email protected]) [email protected])
and and and and
Abstract: Abstract: In In this this paper paper we we consider consider partial partial differential differential equations equations in in aa port-Hamiltonian port-Hamiltonian setting. setting. Abstract: In we consider partial differential in setting. A of class, especially for control is to A crucial crucial property property of this this system class, especially for equations control purposes, purposes, is the the property property to be be Abstract: In this this paper paper wesystem consider partial differential equations in aa port-Hamiltonian port-Hamiltonian setting. A crucial property of this system class, especially for control purposes, is the property to be able to link a power balance relation to the structure of the equations. However, to derive able to link a powerofbalance relation to the structure the equations. to derivetothis this A crucial property this system class, especially forofcontrol purposes,However, is the property be able link balance relation to the equations. However, derive power relation one to into account effects energy flows via the powerto balance relation one has has to take take intostructure account of also the effects of of energy to flows via this the able to balance link a a power power balance relation to the the structure ofalso thethe equations. However, to derive this power balance relation one has to take into account also the effects of energy flows via the boundary. This can be handled in a straightforward manner when the Hamiltonian depends on boundary. This relation can be handled straightforward manner when the Hamiltonian depends on power balance one hasintoa take into account also the effects of energy flows via the boundary. This can be handled in a straightforward manner when the Hamiltonian depends on derivative variables of first order, e.g. by using integration by parts. If higher-order derivatives derivative variables order, by using integration parts. higher-order depends derivatives boundary. This can of be first handled in e.g. a straightforward mannerbywhen theIf Hamiltonian on derivative variables first order, using parts. If appear then by used due appear (higher-order (higher-order field theory) thenby integration by parts partsby cannot be used without withoutderivatives due care, care, derivative variables of offield firsttheory) order, e.g. e.g. byintegration using integration integration bycannot parts. be If higher-order higher-order derivatives appear (higher-order field theory) then integration by parts cannot be used without due thus we suggest an approach by using the so-called Cartan-form. In this paper we concentrate thus we(higher-order suggest an approach by using so-called by Cartan-form. In be thisused paper we concentrate appear field theory) thenthe integration parts cannot without due care, care, thus we approach using Cartan-form. this concentrate on theories in port-Hamiltonian framework and the derivation of on second-order second-order theories in a a by port-Hamiltonian framework and we weIn visualize the we derivation of aa thus we suggest suggest an an approach by using the the so-called so-called Cartan-form. Invisualize this paper paper we concentrate on second-order theories in a port-Hamiltonian framework and we visualize the derivation of power balance relation by using mechanical examples such as plates modeled as a first-order power balance relation mechanical examples suchand as we plates modeled a first-order on second-order theoriesby in using a port-Hamiltonian framework visualize the as derivation of aa power balance relation by using mechanical examples such as plates modeled as a first-order and as second-order field theory. and as balance second-order fieldbytheory. power relation using mechanical examples such as plates modeled as a first-order and as second-order field theory. and as second-order field theory. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Differential Differential geometric geometric methods, methods, Hamiltonian Hamiltonian Systems, Systems, Partial Partial differential differential equations equations Keywords: Differential geometric methods, Hamiltonian Systems, Partial differential Keywords: Differential geometric methods, Hamiltonian Systems, Partial differential equations equations 1. variational 1. INTRODUCTION INTRODUCTION variational calculus calculus whereas whereas in in the the Hamiltonian Hamiltonian picture picture 1. INTRODUCTION variational calculus whereas in the Hamiltonian the focus is on analyzing power flows. the focus is calculus on analyzing power 1. INTRODUCTION variational whereas in flows. the Hamiltonian picture picture the focus power the focus is is on on analyzing analyzing power flows. flows. Our This Our main main tool tool will will be be the the application application of of the the CartanCartanThis paper paper is is focusing focusing on on partial partial differential differential equations equations Our main tool will be the application of the Cartanform approach which is a standard concept in variational This paper is focusing on partial differential equations (PDEs) that allow for a Hamiltonian formulation. It form approach which is a standard concept in variational Our main tool will be the application of the Cartan(PDEs) thatis allow for on a Hamiltonian formulation. It form This paper focusing partial differential equations approach is concept calculus, see Saunders (1989); Giachetta et al. (PDEs) allow Hamiltonian It should noted that weaa have to account calculus, see e.g. e.g.which Saunders (1989); Giachetta et variational al. (1997). (1997). approach which is aa standard standard concept in in variational should be bethat noted thatfor have to take take formulation. account of of the the (PDEs) that allow forwe Hamiltonian formulation. It form calculus, see e.g. Saunders (1989); Giachetta et al. (1997). However, it is well known that the Cartan-form is should be noted that we have to take account of the fact that for PDEs different Hamiltonian formulation exHowever, it is well known that the Cartan-form is not not calculus, see e.g. Saunders (1989); Giachetta et al. (1997). fact thatbefornoted PDEsthat different Hamiltonian ex- However, should we have to take formulation account of the it is well known that the Cartan-form not unique when considering higher-order field theories, is which fact that for PDEs different Hamiltonian formulation exist. For instance, an approach based on a Stokes-Dirac unique when considering higher-order field theories, which However, it is well known that the Cartan-form is not ist. For approach based on formulation a Stokes-Dirac fact thatinstance, for PDEsan different Hamiltonian ex- unique when field which of is connected to in ist. For an on Stokes-Dirac structure can in der Maschke of course course is considering connected higher-order to the the problem problem in performing performing when considering higher-order field theories, theories, which structure can be be found found in van van based der Schaft Schaft and Maschke unique ist. For instance, instance, an approach approach based on a a and Stokes-Dirac of course is connected to the problem in performing repeated integration by parts of higher-order mixed partial structure can be found in van der Schaft and Maschke (2002); Maschke and van der Schaft (2005); Macchelli et al. repeated integration by parts of higher-order mixed partial of course is connected to the problem in performing (2002); Maschke van der Schaft et al. repeated structure can beand found in van der(2005); SchaftMacchelli and Maschke integration by higher-order mixed partial derivatives. This is nature but will (2002); Maschke van (2005); et (2004a,b) and alternative strategy basedMacchelli on jet-bundles jet-bundles derivatives. This problem problem is of ofof intrinsic nature but we we will integration by parts parts ofintrinsic higher-order mixed partial (2004a,b) and an anand alternative strategy based on (2002); Maschke and van der der Schaft Schaft (2005); Macchelli et al. al. repeated derivatives. This problem is of intrinsic nature but we will demonstrate that by choosing a coordinate chart adapted (2004a,b) and an alternative strategy based on jet-bundles has been considered in Ennsbrunner and Schlacher (2005); demonstrate that by choosing a coordinate chart adapted derivatives. This problem is of intrinsic nature but we will has been considered in Ennsbrunner Schlacher (2005); demonstrate (2004a,b) and an alternative strategyand based on jet-bundles that by aa coordinate chart adapted to we able power has been in Ennsbrunner Schlacher Schlacher (2008); and (2012, 2013b,(2005); 2014). demonstrate to the the boundary boundary manifold, we are are able to to derive derive the power thatmanifold, by choosing choosing coordinate chartthe adapted Schlacher (2008); Sch¨ Sch¨ berl and Siuka Siukaand (2012, 2013b, 2014). has been considered considered inooberl Ennsbrunner and Schlacher (2005); to the boundary manifold, we are able to derive the power ports in a unique fashion for second-order field theories. Schlacher (2008); Sch¨ o berl and Siuka (2012, 2013b, 2014). ports in a unique fashion for second-order field theories. to the boundary manifold, we are able to derive the power Schlacher (2008); Sch¨ oberlwill and Siuka (2012, 2013b, 2014). In ap- ports in a unique fashion for second-order field theories. In this this contribution contribution we we will extend extend the the jet-bundle jet-bundle apports in a unique fashion forfollows. second-order field theories. The In this contribution we will extend the jet-bundle approach by explicitly considering second-order field theoThe paper paper is is organized organized as as follows. In In the the second second section section proach explicitly considering second-order field theoIn this by contribution we will extend the jet-bundle ap- The paper is organized as follows. In the second section we introduce the geometric machinery and the notation. proach by explicitly considering second-order field theories, whereas in the cited references above only the firstwe introduce the geometric machinery and the notation. The paper is organized as follows. In the second section ries, whereas in the cited references above onlyfield the theofirst- we proach by explicitly considering second-order introduce the machinery the First-order will in Section 33 and ries, in the references only order case discussed. By the field First-order theory will be be analyzed analyzed in and Section and in in introducetheory the geometric geometric machinery and the notation. notation. orderwhereas case has has been discussed. By above the order order ofthe field we ries, whereas in been the cited cited references above only of theaa firstfirstFirst-order theory will be analyzed in Section 3 and in the fourth section the Mindlin plate example will order has discussed. the of field theory we what order derivative variables the fourth theory section will the be Mindlin plate example3 will be analyzed in Section and be in theorycase we indicate indicate what order of of By derivative variables (jetorder case has been been discussed. By the order order of aa (jetfield First-order the fourth section the Mindlin example be examined. Section is then devotedplate to second-order second-order theory theory we indicate order (jetvariables) the Hamiltonian -- the the order of examined. then devoted to theory fourthSection section55 is the Mindlin plate example will will be variables) enter thewhat Hamiltonian ordervariables of the the PDEs PDEs theory we enter indicate what order of of derivative derivative variables (jet- the examined. Section 5 is then devoted to second-order theory and subsequently the Kirchhoff plate example will be variables) enter the Hamiltonian the order of the PDEs can be higher of course. First-order theory has the signifand subsequently the Kirchhoff plate example will be examined. Section 5 is then devoted to second-order theory can be higher course. First-order theory variables) enterof the Hamiltonian - the orderhas of the the signifPDEs and subsequently the Kirchhoff plate example will be presented. The contribution closes with conclusions and can be higher of course. First-order theory has the significant advantage that based on the Hamiltonian and the presented. The contribution closes with conclusions and and subsequently the Kirchhoff plate example will be icant advantage that based on the Hamiltonian and the can be higher of course. First-order theory has the signif- presented. an icant that Hamiltonian and PDEs aa power can be derived in an outlook. outlook.The The contribution contribution closes closes with with conclusions conclusions and and PDEs advantage power balance balance can on be the derived in aa straightforstraightforicant advantage that based based on the Hamiltonian and the the presented. an outlook. PDEs a power balance can be derived in a straightforward manner, by using integration by parts together with an outlook. ward manner, using integration by parts with PDEs a powerbybalance can be derived in atogether straightfor2. ward manner, by using parts together with Stokes’ theorem. this is longer possible in 2. MATHEMATICAL MATHEMATICAL FRAMEWORK FRAMEWORK Stokes’ theorem. Unfortunately thisby is no no longer possible in ward manner, byUnfortunately using integration integration by parts together with 2. MATHEMATICAL FRAMEWORK Stokes’ theorem. Unfortunately this is no longer possible in the second-order scenario as integration by parts cannot 2. MATHEMATICAL FRAMEWORK the second-order scenario as integration by parts cannot Stokes’ theorem. Unfortunately this is no longer possible in the second-order scenario by parts this paper paper we we will will apply apply differential differential geometric geometric methods methods be without for partial In this be used used without cautiousness cautiousness for repeated repeated mixed partial In the second-order scenario as as integration integration by mixed parts cannot cannot In this paper we will apply differential geometric methods be used without cautiousness for repeated mixed partial and to keep the formulas short and readable we will derivatives. Based on Sch¨ o berl and Schlacher (2015) where andthis to paper keep the formulas and readable we methods will use use derivatives. Basedcautiousness on Sch¨ oberl and Schlachermixed (2015)partial where In we will applyshort differential geometric be used without for repeated and to keep the formulas short and readable we will derivatives. Based on Sch¨ o berl and Schlacher (2015) where tensor notation and especially Einsteins convention on we have analyzed this problem mainly from a Lagrangian tensor and especially Einsteins convention on we have analyzed problem mainly from a(2015) Lagrangian to notation keep the formulas short and readable we will use use derivatives. Based this on Sch¨ oberl and Schlacher where and notation and especially on we have analyzed problem from a sums we indicate the range rangeconvention of the the indices indices point of (the Hamiltonian perspective been sums where where we will will not indicate Einsteins the of point of view view (the this Hamiltonian perspective has been only only tensor tensor notation andnot especially Einsteins convention on we have analyzed this problem mainly mainly from has a Lagrangian Lagrangian we will not the the point of (the perspective been when they the In we touched) we work that is to when where they are are clear from the context. context. In the theof following we touched) we aim aim toHamiltonian work out out aa method method thathas is suitable suitable to sums sums where weclear will from not indicate indicate the range range offollowing the indices indices point of view view (theto Hamiltonian perspective has been only only when they are clear from the context. In the following we touched) we aim to work out a method that is suitable to will introduce the notation and mathematical objects that study second-order field theories in a Hamiltonian setting. will introduce notation andcontext. mathematical objects that study second-order in a Hamiltonian setting. they arethe clear from the In the following we touched) we aim to field worktheories out a method that is suitable to when and objects study second-order field theories in be the sequel we Sch¨ oothat berl The and the Hamiltonian approach can be used used in inthe thenotation sequel where where we closely closely follow follow Sch¨ berl The Lagrangian Lagrangian and the Hamiltonian approach setting. can be be will will introduce introduce the notation and mathematical mathematical objects that study second-order field theories in a a Hamiltonian Hamiltonian setting. will be used in the sequel where we closely follow Sch¨ o The Lagrangian and the Hamiltonian approach can be and Schlacher (2015), as the mathematical framework based on the same mathematical machinery, but of course and be Schlacher (2015), the we mathematical framework basedLagrangian on the sameand mathematical machinery, but ofcan course used in the sequelaswhere closely follow Sch¨ oberl berl The the Hamiltonian approach be will (2015), as mathematical framework based on but in Lagrangian and Hamiltonian setting for the of differs as in the theSchlacher Lagrangian and the the Hamiltonian setting for field field the interpretation interpretation of the the results resultsmachinery, differs significantly, significantly, as and and Schlacher (2015), as the the mathematical framework based on the the same same mathematical mathematical machinery, but of of course course in the Lagrangian and the Hamiltonian setting for the interpretation of the results differs significantly, as theories is similar. in the Lagrangian picture these concepts are used for theories is similar. and the Hamiltonian setting for field in the Lagrangian ofpicture these differs concepts are used for the Lagrangian field the interpretation the results significantly, as in theories is similar. in in the the Lagrangian Lagrangian picture picture these these concepts concepts are are used used for for theories is similar. 244 Hosting by Elsevier Ltd. 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We use ∧ for the exterior product (wedge product), d is the exterior derivative and ⌋ denotes the natural contraction between tensor fields. Furthermore, C ∞ (·) denotes the set of smooth functions on the corresponding manifold. The Lie-derivative of a differential-form ω with respect to a vector field is denoted as Lv ω. The following relation (Cartan’s magic formula) Lv ω = v⌋dω + d(v⌋ω) (1) will be heavily used in the forthcoming sections. We will consider bundle structures in order to be able to separate dependent and independent coordinates. Let us consider the bundle X → D with coordinates (X i , xα ) for X and coordinates (X i ) for D where xα , α = 1, . . . , m are the dependent and X i , i = 1, . . . , r the independent coordinates on the domain D. A section of the bundle X → D is a map s : D → X , i.e. xα = sα (X i ). The k−th order partial derivatives of a section s are ∂k sα = ∂[J] sα = sα [J] 1 j (∂X ) 1 · · · (∂X r )jr where J is an ordered multi index with ji = k = #J. Hence J = j1 . . . jr and it should be noted that an index ji indicates that the derivative with respect to the independent variable X i is performed ji times. Consequently, a section s can be prolonged to the n-th jet which is denoted as j n (s), and the geometric object that contains this information is the n-th order jet-manifold J n (X ) with adapted coordinates (X i , xα [J] ) with #J = . 0, . . . , n where xα = xα [0...0] Example 1. Let us consider the case where r = 2 and m = 1, i.e. we have the coordinates (X 1 , X 2 , x1 ). Consequently, the first jet-manifold J 1 (X ) is equipped with the coordinates (X 1 , X 2 , x1[00] , x1[10] , x1[01] ) , x1 = x1[00] . The second jet-manifold J 2 (X ) possesses the additional coordinates (x1[20] , x1[11] , x1[02] ). A section s is an assignment (X 1 , X 2 ) → (X 1 , X 2 , s1 (X 1 , X 2 )) and the jet-manifold can be seen as a container for the partial derivatives of s and in classical notation e.g. ∂x1 ∂ 2 x1 ∂ 2 x1 1 1 , x = , x = x1[10] = [11] [02] ∂X 1 ∂X 1 ∂X 2 ∂(X 2 )2 1 1 1 2 is met, where the assignment x = s (X , X ) is implicitly assumed. Multi-indices J and I can be added component wise and the special index 1i denotes a multi index with zeros except the i−th entry is one. Hence, J + 1i increases ji by one. We will treat so-called energy densities in the sequel (a quantity that can be integrated), where we pay special attendance to densities of the form H = HΩ with H ∈ C ∞ (J n (X )) where Ω denotes the volume element on the manifold D, i.e. Ω = dX 1 ∧ . . . ∧ dX r with dim(D) = r. Additionally, we denote by H(s) = D (j n s)∗ (H) the integrated quantity, with the n-th jet-prolongation of the map x = s(X). Remark 1. (Notation) ∗ To simplify notation we will denote n ∗ (j s) (H) as s (H) whenever the order of the jet D D prolongation of s follows form the context. Furthermore, it should be noted that with H we denote a density (differential-form), with H the function part of the density, 245
245
and with H the integrated quantity that depends on a section s. The total derivative d[1i ] with respect to the independent variable X i reads as ∂ ∂ [J] d[1i ] = ∂i + xα (2) , ∂α[J] = [J+1i ] ∂α , ∂i = ∂X i ∂xα [J] and (d[1i ] f ) ◦ j n+1 (s) = ∂i (f ◦ j n (s)) is met by construction for all f ∈ C ∞ (J n (X )). The dual elements to the total derivatives are the so-called contact forms. In local coordinates we have α α i ω[J] = dxα (3) [J] − x[J+1i ] dX , #J ≥ 0. Example 2. For the same configuration as in example 1 the contact form including first-order jet-variables is 1 = dx1[00] − x1[10] dX 1 − x1[01] dX 2 ω[00] and for second-order jet-variables we have 1 = dx1[10] − x1[20] dX 1 − x1[11] dX 2 ω[10] 1 ω[01] = dx1[01] − x1[11] dX 1 − x1[02] dX 2 .
Furthermore, we introduce the horizontal projection hor α i as hor(dX i ) = dX i , hor(dxα [J] ) = x[J+1i ] dX which has α the property that hor(ω[J] ) = 0. Finally, we need so-called vertical vector fields and their jet-prolongation. A vertical vector field v ∈ V(X ) on the bundle X → D is of the form v = v α (X, x)∂α , hence it possesses only components into the fiber direction. Remark 2. In contrast to v = v α (X, x)∂α which is a vertical vector field on X , a vector field on X (a section of T (X )) is of the form w = wi (X, x)∂i + wα (X, x)∂α with ∂i = ∂/∂X i and ∂α = ∂/∂xα . Hence, vertical vector fields meet X˙ i = wi = 0. The n-th jet-prolongation of an element v ∈ V(X ) is given as j n (v) = v + d[J] (v α )∂α[J] , d[J] = (d[11 ] )j1 . . . (d[1r ] )jr (4) where 1 ≤ #J ≤ n, see e.g. Olver (1986); Saunders (1989). 3. FIRST-ORDER HAMILTONIAN DENSITY Let us consider a Hamiltonian of first-order defined on X → D. Hence, we have H = HΩ with H ∈ C ∞ (J 1 (X )). Depending on a chosen section s we are led to compute H(s) = D s∗ (H). Let us consider now an evolutionary vector-field v = v α ∂α , corresponding to the PDEs x˙ α = v α , v α ∈ C ∞ (J 2 (X )) (5) and we are interested in ∂t H(s) where s is a formal solution of the PDEs parameterized in t, i.e. ∂t s(X) = v(X, ∂[J] s(X)) , #J ≤ 2 where we have suppressed the explicit dependence of s with respect to the evolution parameter. We compute ˙ H = ∂t H(s) = s∗ (Lj 1 (v) H) (6) D
and using the explicit formula (4) for the first jetprolongation of v, we obtain Lj 1 (v) H = (v α ∂α H + (d[1i ] v α )∂α[1i ] H)Ω.
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Integration by parts of (6) gives s∗ v α (∂α − d[1i ] ∂α[1i ] )HΩ + d[1i ] (v α ∂α[1i ] H)Ω D
i with ω α = dxα − xα [1 ] dX such that i s∗ (H) = s∗ (C). D
D
and finally with Stokes’ theorem we have the decomposition ∂t H(s) = s∗ (v α (δα H)Ω) + s∗ (v α (∂α[1i ] H)Ωi ) (7)
Obviously hor(C) = H is met and it remains to choose the [1 ] functions ρα j properly, in order to be able to separate the domain and the boundary impact.
where Ωi = ∂i ⌋Ω, see also Schlacher (2008); Sch¨oberl and Siuka (2013b), where we have used the variational [1 ] derivative δα = ∂α − d[1i ] ∂α i . Thus, we have obtained a power balance relation when H = HΩ corresponds to an energy density. Obviously, in this power balance relation we have a contribution in the domain and one on/over the boundary. We can restate (7) to obtain H˙ = s∗ (v⌋δH) + s∗ (v⌋δ ∂ H) (8)
Hence, we consider the relation Lj 1 (v) C = j 1 (v)⌋dC + d(j 1 (v)⌋C)
D
∂D
D
∂D
with δH = (δα H)dxα ∧Ω and δ ∂ H = (∂αi H)dxα ∧Ωi . Hence choosing v = (J − R)(δH) + u⌋G (9) where J is a skew- symmetric map and R is positive semidefinite, we can express the impact on the domain as s∗ (−R(δH)⌋δH + y⌋u) = D s∗ (−(δα H)Rαβ (δβ H) + yξ uξ )Ω D
with y = G ∗ (δH) and the input map G together with its adjoint G ∗ . The boundary impact (depending on the boundary conditions of the PDEs) is determined by the expression v⌋δ ∂ H. Corollary 3. Given evolutionary PDEs of the form (5) with the generalized vector field v as in (9), the formal change of the first-order Hamiltonian H along solutions of (5) can be decomposed in a dissipative term in the domain −R(δH)⌋δH, in a power port in the domain y⌋u and in a power port on the boundary v⌋δ ∂ H. In the following we wish to derive this decomposition (7) without the explicit use of integration by parts. Based on this alternative strategy we are able to generalize the concept to second-order theories. 3.1 The Cartan-form approach There is also a different possibility to derive this decomposition which does not make explicit use of integration by parts. Using (1) we can rewrite the Lie-derivative Lj 1 (v) H as it appears in (6) in the form Lj 1 (v) H = j 1 (v)⌋dH + d(j 1 (v)⌋H). Hence, we have a natural decomposition into a domain and a boundary term, as by Stokes’ theorem d(j 1 (v)⌋H) contributes to the boundary. Unfortunately we cannot pick the domain impact directly as j 1 (v)⌋dH may depend on derivatives of the field v that induces further terms on the boundary. To solve this problem we try to reformulate the problem such that no prolongation of v is necessary - this can be performed by using the contact forms (3). Due to α the condition s∗ (ω[J] ) = 0 , #J = 0 we can modify the integrand using the Cartan-form C with j] α C = HΩ + ρ[1 (10) α ω ∧ Ωj 246
(11)
and instead of (6) we compute s∗ (Lj 1 (v) C). D
Using (11), j 1 (v)⌋dC will deliver the domain impact and from j 1 (v)⌋C the boundary impact can be extracted, if we [1 ] choose the functions ρα i such that hor(j 1 (v)⌋dC) = hor(v⌋dC) is met. In the latter expression the horizontal projection has to be used, since we have to take into account that the pullback with s cancels out contact parts. Hence, we compute hor(j 1 (v)⌋dC) and with (10) we obtain α [1i ] α i] α −ρ[1 (∂α Hv α +∂α[1i ] Hv[1 α v[1i ] )Ω−hor(dρα )∧v Ωi (12) i]
where we have used the fact that hor(ω α ) = 0. This expression (12) is independent of the jet-prolongation of α the vector field v[1 iff i] [1i ] i] ρ[1 α = ∂α H.
(13)
Using [1i ] [1i ] i] hor(dρ[1 α ) ∧ Ωi = d[1i ] (ρα )Ω = d[1i ] (∂α H)Ω
and (13) we can restate (12) as α i] hor(j 1 (v)⌋dC) = v α (∂α H−d[1i ] ρ[1 α )Ω = v (δα H)Ω. (14) From (14) we see that we have recovered the domain impact on D (compare with the first integrand in (7)). The boundary impact on ∂D follows from i] j 1 (v)⌋C = v α ρ[1 α Ωi and using (13) we have
(15)
j 1 (v)⌋C = v α (∂α[1i ] H)Ωi (16) corresponding to the second term in (7). Hence, we see that for first-order theories we can easily derive the boundary ports based on the knowledge of H by computing the [1 ] derivatives with respect to the jet variables, i.e. ∂α i H [1i ] with ∂α = ∂/∂xα [1i ] . This will be demonstrated by the following example. 4. EXAMPLE - THE MINDLIN PLATE The example of the Mindlin plate has been analyzed using a Stokes-Dirac Structure in Macchelli et al. (2005) and using variational calculus in Sch¨oberl and Siuka (2013a). We again investigate this example in the setting described in the previous chapter, mainly to demonstrate the derivation of the boundary ports and then also for the purpose of comparison with the Kirchhoff plate example. Let us consider a manifold X with the coordinates (X 1 , X 2 , x1 , . . . , x6 ). The spatial (independent) variables are X 1 and X 2 , whereas x1 , x2 and x3 are the plate
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deflection and the two rotation angles of a cross section. Additionally, x4 , x5 and x6 are the corresponding momenta, where we use the identification x4 = p1 , x5 = p2 and x6 = p3 . The Hamiltonian density reads as� H = HΩ, � H = T + V with 2T = (p1 )2 + α (p2 )2 + (p3 )2 and 1−ν 2 (x[01] + x3[10] )2 2 +(x2[10] )2 + 2νx3[01] x2[10] + (x3[01] )2
2V = (x1[10] − x2 )2 + (x1[01] − x3 )2 +
It should be noted that at each part of the boundary three ports appear - a pairing including the shear force and two pairings corresponding to bending/twisting moments. The collocated quantities are the velocity and the angular velocities. Remark 4. It can be easily verified that e.g. [10]
[10]
[10]
QX = ∂1 H , MX = −∂2 H , MXY = −∂3 H is met.
and corresponds to the energy in the system. For simplicity we set all the plate parameters to one apart from the Poisson’s ratio ν and a constant parameter α ∈ R+ . The variational derivatives that will be used in this setting read as [10] [01] δ1 = −d[10] ∂1 − d[01] ∂1 [10] [01] δ2 = ∂2 − d[10] ∂2 − d[01] ∂2 . (17) [10] [01] δ3 = ∂3 − d[10] ∂3 − d[01] ∂3 δi = ∂i , i = 4, 5, 6 Introducing the shearing forces QX = (x1[10] − x2 ) , QY = (x1[01] − x3 ) and the bending moments MX = −(x2[10] + νx3[01] ) , MY = −(x3[01] + νx2[10] ), 2 3 as well as MXY = − 1−ν 2 (x[01] + x[10] ), we can formulate the evolutionary PDEs as x˙ 1 = p1 , x˙ 2 = αp2 , x˙ 3 = αp3 and
p˙ 1 = −d[10] QX − d[01] QY p˙ 2 = −QX + d[10] MX + d[01] MXY
247
(18)
p˙ 3 = −QY + d[01] MY + d[10] MXY . Thus, the PDEs have the form x˙ = v = J (δH) (19) where J is the canonical skew-symmetric mapping such that 1 x˙ δ1 H 0 0 0 1 0 0 x˙ 2 0 0 0 0 1 0 δ2 H 3 x˙ 0 0 0 0 0 1 δ3 H = −1 0 0 0 0 0 δ H 4 p˙ 1 p˙ 0 −1 0 0 0 0 δ5 H 2 0 0 −1 0 0 0 δ6 H p˙ 3 with the variational derivatives δi , i = 1 . . . 6 as given in ∂ ∂ (17) where δi = ∂i = ∂x for i = 4 . . . 6. i = ∂p i−3 Hence, due to (19) and using (8) the domain impact vanishes. The boundary impact follows from the second � term in (8), namely ∂D s∗ (v⌋δ ∂ H). 4.1 The boundary impact
[1 ]
Consequently, since v⌋δ ∂ H = v α (∂α i H)Ωi , see (16), we [1 ] have to compute ∂α i H using the Hamiltonian H = T + V and since no jet variables concerning the momenta pi appear we derive the power balance as � � 1 � x˙ QX − x˙ 2 MX − x˙ 3 MXY dX 2 H˙ ∂D = ∂D � � 1 � x˙ QY − x˙ 2 MXY − x˙ 3 MY dX 1 . − ∂D
247
5. SECOND-ORDER HAMILTONIAN DENSITY In this section we will generalize the Cartan-form solution to the case of a second-order Hamiltonian. To derive the domain and the boundary impact with respect to a power balance relation we will proceed similarly as in the firstorder case. However, it will turn out that in the secondorder case the derivation of the boundary impact is much more involved, as it is no longer possible to obtain the boundary ports from a given Hamiltonian H without additional assumptions. 5.1 The domain impact We now consider a second-order Hamiltonian density with H = C ∞ (J 2 (X )) and we additively modify it with contact forms of the type α α i ω[I−1 = dxα [I−1i ] − x[I] dX , 1 ≤ #I ≤ 2 i] such that C = HΩ +
�
α ρ[αI−1i |I] ω[I−1 ∧ Ωi i]
is met. Remark 5. It should be noted there is no unique assignment of a contact form with respect to a jet-variable xα [I] (in contrast to the first-order case). Following Ennsbrunner (2006), we use a double multiindex for ρα in order to indicate which contact form corresponds to the jet-variable xα [I] . Furthermore, we take the sum over all possible contact forms. Similarly as in the first-order case, we compute � α dC = dH ∧ Ω + dρ[αI−1i |I] ∧ ω[I−1 ∧ Ωi i] � α + ρ[αI−1i |I] ∧ dω[I−1 ∧ Ωi i]
(using a second-order jet prolongation of v) and consequently we derive the following expression for hor(j 2 (v)⌋dC) � � � α [ I−1i |I] α α ∂α[0] Hv[0] + ∂α[I] Hv[I] − ρα v[I] Ω � α ) ∧ Ωi −hor( d(ρ[αI−1i |I] )v[I−1 i]
with the zero multi-index 0 = 0 . . . 0 and 1 ≤ #I ≤ 2. Remark 6. It should be noted that this latter expression simplifies to (12) in the first-order case. In order to cancel out the jet-prolongation of the vector field v, we have to choose �
ρ[αI−1i |I] = ∂α[I] H ,
#I = 2
(20)
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and for #I = 1, I = 1l we have to assign [ 0|1l ] [ 1l |1l +1i ] ρα = ∂α[1l ] H − d[1i ] (ρα ).
6. EXAMPLE - KIRCHHOFF PLATE (21)
2
Hence, hor(j (v)⌋dC) simplifies to α hor(j 2 (v)⌋dC) = ∂α[0] H − d[1i ] (ρ[α0|1i ] ) v[0] Ω
where we have used the fact that all expressions v[I] for #I �= 0 are canceled out. Finally, we obtain α hor(j 2 (v)⌋dC) = (δα H)v[0] Ω
with δα H =
2
(−1)#I d[I] ∂α[I] H
2V = (x1[20] )2 + (x1[02] )2 + 2νx1[20] x1[02] + 2(1 − ν)(x1[11] )2 (22)
#I=0
where we have set the physical parameters to one apart from the Poisson’s ratio ν. The PDEs 1read as δ1 H 0 1 0 1 x˙ = , J = −1 0 −1 0 δ1H p˙ 1
where we have used [ 1i |1k +1i ] ) −d[1i ] (ρ[α0|1i ] ) = −d[1i ] ∂α[1i ] H + d[1i ] d[1k ] (ρα together with (20) and (21).
with
From this calculation we see that, although the choice for pα is not unique (see (20)) we again derive the domain impact of the form 2 s∗ (v⌋δH) , δH = ( (−1)#I d[I] ∂α[I] H)dxα ∧ Ω D
Let us consider the case of a (rectangular) Kirchhoff plate with the two independent variables X 1 = X and X 2 = Y , such that we have two spatial variables and the time serves as an evolution parameter. Hence, X is equipped with the coordinates (X 1 , X 2 , x1 , p1 ) where x1 again denotes the deflection and x2 = p1 the linear momentum. The Hamiltonian is given as H = T + V with T = 21 (p1 )2 and
#I=0
with the only difference that now δ is a second order operator. Thus, we see again that if we choose v as in (9) (with a second-order Hamiltonian density) the same reasoning according to the dissipation and the domain port is appropriate. The boundary impact needs a more careful analysis - this is done in the forthcoming section. 5.2 The boundary impact We will exploit the degree of freedom in choosing ρ in such a way that we obtain the minimal number of boundary ports. To reduce the different possibilities for ρ (see again (20)) we parameterize the boundary of the manifold D which is denoted as ∂D as X r = const. which is always possible (at least locally) by a change of the independent coordinates. The boundary volume form in adapted coordinates can be introduced as Ωr = ∂r ⌋Ω. Hence, the boundary conditions are derived by means of [ 0|1r ] α α j 1 (v)⌋C = ρα Ωr v[0] + ρ[α1k |1k +1r ] v[1 (23) k]
(with a summation over k), which is the generalization of (15) to the second order case. In order to guarantee a minimal number of ports, we set
ρ[α1k |1k +1r ] = 0 , 1k �= 1r or equivalently in the relations (20) we have to choose [ 1 |1 +1 ] ρα r k r . This is always possible as for the multi-index I = 1k + 1r , #I = 2, 1k �= 1r there are always two possible [ I−1 |I] [ 1 |1 +1 ] choices for the functions ρα i namely ρα k k r or [ 1r |1k +1r ] . ρα Remark 7. It should be noted that the products of expressions with ρ and with v (and derivatives of v) are candidates for possible power ports. In a Lagrangian scenario the vector field v corresponds to a variation, and one can argue that using adapted coordinates only v[0] and v[0...01] = v[1r ] are independent, see Sch¨ oberl and Schlacher (2015). 248
δ1 =
∂ [1 ] [J] , δ1 = ∂1 − d[1i ] ∂1 i + d[J] ∂1 , #J = 2 ∂p1
which is x˙ 1 = p1 , p˙ 1 = −(x1[40] + x1[04] + 2x1[22] ). Remark 8. The PDEs can be derived using a Lagrangian L = T − V using the Euler-Lagrange operator, see e.g. Sch¨oberl and Schlacher (2015). The Hamiltonian includes second-order jet variables, and the application of the variational derivative implies then forth-order PDEs in our evolutionary setting. Since we use a skew-symmetric map J we can conclude that H˙ D in H˙ = H˙ D + H˙ ∂D is equal to zero and that only a boundary impact has to be considered. Remark 9. From the potential energy V we see, that we have one term involving mixed partial derivatives, namely (1 − ν)(x1[11] )2 . Calculation of the Lie-derivative with respect to an evolutionary field v results in 1 2(1 − ν)x1[11] v[11] dX 1 ∧ dX 2 . Now we have two possibilities to apply the integration 1 by parts regarding v[11] . Both possibilities lead to the same conditions on the domain (PDEs). In the following paragraph we will demonstrate by using the Cartan-form approach, that to obtain the minimal number of boundary (corresponding the mechanical meaningful ones) ports on the boundary X 2 = Y = const. we have to consider the integration by parts with respect to the variable X 1 first. 6.1 The boundary impact To obtain the boundary impact on the boundary X 2 = Y = const. we need to consider the relation (23) which in our example with 1r = [01] leads to [ 00|01] 1 v[00]
(ρ1
[ 10|11] 1 v[10]
+ ρ1
[ 01|02] 1 v[01] )Ωr
+ ρ1
(24)
since r = 2. The non-zero components for ρ1 of jet order #I = 2 are [01,02]
ρ1
[10,11] ρ1
[02]
[10,20]
= ∂1 H , ρ1 [01,11] ρ1
[11] ∂1 H
[20]
= ∂1 H
(25)
+ = where the remarkable as well as observation is that for the mixed spatial derivative we get two possible contact forms. For #I = 1 we obtain
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[00,01]
= ∂1 H − d[01] ρ1
[00,10]
= ∂1 H − d[10] ρ1
ρ1 ρ1
[01]
[01,02]
− d[10] ρ1
[01,11]
[10]
[10,20]
− d[01] ρ1
[10,11]
(26) .
1 is not admissible on the boundary of interest we Since v[10] [ 10|11]
can choose ρ1 follows.
[01,11]
= 0 and consequently ρ1
[11]
= ∂1 H
Hence, the boundary impact follows from (24) [ 01|02] 1 [ 00|01] 1 v[01] )Ωr , r = 2 v[00] + ρ1 (ρ1 ∂D
[ 10|11]
as we have set ρ1
= 0. [ 10|11]
1 Remark 10. The expression ρ1 v[10] on the boundary 2 X = Y = const. would be obtained by integration by parts with respect to X 2 , see Remark 9. [ 00|01]
We need to compute the coefficients ρ1 and using (25) and (26) we obtain [01,02]
ρ1
[ 01|02]
and ρ1
[02]
= ∂1 H = x1[02] + νx1[20]
as well as [01,11] [00,01] [01,02] ρ1 = −(x1[03] +(2−ν)x1[21] ), −d[10] ρ1 = −d[01] ρ1 (27) where we have used [01,11] [11] = ∂1 H = 2(1 − ν)x1[11] . ρ1 Finally, on the boundary X 2 = const. we have [ 00|01] [ 01|02] 1 1 ˙ ρ1 + v[01] ρ1 )dX 1 H∂D = − (v[00] = x˙ 1 FS + x˙ 1[01] MB dX 1
with the shear force FS = (x1[03] + (2 − ν)x1[21] ) appearing in the power port x˙ 1 FS and the bending moment MB = −(x1[02] + νx1[20] ) in x˙ 1[01] MB . The collocated quantities are 1 1 x˙ 1 = v[00] (angular velocity). (velocity) and x˙ 1[01] = v[01] In contrast to the Mindlin plate only two ports appear (the minimal number). From the equation (24) it can be deduced that a wrong choice for ρ1 would give again three ports, which is wrong from a mechanical point of view. Hence, it is vitally important to perform the integration by parts properly, which in our language amounts for a correct selection of the coefficients of the Cartan-form in accordance with the parameterization of the boundary. 7. CONCLUSION We have provided a strategy to compute the domain impact and the boundary ports for second-order field theories in a port-Hamiltonian fashion suitable for the computer algebra implementation. The main difficulty in the second-order case is the fact that the derivation of the boundary ports is not straightforward, as for mixed partial derivatives integration by parts cannot be uniquely performed. This is also reflected in the non uniqueness of the Cartan-form. However, taking into account the parameterization of the boundary manifold it is possible to uniquely determine the minimal number of ports. In future investigations we shall be focusing on generalizing the results of Sch¨ oberl and Siuka (2013b) regarding control concepts from the first-order to the second-order case. 249
249
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