Canonical variational completion of differential equations
aa r X i v : . [ m a t h - ph ] O c t Canonical variational completion of differentialequations
Nicoleta Voicu”Transilvania” University, Brasov, RomaniaDemeter KrupkaLepage Research Institute, Czech Republic
Abstract
Given a non-variational system of differential equations, the simplestway of turning it into a variational one is by adding a correction term. Inthe paper, we propose a way of obtaining such a correction term, based onthe so-called Vainberg-Tonti Lagrangian, and present several applicationsin general relativity and classical mechanics.
Keywords: jet bundle, source form, variationality conditions, Einstein fieldequations, canonical energy-momentum tensor
MSC 2010:
For a given non–variational system of differential equations, there are multipleways of transforming it into a variational one - among these, variational multi-pliers (or variational integrating factors), [1], are maybe the most well known.Another possibility is to simply add a correction term.In the paper, we consider systems of ordinary or partial differential equa-tions - represented by source forms, or source tensors, similar to Euler-Lagrangesystems for extremals of integral variational functionals in the calculus of vari-ations. We propose a way of obtaining such a correction term - which we calla variational completion, as follows. Any ordinary or partial differential systemcan be expressed as the vanishing of some source form ε on sections of an ap-propriate jet bundle. Further, to this source form, one can naturally attach aLagrangian λ ε , called the Vainberg-Tonti Lagrangian of ε , [7]; this Lagrangianhas the property that the difference τ := E ( λ ε ) − ε (1)between its Euler-Lagrange form E ( λ ε ) and ε offers a measure of the non-variationality of ε . Using τ in (1) as the correction term, the system ε + τ = 0becomes variational. 1he method appears to have several interesting applications. We presenthere three of them.1) Einstein tensor obtained from variational completion of the Ricci ten-sor.
Historically, the first variant of gravitational field equations proposed byEinstein was: R ij = 8 πκc T ij , (2)where: R ij is the Ricci tensor of a 4-dimensional Lorentzian manifold ( X, g ) , T ij is the energy-momentum tensor, while κ and c are constants, [8]. This variantcorrectly predicted some physical facts, but failed to fulfil another request: localenergy-momentum conservation. This led Einstein to adding in the left handside the ”correction term” − Rg ij (by a reasoning based on Bianchi identities),thus leading to the nowadays famous: R ij − Rg ij = 8 πκc T ij . (3)The variational deduction of (3), due to Hilbert, relies on a heuristic argument- simplicity. Hilbert chose to construct the action for the left hand side using the”simplest scalar” (i.e., simplest differential invariant) which can be constructedfrom the metric tensor and its derivatives alone. Happily, the Euler-Lagrangeexpression ensuing from this simplest scalar - which is the scalar curvature R -coincides with the left hand side of (3).There is, still, another way of finding this correction term. Equation (2)is not variational. Actually, the term which fails to be variational is R ij ; inthe paper, we prove that the Hilbert Lagrangian is (up to multiplication by anon-essential constant), nothing else than the Vainberg-Tonti Lagrangian cor-responding to R ij . Accordingly, the correction term − Rg ij can be obtainedfrom R ij as a canonical variational completion.2) Energy-momentum tensors . In special relativity, energy-momentum ten-sors are obtained by adding to the Noether current corresponding to the in-variance of the matter Lagrangian to space-time translations a symmetrizationterm. The way of obtaining the symmetrization term is subject to an old de-bate, [2], [3]. The canonical variational completion method offers the possibilityof recovering the expression of a full, symmetric energy-momentum tensor fromjust one of its terms - e.g., from a non-symmetrized Noether current. In partic-ular, the energy-momentum tensor of the electromagnetic field can be obtainedthis way.3) In classical mechanics , equations of damped small oscillations are knownto be non-variational. Without aiming to give a general physical interpreta-tion of the obtained correction term, we determine the canonical variationalcompletion of these equations.In Sections 2 and 3, we briefly present some known notions and results tobe used in the following. 2
Differential forms on jet bundles
The mathematical background for a modern formulation of both field theoryand mechanics are fibered manifolds and their jet bundles.Consider a fibered manifold Y of dimension m + n, with n -dimensional base X and projection π : Y → X. Fibered charts (
V, ψ ), ψ = ( x i , y σ ) on Y induce thefibered charts ( V r , ψ r ) , ψ r = ( x i , y σ , y σj , ..., y σj j ...j r ) on the r -jet prolongation J r Y of Y and ( U, φ ) , φ = ( x i ) on X. The manifold J r Y can be regarded as afibered manifold in multiple ways, by means of the projections: π r,s : J r Y → J s Y, ( x i , y σ , y σj , ..., y σj j ...j r ) ( x i , y σ , y σj , ..., y σj j ...j s ) , where r > s, J Y := Y and: π r : J r Y → X. The set of C ∞ -smooth sections γ : X → Y , locally expressed by somefunctions ( x i ) γ ( x i ) = ( x i , y σ ( x i )) is denoted by Γ( Y ) . Given a sec-tion γ ∈ Γ( Y ) , its prolongation to J r Y is: J r γ : ( x i ) J r γ ( x i ) =( x i , y σ ( x ) , y σ,j ( x ) , ..., y σ,j j ...j r ( x )), where the symbol ,j denotes partial differ-entiation with respect to x j . In field theoretical applications, the coordinates x i play the role of space-time coordinates, while y σ are ”field” coordinates (to be accurate, real fieldsare encoded in sections y σ = y σ ( x i )). The case of mechanics is characterizedby dim X = 1; in this case, the coordinates on J r Y are usually denoted by( t, q σ , ˙ q σ , ¨ q σ , ..., q ( r ) ) and are interpreted as: time, generalized coordinates, gen-eralized velocity etc.By Ω rk W, we denote the set of k -forms of order r over an open set W ⊂ Y, i.e., the set of k -forms over the r -th prolongation J r W ⊂ J r Y . In particular, F ( W ) := Ω r W is the set of real-valued smooth functions over J r W. The subset of Ω rk W consisting of k -forms: ρ = 1 k ! a i i ...i k dx i ∧ dx i ∧ ... ∧ dx i k , (4)(where a i i ...i k , k ≤ n, are smooth functions of the coordinates x i , y σ , y σj , ..., y σj j ...j r ) is called the set of ( π r -) horizontal k - forms of order r ; similarly, one can speak about π r,s -horizontal forms of order r as forms gen-erated by exterior products of the differentials dx i , dy σ , ..., dy σj ...j s . Examples of π r -horizontal forms are volume forms and Lagrangians.For X = R n , the Euclidean volume form is: ω = dx ∧ dx ∧ ... ∧ dx n . (5)On pseudo-Riemannian manifolds ( X, g ij ), a coordinate-invariant volume formis locally given by: dV = p | g | ω , where g := det( g ij ) . Lagrangian of order r is defined as a π r -horizontal n -form of order r : λ = L ω , L = L ( x i , y σ , ..., y σi ...i r ) . (6)A form θ ∈ Ω rk Y is a contact form if it is annihilated by all jets J r γ ofsections γ ∈ Γ( Y ) . Important examples are the basic contact 1-forms on J r Y defined on a coordinate neighborhood by: ω σ = dy σ − y σj dx j , ω σi = dy σi − y σi j dx j , ... (7) ω σi i ...i r − = dy σi i ...i r − − y σi i ...i r − j dx j . A differential form is called p -contact if it is generated by p -th exterior powersof contact forms. A source form of order r on a fibered manifold Y, [5], is a π r, -horizontal, 1-contact ( n + 1)-form on J r Y . In local coordinates, any source form is expressedas: ε = ε σ ω σ ∧ ω , ε σ = ε σ ( x i , y σ , y σi , ..., y σj ...j r ) . (8)The set of source forms of order at most r over Y is closed under addition andunder multiplication with functions f ∈ F ( J r Y ) . The most notable example of a source form is the
Euler Lagrange form E ( λ )of a Lagrangian λ = L ( x i , y σ , ..., y σi ...i r ) ω ∈ Ω rn ( Y ) : E ( λ ) := E σ ω σ ∧ ω ,E σ = ∂ L ∂y σ − d k ∂ L ∂y σk + ... + ( − r d k ...d k r ∂ L ∂y σk ...k r . A section γ : X → Y is critical for the Lagrangian λ if and only if the E ( λ )is annihilated by the r -jet of γ, i.e., E σ ( λ ) ◦ J r γ = 0 , σ = 1 , ..., m. A source form ε is called:a) locally variational if around any point of the fibered manifold Y, thereexists a local fibered chart ( V, ψ ) and a Lagrangian λ on some jet prolongation V r ( r ∈ N ) of V, such that, on V r , ε = E ( λ );b) globally variational if there exists a Lagrangian λ on the whole manifold Y such that ε = E ( λ ) . Local variationality of a source form ε = ε σ ω σ ∧ ω of order r is equivalentto a generalization of classical Helmholtz conditions, [6]: H j ...j k σν ( ε ) = 0 , k = 0 , ..., r, (9)where: 4 j ...j k σν ( ε ) = ∂ε σ ∂y νj ...j k − ( − k ∂ε ν ∂y σj ...j k − (10) − r X l = k +1 ( − l ( lk ) d i k +1 d i k +2 ...d i l ∂ε ν ∂y σj ...j k i k +1 ...i l locally describe the Helmholtz form H ε = 12 r P k =0 H j ...j k σν ( ε ) ω νj ...j k ∧ ω σ ∧ ω . By variational completion of a given source form ε on Y, we will mean anysource form τ on Y with the property that ε + τ is variational. Of course, onecan speak about local and about global variational completions.In the following, we will only study local variational completions.Clearly, every source form has infinitely many variational completions: in-deed, any Lagrangian λ induces the completion τ := E ( λ ) − ε . Thus, the ques-tion is how to choose the Lagrangian λ in a meaningful way. In the following,we will try to give an answer to this question.Given an arbitrary source form ε = ε σ ω σ ∧ ω ∈ Ω rn +1 Y of order r , a localLagrangian attached to ε is the Vainberg-Tonti Lagrangian λ ε = L ε ω , [7], [4],defined by: L ε ( x i , y σ , ..., y σj ...j s ) = y σ Z ε σ ( x i , uy σ , ..., uy σj ...j s ) du. (11)The Euler-Lagrange form E ( λ ε ) = E ν ω ν ∧ ω of the Vainberg-Tonti La-grangian λ ε is given, [7], by: E ν = ε ν − Z u { y σ ( H νσ ◦ χ u ) + y σj ( H jνσ ◦ χ u ) + ... + y σj ...j r ( H j ...j r νσ ◦ χ u ) } du, where χ u : J r Y → J r Y denotes the homothety ( x i , y σ , y σj , ..., y σj ...j r ) ( x i , uy σ , uy σj , ..., uy σj ...j r ) and the coefficients H j ...j k σν are as in (9).From (9), it follows that the coefficients H j ...j k σν above have the meaningof ”obstructions from variationality” of the source form ε. In particular, if thesource form ε is variational, then E ( λ ε ) = ε. It thus appears as natural
Definition 1
The canonical variational completion of a source form ε ∈ Ω rn +1 ( Y ) , is the source form τ ( ε ) given by the difference between the Euler-Lagrange form of the Vainberg–Tonti Lagrangian of ε and ε itself: τ ( ε ) = E ( λ ε ) − ε. (12)5he local coefficients τ ν of the canonical variational completion τ ( ε ) = τ ν ω ν ∧ ω can be directly expressed in terms of the coefficients H j ...j k νσ : τ ν = − Z u { y σ ( H νσ ◦ χ u ) + y σj ( H jνσ ◦ χ u ) + ... + y σj ...j r ( H j ...j r νσ ◦ χ u ) } du. Remark.
Generally speaking, the Vainberg-Tonti Lagrangian and, accord-ingly, the canonical variational completion of a source form of order r, are oforder 2 r. Still, under certain conditions, [4] (which are fulfilled by a large numberof equations in physics), the Vainberg-Tonti Lagrangian is actually equivalentto a Lagrangian of order r. Consider a Lorentzian manifold (
X, g ij ) of dimension 4, with local charts ( U, φ ), φ = ( x i ) i =0 , and Levi-Civita connection ∇ . We denote by R ij the Ricci tensorof ∇ and by R = g ij R ij , the scalar curvature. We assume in the following thatmeasurement units are chosen in such a way that c = 1 . Indices of tensors willbe lowered or raised by means of the metric g ij and its inverse g ij .Einstein field equations (3) arise by varying with respect to the metric tensorthe Lagrangian λ = λ g + λ m , where: i) λ g = − πκ R p | g | ω (with ω = dx ∧ dx ∧ dx ∧ dx ) is the HilbertLagrangian; ii) the matter Lagrangian λ m = L m p | g | ω , is given by a differential invari-ant L m = L m ( g ij , g ij,h , ... ; y σ , y σj , ..., y σj ...j r ) depending on the metric tensorcomponents and their derivatives up to a certain order s ∈ N and on the r -jetof a field y σ . Typically, in classical general relativity, s = 0 . In the case of vacuum Einstein equations R ij − Rg ij = 0 , (13)the ”field components” to be varied are the metric tensor components g ij (or,more commonly, the inverse metric components g ij ), hence the fibered manifold Y is the bundle of metrics M et ( X ) , defined as the set of symmetric nondegen-erate tensors of type (0,2) on X. Since both R ij and R are of second order in g ij , the space we have to work on is the second order jet bundle J M et ( X ).We denote the local charts on M et ( X ) by ( V, ψ ) , with ψ = ( x i , g jk ) and theinduced fibered chart on J M et ( X ) , by ( V , ψ ) , with ψ = ( x i , g jk ; g jk,i ; g jk,il ) . We will also use the notations: ω jk = dg jk − g jk,i dx i ; ω jk,l = dg jk,l − g jk,li dx i for the basic contact forms on J M et ( X ). The Riemann tensor, the Ricci tensorand the Ricci scalar thus become objects on J M et ( X ) . .1 Canonical variational completion of the Ricci tensor We will prove in the following that vacuum Einstein equations (13) can beobtained by means of the canonical variational completion of the source formwith components R ij . Take the following source form on J M et ( X ) : ε := αR ij p | g | ω ij ∧ ω , (14)where α is a (momentarily) arbitrary constant. Its components ε ij = ε ij ( g kl ; g kl,i , g kl,ij ) are given by ε ij = αR ij p | g | . The Vainberg-Tonti Lagrangian λ ε = L ε ω is defined as: L ε = g ij Z ε ij ( ug kl ; ug kl,i ; ug kl,ij ) du. Let us study the behavior of the integrand with respect to homotheties χ u : ( g kl ; g kl,i ; g kl,ij ) ( ug kl ; ug kl,i ; ug kl,ij ) . These homotheties induce thetransformation g kl u − g kl of the inverse metric tensor components. TheChristoffel symbols Γ ijk = 12 g ih ( g hj,k + g hk,j − g jk,h )are invariant to χ u and hence the curvature tensor components R ij kl = Γ ijk,l − Γ ijl,k + Γ hjk Γ ihl − Γ hjl Γ ihk are also invariant. The Ricci tensor R jk = R ij ki isobtained just by a summation process from R ij kl , which means that it is alsoinsensitive to χ u . That is, R ij = g ih g jl R hl will acquire a u − . It remains to compute the contribution of χ u to the factor p | g | . Each lineof the matrix ( g jk ) is multiplied by u, that is, g = det( g ij ) will acquire a factorof u and finally, p | g ◦ χ u | = u p | g | . Substituting into the expression of L ε , we get this way, L ε = g ij Z u αR ij p | g | du = αg ij R ij p | g | Z u du = αR p | g | . Thus, if we choose α := − πκ , the Vainberg-Tonti Lagrangian λ ε = L ε ω becomes the Hilbert Lagrangian λ g : λ ε = λ g . (15)7e know, however, that the Euler-Lagrange expressions of R p | g | with re-spect to g ij are given by (minus) the contravariant components of the Einsteintensor. In differential form writing, this is: E ( λ ε ) = 116 πκ ( R ij − Rg ij ) p | g | ω ij ∧ ω hence, we find the variational completion τ = E ( λ ε ) − ε as τ = 116 πκ (2 R ij − Rg ij ) p | g | ω ij ∧ ω . Remark.
The factor α in (14) is actually unessential, the variationallycompleted equation E ( λ ε ) = 0is still the correct vacuum Einstein equation, regardless of its value. Having one term of an energy-momentum tensor, the canonical variational com-pletion method offers a way of recovering its full expression. We will apply thismethod in the case when the known piece is a (non-symmetrized) Noether cur-rent.In the case of Einstein equations with matter (3), we will have to work ona fibered product Y × X M et ( X ) over X (where Y is a fibered manifold withbase X ) with coordinate charts ( V, ψ ) , ψ = ( x i , y σ , g jk ) . In this case, one canspeak separately about variations with respect to y σ and to g jk and accordingly,about Y -variationality and M et ( X )-variationality, Y - and M et ( X )-variationalcompletions.Consider a first order Lagrangian λ m on Y × X M et ( X ); we suppose inaddition that λ m does not depend on x j and on the derivatives g ij,k . Thus, λ m = L m p | g | ω , where L m = L m ( y σ , y σj , g ij ) . In classical relativity theory, there are two major ways of defining energy-momentum tensors, corresponding to two different contexts:1) The canonical energy-momentum tensor , corresponding to special relativ-ity (where X = R and the metric tensor is fixed as η ij = diag (1 , − , − , − λ m = L m ω , which is invariant to the group of space-time trans-lations ˜ x i = x i + a i , a i = const., gives rise to a system of conserved Noethercurrents, called the canonical energy-momentum tensor (invariance to space-time translations amounts to the above assumption that L m does not explicitlydepend on x i ). These Noether currents are given by, [8]:˜ T ij = η ik ( y σ,k ∂L m ∂y σ,j − δ jk L m ) . (16)8he canonical energy-momentum tensor ˜ T ij is, generally, not symmetric -which is inconvenient, since symmetry is required on physical grounds (angularmomentum conservation). This is usually solved by adding a divergence-freeterm, thus obtaining a tensor can T ij which is symmetric and still conserved, i.e., can T ij,j = 0 . There are multiple possibilities of choosing the symmetrization term,[2]. 2) In general relativity (where (
X, g ij ) is an arbitrary Lorentzian manifold),energy-momentum tensors ( Hilbert, or metric energy-momentum tensors ) met T ij are defined by means of functional derivatives of the matter Lagrangian λ m = L m ω , L m = L m p | g | , with respect to g ij : − T ij := δ L m δg ij ; met T ij := 1 p | g | T ij . (17)Here, L m = L m ( y σ , y σj , y σj ; g ij ) is a differential invariant (a ” scalar ”), hence theLagrangian λ m is invariant to (transformations on J r Y induced by) arbitrarydiffeomorphisms on X . As a result, met T ij obeys on-shell the covariant conser-vation law met T ij ; j = 0 and also, has gauge invariance properties, [2]. Moreover, met T ij is, by construction, symmetric.The two procedures of defining the energy-momentum tensor are fundamen-tally different and obviously require a thorough geometric analysis. Just as afirst remark, they generally do not even make sense at the same time: in specialrelativity, where the metric is fixed, it makes no sense to speak about variationsof a Lagrangian with respect to the metric. On the other hand, in general rel-ativity, where X is an arbitrary manifold, space-time translations ˜ x i = x i + a i ,a i = const., cannot be defined geometrically. However, there is a realm (see,e.g., [10]) where both procedures can be applied, namely, when: X = R , g ij − arbitrary (18)(actually, in [10], it is pointed out the particular case of weak metrics – in whichthe author studies the equivalence between the two definitions. Still, for ourpurposes, we do not need the assumption that the metric is weak).For a special-relativistic Lagrangian λ m = L m ω , L m = L m ( y σ , y σi , g ij = η ij ) , (19)the canonical variational completion offers a recipe of symmetrization of theNoether current ˜ T ij . We will do this in three steps: Step 1.
We leave for the moment the special relativistic context and for-mally allow g ij to vary. Abiding by the principle of general covariance, [8], astraightforward generalization of (16) to the new context is given by the tensordensity: ˜ T ij = g ik ( y σ ; k ∂L m ∂y σ ; j − δ jk L m ) p | g | , (20)9here the semicolon ; k denotes (formal) covariant differentiation with respectto ∂/∂x k . Note:
In the above, y σ are tensors of some unspecified rank (the upper posi-tion of the index is chosen just for convenience; y σ can very well be componentsof, e.g., a scalar, a covector field or of a tensor of type (0,2)). Step 2.
Taking into account (17), we consider the source form ε = α ˜ T ij ω ij ∧ ω on J ( Y × X M et ( X )) , with components ε ij = α ˜ T ij ( y σ , y σj , g kh ), where α ∈ R is a constant. Its M et ( X )-Vainberg-Tonti Lagrangian λ ε := L ε ω is: L ε = αg ij Z ( ˜ T ij ◦ χ u ) du, where χ u ( y σ , y σi , g ij ) := ( y σ , y σi , ug ij ) only affects the metric components. Sub-stituting ˜ T ij from (20) and taking into account that χ u leaves Christoffel sym-bols invariant and that δ ii = dim( X ) = 4 , we have: L ε = α Z u ( y σ ; i ∂ ( L m ◦ χ u ) ∂y σ ; i − L m ◦ χ u ) p | g | du. (21)Further, we calculate the Hilbert energy-momentum tensor of λ ε as: − met T ij := 1 p | g | δ L ε δg ij . (22) Step 3.
Finally, particularize in (22) g ij as η ij and define met T ij : | g ij = η ij =: T ij . This way, met T ij is defined up to multiplication by the constant α . This constantcan then be adjusted, for instance, in such a way that the obtained symmetriza-tion term τ ij := T ij − ˜ T ij (23)is independent from ˜ T ij (it does not contain any multiple of ˜ T ij ).The covariant conservation law of met T ij (obtained as a consequence of thefact that met T ij is a Hilbert energy-momentum tensor) now transforms into theusual conservation law: T ij,j = 0 . Thus, the obtained energy-momentum tensor T ij is, as required, both symmetric and conserved. Moreover, the symmetrzationterm τ ij offers a measure of the non- M et ( X )-variationality of ˜ T ij . Example: energy-momentum tensor of the electromagnetic field.
The electromagnetic field is described by the potential 1-form A = A i dx i on X and by the 2-form F := dA = 12 F ij dx i ∧ dx j .
10n the special relativistic case g ij = η ij , we have F ij = A j,i − A i,j , or, in termsof the contravariant components A i : F ij = η jk A k,i − η ik A k,j . The Lagrangianof the electromagnetic field is λ f = L f ω with L f = − π F ij F ij ; (24)Translational invariance of λ f leads to the Noether current, [8]:˜ T ij = − π η ih ∂A l ∂x h F jl + 116 π η ij F kl F kl . (25)The curved space generalization of ˜ T ij in (25) is the tensor density:˜ T ij = ( − π g ih A l ; h F jl + 116 π g ij F kl F kl ) p | g | (26)where, this time: F ij = g jk A k ; i − g ik A k ; j . (27)Further, we calculate the Vainberg-Tonti Lagrangian of the source form ε = α ˜ T ij ω ij ∧ ω , where ˜ T ij = ˜ T ij ( A k ; A k,l ; g kl ; g kl,h ). We prefer to use A k rather than A k := g kl A l as the field variables for a reason which will become transparent below. Thisway, χ u acts as follows: g ij ◦ χ u = ug ij , g ij ◦ χ u = u − g ij , while χ u does not affect the field variables y σ = A k . Again, the Christoffel sym-bols Γ ijk are invariant to χ u . Expressing F ij as in (27), we can now determinethe effect of χ u on each term of ˜ T ij : A l ; i ◦ χ u = A l ; i ; F jl ◦ χ u = uF jl ; F kl ◦ χ u = u − F kl , p | g | ◦ χ u = u p | g | . All in all, we have: ˜ T ij ◦ χ u = u ˜ T ij and hence, the Vainberg-Tonti Lagrangian λ ε = L ε ω is given by: L ε = g ij Z uα ˜ T ij du = α g ij ˜ T ij , that is, L ε = α ( − π A l ; k F kl + 18 π F kl F kl ) p | g | . (28)11aking into account that F kl = − F lk , the term A l ; k F kl in the above can bere-expressed as: A l ; k F kl = 12 ( A l ; k − A k ; l ) F kl = 12 F kl F kl ; substituting into (28),we finally obtain the M et ( X )-Vainberg-Tonti Lagrangian of (26) as: λ ε = α π F kl F kl p | g | ω = − αλ f . (29)But, variation of λ f with respect to g ij is well-known, [8]; namely, we willget for λ ε = − αλ f the Hilbert energy-momentum tensor met T ij ( α ) = − α ( − π F il F jl + 116 π g ij F kl F kl ) . Particularizing now g ij = η ij , we get the symmetrized energy-momentumtensor: T ij := met T ij = − α ( ˜ T ij + 14 π A i,l F jl ) . Taking α := − met T ij = ˜ T ij + independent term ), we obtain λ ε = λ f and the symmetrization term: τ ij = 14 π A i,l F jl , or, in covariant writing, τ ij = 14 π A i,l F lj as the correction term. This is the classical symmetrization term, [8], yet,obtained here by a completely different reasoning. Remarks.
1) If, for a given (symmetrized or not) energy-momentum tensor ˜ T ij , theLagrangian λ m is not known, a Lagrangian can be constructed as the M et ( X )-Vainberg-Tonti Lagrangian (21); if a Lagrangian λ m is already known, the abovegives an alternative construction.1) If the given matter Lagrangian density L m is homogeneous both in themetric components and in the derivatives y σ ; i (the homogeneity degrees need notcoincide) then, applying Euler’s theorem in (21), we see that the Vainberg-TontiLagrangian density L ε in (21) actually coincides, up to multiplication by someconstant, with the matter Lagrangian density L m . In this case, we can alwayschoose α such that L ε = L m . In this case, the symmetrization term coincideswith the one in [3], yet, it is found just by considerations of variationality.2) If we had worked with the potential 1-form components A i (instead ofthe vector field components A i ) as our field variables, we would have had F ij = A j ; i − A i ; j - invariant to χ u and by a similar reasoning to the above, we wouldhave got ˜ T ij ( ug kl ) = u − ˜ T ij ( g kl ) and, consequently, to L ε = ( g ij ˜ T ij ) R u − du. But, since the latter integral does not have a finite value, we could not have12alculated L ε this way. Hence, it appears that, at least in this case, the 4-potential vector field components A i are a more advantageous choice for ourdynamical variables. Take Y = R × R n , with local coordinates ( t, q σ ); on the second jet prolongation J Y, we denote the induced local coordinates by ( t, q σ , ˙ q σ , ¨ q σ ) . Consider the second order source form ε = ε σ ω σ ∧ dt,ε σ = m σν ¨ q ν + k σν q ν + ∂F∂ ˙ q σ , (30)where:- m σν , k σν are constant and symmetric;- F = F ( ˙ q σ ) is homogeneous of some degree p ≥ q σ . The ODE system ε σ = 0 is generally non-variational. Let us determine itscanonical variational completion. The Vainberg-Tonti Lagrangian attached to ε is λ ε = L ε dt, with L ε = q σ Z ε σ ( t, uq ν , u ˙ q ν , u ¨ q ν ) du = q σ Z ( m σν u ¨ q ν + k σν uq ν + ∂F∂ ˙ q σ ( u ˙ q ν )) du. Taking into account the homogeneity degree of F, this is: L ε = q σ Z [ u ( m σν ¨ q ν + k σν q ν ) + u p − ∂F∂ ˙ q σ ] du == 12 ( m σν ¨ q ν q σ + k σν q σ q ν ) + 1 p q σ ∂F∂ ˙ q σ . The term 12 m σν ¨ q ν q σ differs by a total derivative d t ( 12 m σν ˙ q ν q σ ) from − m σν ˙ q ν ˙ q σ , hence the two expressions are dynamically equivalent. We willthus prefer to take the latter, which is of lower order and thus, we obtain thefollowing Lagrangian function, which is equivalent to the Vainberg-Tonti one: L = 12 ( − m σν ˙ q ν ˙ q σ + k σν q σ q ν ) + 1 p q σ ∂F∂ ˙ q σ . (31)Let us determine the Euler-Lagrange form of L . We have, on one hand: ∂ L ∂q ρ = k σρ q σ + 1 p ∂F∂ ˙ q ρ ∂ L ∂ ˙ q ρ = − m σρ ˙ q ρ + 1 p ∂ F∂ ˙ q σ ∂ ˙ q ρ q σ ,d t ( ∂ L ∂ ˙ q ρ ) = − m σρ ¨ q ρ + 1 p ∂ F∂ ˙ q σ ∂ ˙ q ρ ∂ ˙ q ν ¨ q ν q σ + 1 p ∂ F∂ ˙ q σ ∂ ˙ q ρ ˙ q σ ;taking again into account that ∂F∂ ˙ q ρ is homogeneous of degree p − , the latterterm is: 1 p ∂ F∂ ˙ q σ ∂ ˙ q ρ ˙ q σ = p − p ∂F∂ ˙ q ρ and, finally, E ρ ( L ) = ( m σρ ¨ q σ + k σρ q σ ) + 2 − pp ∂F∂ ˙ q ρ − p ∂ F∂ ˙ q σ ∂ ˙ q ρ ∂ ˙ q ν ¨ q ν q σ . We find the variational completion τ = τ ρ ( t, q, ˙ q ) ω ρ ∧ dt as: τ ρ = 2( 1 p − ∂F∂ ˙ q ρ − p ∂ F∂ ˙ q σ ∂ ˙ q ρ ∂ ˙ q ν ¨ q ν q σ . (32) Particular cases:
1) If F = 0 , the system ε σ = 0 is equivalent to m σν ¨ q ν + k σν q ν = 0 . These equations characterize free small oscillations with multiple degrees offreedom, [8]. They are known to be variational; their Lagrangian function L =12 ( − m σν ˙ q ν ˙ q σ + k σν q σ q ν ) coincides (as expected), with (31).2) If p = 2 and F is quadratic in ˙ q : F = 12 α σν ˙ q σ ˙ q ν , (where α σν = α νσ ∈ R ) the ODE system ε σ = 0 characterizes, [9], Section 25,linearly damped oscillations. In this case, the function F is called the Rayleighdissipation function and is interpreted as the rate of energy dissipation in thesystem. In (30), the last term (with a minus in front) − ∂F∂ ˙ q σ = − a σν ˙ q ν isinterpreted as a friction force. In this case, the canonical variational completion(32) is given by τ ρ = − ∂F∂ ˙ q ρ and the variationally completed equations are: m ρν ¨ q ν + k ρν q ν = 0 , (33)14hich are precisely the equations of ”undamped” oscillations. That is, the fric-tion force ∂F∂ ˙ q ρ has, in this case, the meaning of obstruction from variationalityof the equations. Remark.
In other cases (e.g., when − ∂F∂ ˙ q ρ is quadratic or cubic in ˙ q σ ), thevariationally completed equations will not coincide anymore with the equations(33) of undamped oscillations. Acknowledgment.
The work was partially supported by the Grant ofTransilvania University 2013 (
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