Bibbona, Enrico

Fatibene, Lorenzo

Francaviglia, Mauro

English

Fatibene, L.

Francaviglia, M.

PORTO Publications Open Repository TOrino

Gauge-natural PVPs, Superpotentials and Trivial Lagrangians

BIBBONA, ENRICO

L. FATIBENE

M. FRANCAVIGLIA

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

Institutional Research Information System University of Turin

This is an author version of the contribution published on:E. Bibbona, L. Fatibene, and M Francaviglia.Gauge-natural PVPs, superpotentials and trivial Lagrangians..XV International Workshop on Geometry and Physics,Publ. R. Soc. Mat. Esp., 11, p. 231–236, Madrid, 2007.Proceedings of the XV International Workshop onGeometry and PhysicsPuerto de la Cruz, Tenerife, Canary Islands, SpainSeptember 11–16, 2006Publ. de la RSME, Vol. 11 (2007), 229–234Gauge-natural PVPs, Superpotentials andTrivial LagrangiansEnrico Bibbona1, Lorenzo Fatibene2 and MauroFrancaviglia31Dipartimento di Matematica, Universita` degli Studi di Torino, Italy2INFN, Sezione di Torino, Iniziativa Specifica Na12, Italy3ESG, Universita` della Calabria, ItalyAbstractWe define the superpotentials for gauge-natural Parametrized Variational Prob-lems. In view of a definition of augmented PVPs, we show that a pure divergenceLagrangian has trivial equations also in the case of PVPs but, as it happens in thefree case, it has a non-vanishing superpotential. We compute it.Keywords: Parametrized Variational Problem.2000 Mathematics Subject Classification: 70S05, 58A20.1. IntroductionIn [1] we defined and studied the so-called gauge-natural Parametrized VariationalProblems (PVPs) extending the natural case studied by Garc´ıa and coworkers in [2].They are Variational Problems where variations of the fields are not taken arbitrarily,but along given parametrizations.In General Relativity different notions of energy were given (see [5] for a review).Two of the authors (L.F. and M.F.) contributed to the debate in particular witha variational pathway to this definition. The major source of difficulty is that thenaive definition of conserved quantities as integrals of the superpotential suffers thefollowing ambiguity: adding a divergence to the Lagrangian nothing should change inthe Dynamics, nevertheless the conserved quantities are changed by a summand (see[3, 4]). In General Relativity, furthermore, the naive prescription gives anomalous232 Gauge-natural PVPsresults and a boundary correction for the superpotential was independently found bymany authors (see for example [6, 7, 8] and reference quoted therein). In [3] a generalsetting was proposed in which for a given Field Theory the corrected superpotentialcan be derived from an augmented Lagrangian .The aim of the present paper is to introduce a general definition of superpoten-tials for gauge-natural PVPs and to study their behavior with respect to the additionof a divergence to the Lagrangian, in view of the possibility of generalizing the pre-scriptions for the augmented Lagrangians to the case of PVPs.Due to the very strict limitations in length we are compelled to skip some details.We remark however that the reader can find in [1, 4] all the preliminary material withan identical notation.2. PVPsLet Cπ−→M be the configuration bundle whose (global) sections Γ(C) are the fields(by an abuse of language we will often denote bundles with the same label as their totalspaces). Let moreover (xµ, yi) be a fibered coordinate system on C and m = dimM .Definition 1 A first order Lagrangian C is a fibered morphismL : J1C −→ ∧m T ∗M.In local coordinates we write L as L(xµ, yi, yiµ) ds.Definition 2 A parametrization of order 1 and rank 1 of the set of constrained vari-ations is a couple (E,P), where E is a vector bundle EπE−→ C, while P is a fiberedmorphism (section)P : J1C −→ (J1E)∗ ⊗J1C V C.If (xµ, ya, εA) are local fibered coordinates on E and {∂a} is the induced fiberwisenatural basis of V C, a parametrization of order 1 and rank 1 associates to any sectionya(x) of C and any section εA(x, y) the section[paAεA + pa µA dµεA]∂a (1)of V C, where paA and pa µA are functions of(xν , yb(x), ∂νyb(x)), while εA depends on(xν , yb(x)).Definition 3 The set {C,L,P}, where C is the configuration bundle, L the La-grangian and P the parameterization, is called a “Parametrized Variational Problem”(PVP).The role of the parametrization is to select a subset of variations in order todefine criticality of a section with respect to this set.E. Bibbona, L. Fatibene and M. Francaviglia 233Definition 4 We define critical for {C,L,P} those sections of C for which, forany compact D ⊂ M and for any section εA(x, y) such that both εA(x, y(x)) anddµεA(x, y(x)) vanish for all x ∈ ∂D, we have∫D[∂L∂ya(paAεA + pa µA dµεA) +∂L∂yaµdµ(paAεA + pa νA dνεA)]ds = 0. (2)To set up a characterization of critical sections in terms of a set of differentialequations let us split the integrand of (2) into a first summand that factorizes εA(without any derivative) plus the total derivative of a second term (a general theoremensures that this splitting is unique; see [1]). To do this, we integrate by parts thederivatives of ε in the integrand of equation (2). What we get is∫D(EAεA + dµ(FµAεA + FµνA dνεA))ds = 0withEA =(∂L∂ya− dν ∂L∂yaν)paA − dµ[(∂L∂ya− dν ∂L∂yaν)paAµ](3)FµA =(∂L∂ya− dν ∂L∂yaν)paAµ +∂L∂yaµpaA and FµνA =∂L∂yaµpaAν .In [1] we have shown that the coefficients EA, FµA and FµνA are the componentsof two global morphismsE(L,P) : J3C −→ E∗ ⊗C ΛmT ∗MF(L,P, γ) : J2C −→ (J1E)∗ ⊗J1C ΛmT ∗Mand that a global meaning in terms of variational morphisms and global operationsbetween them can be given to the whole procedure. The same can also be done forhigher order Lagrangians and for higher rank and higher order parametrizations.3. Gauge-natural PVP and SuperpotentialsFor the definitions of gauge-natural bundles and morphisms we refer the reader to[1, 4] and references quoted therein.Definition 5 A gauge-natural variational problem with parametrized variations isdefined by the set (C(P ), L,F,P, J, ω¯) of the following objects:1. a gauge-natural configuration bundle C(P )π−→M of order (r, s) with structureG-bundle Pη−→M ;2. a variationally gauge-natural (local) Lagrangian morphism L of order k;3. a vector bundle FπF−→ C(P ) such that the composite projection F π◦πF−→ M is agauge-natural bundle F(P ) of order (p, f) called bundle of parameters;234 Gauge-natural PVPs4. a gauge-natural parametrization P : J1C(P ) −→ (JqF(P ))∗ ⊗JqC(P ) V C(P )(q ∈ {0, 1});5. a gauge-natural morphism J : J1C −→ (Jr−qIGA(P ))∗ ⊗J1C F(P ) such that wehave〈P | jq < J|jr−qΞ > 〉 ◦ j1σ = £Ξσ; here IGA(P ) denotes the bundle ofInfinitesimal Generator of Automorphisms (see [4], Ch. 3);6. a morphism ω¯ : JkC(P ) −→ C(M) ×M C(P ) that associates to any configura-tion a couple (γ,ω) where γ is a linear connection on M and ω is a principalconnection on P .In [1] we proved that if we have a gauge-natural PVP we can associate to anygauge symmetry Ξ a conserved currentE : J2k+l+s−1C(P ) −→ Jr−q−1IGA(P )⊗ Λm−1T ∗Msuch thatd(< E | jr−q−1Ξ > ◦ j2k+l+s−1σ) = 0.The analytic expression of the current is the following:< E | jr−q−1Ξ >=< F | jl+k−1J ◦ Ξ > −Tη Ξ!L (4)The conserved current is thence a variational morphism. General theorems onvariational morphisms ensure (see [4], Chapter 5 and Chapter 8) that the current canbe uniquely (given ω¯) split into a reduced current E˜ that vanishes on-shell plus thedivergence of another variational morphism U called superpotential. For any gauge-natural PVP we have thence< E | jr−q−1Ξ >=< E˜ | jr−q−1Ξ > + Div < U | jr−q−2Ξ > . (5)The integralQD(Ξ,σ) =∫∂D< U | jr−q−2Ξ > ◦j2k+l+s+r−q−3σcan be naively interpreted as the conserved current associated to the symmetry Ξ,the (m − 1)-surface D and the solution σ. In ordinary Variational Problems thisinterpretation is affected by the following ambiguity: if one adds a pure divergence tothe Lagrangian of the starting problem the equations (i.e. the true observable things)are left unchanged. One expects that so happens also for the conserved quantities,whereas in fact they gain a summand. A way to avoid this ambiguity was studied forordinary variational problems in [3]. Let us study what happens with PVPs.4. Trivial LagrangiansLet us consider a gauge-natural variational morphismsα : C(P ) −→ Λm−1T ∗ME. Bibbona, L. Fatibene and M. Francaviglia 235with coordinate expression αµdsµ. Let the (first-order) Lagrangian L be L = dµαµds.Constrained variations are a subset of free variations, thus if any section σ is criticalfor L in the free problem, then so it is also in the parametrized case. Anyway for k = 1one can directly see from expressions (3) that if the free Euler-Lagrange equations forL are satisfied so are the parametrized ones. Thanks to Lemma 17 in [1] (but one canalso directly compute it) one has that for any variation δy< δL | j1δy >= Div < δα | δy >where δα is a variational morphism δα : C(P ) −→ (Jk−1V C(P ))∗ ⊗Jk−1C Λm−1T ∗Mthat, using the same notation as before, has coordinate expression< δα | δy >= ∂αµ∂yiδyi dsµ.If we now restrict to the admissible variations δy that can be derived from the para-metrization P we find that if ε is a section of the bundle of parameters F(P ) then theP-Poincare´-Cartan morphism is< F | jqε >:=〈δα | < P|jqε >〉A gauge transformation Ξ that projects onto a vector field ξ = ξµ∂µ on the baseis a symmetry for L. Taking equation (4) and the properties of the morphism J (see5. in Definition 5) into account, one has that the conserved current is< E | jrΞ >=〈δα |£Ξy〉− ξ!LNow by gauge-naturality of α (see again [1]) one means〈δα |£Ξy〉= £ξαand thanks to Cartan formula one easily gets< E | j1Ξ >= dµ(ξ[ναµ]).Being the splitting (5) unique one has (here the order is much lower than in thegeneral case)< U | Ξ >= ξ[ναµ] dsνµthat is the same result that one gets in the non parametrized case (see [3]). Theanalogy between parametrized and free case still holds when if α depends on thederivatives of the field, some more care however is needed in the computations.5. ConclusionsAs a first step toward the definition of Augmented Parametrized Variational Problemswe defined the superpotentials for PVPs and studied their behavior under the additionof a pure divergence to the Lagrangian. The result obtained is that the superpotentialis modified by a summand that is identical to what one gets in the case of freevariations.236 Gauge-natural PVPsReferences[1] E. Bibbona and L. Fatibene and M. Francaviglia, Gauge-natural para-metrized variational problems, vakonomic field theories and relativistic hydro-dynamics of a charged fluid, Int. J. Geom. Methods Mod. Phys., vol. 3, No. 8(2006), 1-36. Preprint: math-ph/0604053.[2] A. Ferna´ndez and P. L. Garc´ıa and C. Rodrigo, Stress-energy-momentumtensors for natural constrained variational problems , J. Geom. Phys. , vol. 49,No. 1 (2004), 1-20[3] L. Fatibene and M. Ferraris and M. Francaviglia, Augmented varia-tional principles and relative conservation laws in classical field theory, Int. J.Geom. Methods Mod. Phys., vol. 2, No. 3 (2005), 373-392[4] L. Fatibene and M. Francaviglia, Natural and gauge natural formalism forclassical Field Theories, Kluwer Academic Publishers, Dordrecht, 2003.[5] L. Szabados, Quasi-Local Energy-Momentum and Angular Momentumin GR: A Review Article, Living Rev. Relativity 7, (2004), 4. URL:http://www.livingreviews.org/lrr-2004-4[6] B. Julia and S. Silva, Currents and superpotentials in classical gauge theories,Classical Quantum Gravity , vol. 17, No. 22 (2000), 4733-4743[7] V. Iyer and R. Wald, Some properties of the Noether charge and a proposalfor dynamical black hole entropy , Phys. Rev. D , vol. 50, No. 2 (1994), 846-864[8] J. Katz and J. Bicˇa´k and D. Lynden-Bell, Relativistic conservation lawsand integral constraints for large cosmological perturbations , Phys. Rev. D ,vol. 55, No. 10 (1997), 5957-5969

Gauge-natural PVPs, superpotentials and trivial Lagrangians

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Gauge-natural PVPs, superpotentials and trivial Lagrangians

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