Lie Symmetry Group Theory for Turbulence Modelling and Simulation

Lie symmetry group theory for turbulence modelling and simulation

Aziz Hamdouni∗, Dina Razaﬁndralandy∗, Marx Chhay†, and Nazir Al Sayed‡ ∗Laboratoire des Sciences de l’Ingénieur pour l’Environnement – UMR 7356 Université de La Rochelle – France †Laboratoire Optimisation de la Conception et Ingénieurie de l’Environnement – UMR 5271 Université de Savoie – France ‡Laboratoire de Mécanique des Structures Industrielles Durables – UMR EDF/CNRS/CEA 8193 EDF R&D – France

Abstract—In this paper, we present some applica- II. Lie symmetry group theory and application tions of the Lie-group theory to fluid mechanics prob- to the non-isothermal Navier-Stokes equations lems. We present how to use the Lie-group theory to deduce conservation laws of the Navier-Stokes equations Instead of presenting the Lie symmetry group theory and scaling laws of turbulent flows. We also develop on a general equation, we apply it directly to the non- symmetry preserving turbulent models, and show how isothermal Navier-Stokes equations: the symmetry theory can contribute to build robust  numerical schemes. ∂u 1  + div(u ⊗ u) + ∇p − ν∆u − βgθ e = 0  ∂t ρ 3    I. Introduction ∂θ (1)  + div(θu) − ∆θ = 0  ∂t  The literature shows that the Lie-symmetry group the-   ory constitutes a powerful modelling tool in engineering  div u = 0 science. They allow, for instance, the computation of A continuous symmetry of equations (1) is a transforma- Green function of linear equations. In addition, symme- tion tries are extensively used in literature to compute self- ˆ similar solutions of various equations. In turbulence, vor- T : q = (t, x, u, p, θ) 7−→ qˆ = (t,ˆ xˆ, uˆ, p,ˆ θ), (2) tex solutions of the Navier-Stokes equations was found depending smoothly on the parameter , and which leave as special self-similar solutions. Finally, we mention that the set of solutions of (1) invariant. In other words, T is the symmetries may give an information on the large-time a symmetry of (1) if behaviour of the solution. To some extent, the symmetries traduce the physics of the equations. NS(q) = 0 =⇒ NS(qˆ) = 0 (3) In this communication, we present some applications to where NS designate equations (1). A Lie symmetry group fluid mechanics, and especially to the modelling and sim- of NS is a set of continuous transformations which has a ulation of turbulent flows. In section II, we recall briefly Lie group structure. the theory of Lie-symmetry group and how to compute Definition (3) permits to compute some but not all them. We then list the Lie-symmetry transformations of symmetry groups of NS. In order to be exhaustive, one the Navier-Stokes equations in the non-isothermal case. introduces the tangent vector: Next, we recall the Nœther’s theorem which links the symmetries of an equation to conservation laws in the case ∂ ∂ ∂ ∂ ∂ X = ξ + ξ · + ξ · + ξ + ξ (4) of a Lagrangian-described system. We then show how to t ∂t x ∂x u ∂u p ∂p θ ∂θ extend this theorem to non-Lagrangian systems and how where to find conservation laws of the Navier-Stokes equations. ∂qˆ ξ = . (5) In section IV, we are especially interested in turbulent q ∂ flows. We use the Lie-group theory to find wall and scaling =0 laws of non-isothermal turbulent flows. Next, we develop A tangent vector of a Lie group is also called its generator symmetry preserving turbulence models and show their since, knowing the components ξq of X, one can determine numerical performance. the elements T of the group by solving the equation In section V, we build robust numerical schemes, based dqˆ on symmetry preservation. = ξ(qˆ), qˆ( = 0) = q. (6) d Definition (3) is equivalent to the infinitesimal condition • the group of the first scaling transformations: NS(q) = 0 =⇒ X(2) · NS(q) = 0. (7) (t, x, u, p, θ) 7−→ (e2 t, e x, e− u, e−2 p, e−3 θ) (21) X(2) is a prolongation of X which accounts for the first which shows how u, p and θ change when the spatio- and second order partial derivatives in the equations (see temporal scale is multiplied by (e, e2), [1]–[3]). This condition enables to compute the tangent • and the group of the second scaling transformations: vectors of all the Lie symmetry groups of (1): 2 2 2 ∂ (t, x, u, p, θ, ν, κ) 7→ (t, e x, e u, e p, e θ, e ν, e κ) X = (8) (22) 1 ∂t which shows the consequence of a modification of the ∂ spatial scale. X2 = ζ(t) (9) ∂p Equations (1) own other known non-Lie symmetries, ∂ 1 ∂ which are: X3 = βg x3 + (10) ∂p ρ ∂θ • the reflections about x1 and x2 axes • and the material indifference in the limit of a 2D ∂ ∂ ∂ ∂ horizontal flow in a simply connected domain [4]: X4 = x2 − x1 + u2 − u1 (11) ∂x1 ∂x2 ∂u1 ∂u2 (t, x, u, p) 7→ (t , R(t)xˆ , R(t)u + R˙ (t)x , pˆ) (23) ∂ ∂ ∂ X4+i = αi(t) +α ˙ i(t) − ρ xiαï(t) , i = 1, 2, 3, ∂xi ∂ui ∂p with (12) pˆ = p − 3ωφ + ω2kxk2/2 ∂ ∂ ∂ ∂ ∂ X8 = 2t + x · − u · − 2p − 3θ (13) where R(t) is an horizontal 2D rotation matrix with ∂t ∂x ∂u ∂p ∂θ angle ωt, ω a real parameter, and φ the usual 2D where ζ and the αi’s are arbitrary functions of time. The stream function. dot symbol ( ˙ ) stands for time derivation. In the next section, we recall Nœther’s theorem which We can also consider symmetries (which are sometimes links symmetries to conservation laws. called equivalence transformations) of the form (t, x, u, p, θ, ν, κ) 7−→ (t,ˆ xˆ, uˆ, p,ˆ θ,ˆ ν,ˆ κˆ). (14) III. Symmetry and conservation laws Consider a system described by a Lagrangian action Such symmetries take a solution of (1) into a solution of Z other non-isothermal Navier–Stokes equations with differ- L[w] = L(y, w, w˙ ) (24) ent values of ν and κ. Applying condition (7) leads to the Ω infinitesimal generator and the corresponding Euler-Lagrange equation ∂ ∂ ∂ ∂ ∂ ∂ ∂L ∂L X9 = x· +u· +2p +θ +2ν +2κ . (15) − Div = 0, (25) ∂x ∂u ∂p ∂θ ∂ν ∂κ ∂w ∂w˙ From these generators, the symmetry group of (1) can P ∂f be identified. They are, respectively: where Div f = i for any smooth function f. ∂yi • the group of time translations: Nœther showed that to each Lie group which leaves the action L invariant corresponds a conservation law [5]. More (t, x, u, p, θ) 7−→ (t + , x, u, p, θ), (16) precisely, if • the group of pressure translations: X ·L = 0 (26) (t, x, u, p, θ) 7−→ (t, x, u, p + ζ(t), θ), (17) then Div C = 0. (27) • the group of pressure-temperature translations: The conserved quantity C is defined by (t, x, u, p, θ) 7−→ (t, x, u, p + βg x3, θ + a/ρ), (18) " #T ∂L • the group of horizontal rotations: C = Lξ + (ξ − γξ ) (28) y ∂γ w y (t, x, u, p, θ) 7−→ (t, Rx, Ru, p, θ) (19) where R is a 2D (constant) rotation matrix, j ∂wj where γk = . • the three-parameter group of generalized Galilean ∂yk Unfortunately, Nœther’s theorem does not apply di- transformations: rectly to equations (1) for these later do not have a (t, x, u, p, θ) 7−→ (t, x+α(t), u+α˙ (t), p+ρ xα¨(t), θ), Lagrangian structure. However, one can associate to (1) (20) adjoint equations such that, together, they derive from a “Bilagrangian” function. Indeed, consider, for example, Using the described theory, one can compute the Lie the isothermal case. Then, the Navier-Stokes equations symmetry groups of (33) which, for lack of space, are not ∂u 1 listed here but can be found in [6].  + u · ∇u + ∇p − ν∆u = 0 A scaling law of equations (33) verifies the condition  ∂t ρ (29) X · U = 0, X · Θ = 0 (34)  1  div u = 0 where X is the generator of one of the Lie symmetry and their adjoint equations groups of (33). Solving these equations leads to the fol-  ∂v 1 lowing scaling laws (among others). − + v · ∇u − u · ∇v − ∇s − ν∆v = 0  ∂t ρ • The linear law which can be found in the middle (30) region of a Couette flow and in the viscous sublayer  of a boundary flow:  div u = 0 derive from the Bilagrangian U1 = C1x2 + C3, Θ = C2x2 + C4. (35) ! 1 du dv 1 1 • The classical logarithmic wall law and its correspond- L = · v − u · + s − u · v div u 2 dt dt ρ 2 ing temperature law: (31) 1 U = C ln(x +b)+C , Θ = C [x +b]−1+C . (36) − p div v + ν tr(∇uT ∇v). 1 1 2 3 2 2 4 ρ • The exponential law: In these expressions, v and s are adjoint variables. With this extension of the notion of Lagrangian system, U1 = C1 exp(Cx2) + C3, Θ = C2 exp(2Cx2) + C4. Nœther’s theorem can be applied to the Navier-Stokes (37) equations to find conservation laws. • The power law of velocity discovered by Oberlack In the next section, we show some applications of sym- [7] in the buffer region of a boundary layer and its metry analysis to the understanding and simulation of temperature counterpart: turbulence. a 2a−1 U1 = C1(x2 + b) + C3, Θ = C2(x2 + b) + C4. IV. Symmetry and turbulence (38) In the first subsection, we present the derivation of scal- As we now know, symmetries contain fundamental phys- ing laws of turbulent flows from symmetry. Then we show ical properties, such as conservation laws and wall laws. how to build symmetry-preserving turbulence models. It is then essential not to destroy them when developing A. Scaling laws of non-isothermal fluid flows a model whether it be a theoretical or a numerical one. Consider a turbulent channel fluid flow. We decompose In the next subsection, we present symmetry-preserving the velocity, pressure and temperature into mean and models for non-isothermal fluid flows, and, after that, we fluctuating parts such that the mean parts depend only show how to develop a symmetry-preserving numerical models. on x2 (wall normal coordinate): 0 u = u + U(x2), B. Turbulence modelling 0 p = p + P (x2) − Kx1, (32) A direct simulation of a fluid flow would cost too much 0 θ = θ + Θ(x2) because it requires a very fine grid. Hence, instead of K is a consatant pressure gradient in streamwise direction. solving equations (1), we consider the large-eddy simula- We assume that the mean flow is parallel to the walls. The tion approach [8] which consists in computing an (filtered) equations of the fluctuating quantities are then: approximation (u, p, θ) of (u, p, θ). It verifies the equations  0 0 2 !  ∂u ∂u dU1 ∂ U1 ∂u 1  i + U i + u0 δ − K + ν δ +  + div(u ⊗ u) + ∇p − div(τ − τ ) − βgθ e = 0  1 2 i1 2 i1  ∂t ρ s 3  ∂t ∂x1 dx2 ∂x2   !   ∂P ∂u0 u0 ∂p0 ∂2u0   i k i 0   − βgΘ δi2 + + − ν − βgθ δi2 = 0 ∂θ  ∂x ∂x ∂x ∂x2  2 k i k  + div(θu) − div(h − hs) = 0   ∂t   ∂θ0 ∂θ0u0 ∂2θ0 ∂2Θ ∂Θ ∂θ0   j 0  div u = 0.  + − κ − κ + u + U1 = 0  ∂t ∂x ∂x2 ∂x2 2 ∂x ∂x  j j 2 2 1 (39)   In these equations, τ = 2νS and τs = u ⊗ u − u ⊗ u are  0 respectively the molecular and the subgrid stress tensors, ∂uk  = 0. whereas h = κ∇θ and h = θu − θu are molecular and ∂xk s (33) the subgrid heat fluxes. τs and hs have to be modelled. Many models have been proposed in the literature but 15°C most of them destroy the symmetry properties of equations (1) [9]. So, we propose new ones using the symmetry approach. x It is clear that time translations (16), applied to 2 (t, x, u, p, θ), are symmetries of the filtered equations (39) x1 if τs and hs do not explicitly depend on t. It is also 35°C straightforward to check that the pressure-temperature translations, and the Galilean transformations remain Fig. 1. Geometry of the room symmetries of (39) if τs and hs depend only on S and = ∇θ. From the classical theory of isotropic functions T model [11] which seems very dissipative. The vertical and the invariance under scale symmetries, we deduce a velocity presents the same trend. Figures 3 report the class of symmetry preserving models

d d  d Adj S (T⊗T) 1 0.3 Experimental −τ = νF1S + νF2 + νF3 Invariant  s 3 Smagorinsky  ||S|| ||S|| 0.2  0.8  d d 0.1  [S(T⊗T)] S[(T⊗T)S] 0.6 + νF + νF , 0 5 u2 4 ||S||4 5 ||S|| (40) u1  0.4 -0.1   -0.2  2 0.2 Experimental  S S Invariant  -0.3 −hs = κ F6 + F7 + F8 2 T. Smagorinsky ||S|| ||S|| 0 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 x2 x1 The Fi are arbitrary functions of the invariant quantities 2 Fig. 2. Horizontal (left) and vertical (right) mean velocity profile at det S · S S · S v = , T , T T, T T (41) respectively x1 = 0.501m and x2 = 0.501m, at the half-width ||S||3 ||S||4 ||S||5 ||S||6 temperature profiles along a vertical and an horizontal which naturally arise from the symmetry requirements. d lines, passing through the middle of the room. It can be Superscript ( ) symbolizes the deviatoric operator and observed on these figures that the invariant model predicts Adj S the adjugate of S. the temperature behavior better than the Smagorinsky Many simpler models may be deduced from (40). One model does. However, both models under-estimate the 3 of them which vanishes at walls with a rate O(x2) (if x2 experimental measurements. This may not due to the is the wall normal coordinate) is the following: models but to the (well-known) bad control of boundary  " d # conditions during the experimentation. Indeed, some phe- d −v3 3 −v3 2 −v3 Adj S −τ = Cmν 2 − 2 e −9v e S + 3v e nomena such as radiation or the variation of temperature  s kSk at the wall are hard to take into account.   3 1.2 35  −v Experimental Experimental −hs = Ctκ(1 − e ) . Invariant Invariant T 1 Smagorinsky Smagorinsky (42) 30 0.8 A numerical test on model (42) is presented in the next 0.6 25 subsection. Temperature 0.4 Temperature 20 C. Numerical test 0.2 0 15 Consider an air flow in the ventilated room pre- 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 x2 x1 sented in figure 1 [10]. The dimension of the room is 1.04m×1.04m×0.7m. The inlet and outlet heights are Fig. 3. Mean temperature profiles at x1 = 0.501m (left) and at respectively 0.018 and 0.024m, and the inlet velocity is x2 = 0.501m (right), at the half-width 0.57m/s. The floor is heated at 35◦C while the other In the last section, we present how to preserve the walls are maintained at 15◦C. The Reynolds number, symmetry group of an equation at the discrete scale. based on the inlet height and the inlet velocity, is 678. Figure 2, left, shows that the horizontal velocity given by V. Symmetry-based numerical schemes the invariant model (42) fits very well the experimental Consider a differential equation data. The concordance is particularly striking near the walls. The maximum values near the floor and near the E(x, u) = 0 (43) ceilor are correctly predicted. This good agreement can on a domain Ω. The latter is discretized into discrete points be explained by the non-violation of the wall laws by the x = (x , ..., x ) verifying a relation invariant model. Almost everywhere, the invariant model 1 J provides better results than the classical Smagorinsky Φ( x ) = 0. (44) Φ is called the mesh. Similarly, u is discretized into a • and Galilean boosts: sequence u = (u1, ..., uJ ). We denote zj = (xj, uj) and g5 :(t, x, u) 7−→ (t, x + 5t, u + 5). (55) z = (z1, ..., zJ ). A discretization scheme of equation (43), with an accu- In practice, choosing a moving frame is equivalent to racy order (q1, ..., qM ), is a couple of functions (N, Φ) such deciding the values of the parameters i. that As an example, the described method yields the follow- N(z) = O (∆x1)q1 , ..., (∆xM )qM (45) ing invariant FTCS scheme [14], [15]: [1− (tn+1+ )]un+1+ (xn+1+ )−un Φ(z) = 0 (46) 4 1 j 4 j 2 j n+1 n+1 [1−4(t +1)] m t + 1 as soon as uj = u(xj) for all j = 1, ..., J. ∆x is the " # un − un un − 2un + un step size in the m-th direction, m = 1, ..., M. Φ has been n j+1 j−1 j+1 j j−1 +(uj + 5) n + 4 = ν n 2 . extended to z in (46) such that N and Φ has the same 2(xj+1 + 2 (xj+1 + 2) argument. This extension is also necessary when the mesh (56) changes with the values of u (adaptative mesh, ...). The moving frame corresponds to n n Starting from any usual scheme (N, Φ), it is possible, n n uj+1−uj−1 1 = −tj , 2 = −xj , 4 = − 2∆x with the use of moving frame, to derive a new scheme n n n (57) (N,˜ Φ)˜ which is invariant under the symmetry group of 5 = duj+1 + euj + duj−1 with e + 2d = −1. equation (43). The invariance of the scheme means: ( In a first test, we solve numerically equation (50) on Ne(z) = 0 ⇒ Ne(T (z)) = 0 (47) Ω = {(t, x) ∈ [0, 1] × [−2, 2]} with the exact solution Φ(e z) = 0 ⇒ Φ(e T (z)) = 0 − sinh( x ) for any Lie symmetry T of the equation. u(t, x) = 2ν (58) cosh( x ) + exp(− t ) Indeed, let G be a multi-parameter group G acting 2ν 4ν regularly and freely on a manifold M. A (right) moving and ν = 5.10−3. The space step size is 2.10−2. At each frame is a map ρ : M 7→ G verifying the equivariance time step, the frame of reference is shifted by a Galilean condition [12], [13]: translation ρ[T.y] = ρ[y]T −1, ∀y ∈ M, g ∈ G. (48) (t, x) 7→ (t, x + λt). (59) A fundamental theorem [12] then shows that when G is It is expected that u follows this shifting according to the symmetry group of the equation and ρ a moving frame (55). Figures 4, left, present, in the original frame of then the scheme (N,˜ Φ)˜ defined by reference, how the standard schemes react to this shifting. They show clearly that, when λ is high, the standard N˜(z) = N(ρ[z].z), Φ(˜ z) = Φ(ρ[z].z) (49) schemes produce locally important errors. In particular, is an invariant numerical scheme of the same order for the a blow-up arises with the FTCS scheme when λ = 1. On same equation. the contrary, the invariantized version of these schemes In what follows, we consider Burgers’equation present no oscillation, as can be observed on Figures 4, right. Consistency analysis shows that, as the original ∂u ∂u ∂2u + u = ν . (50) schemes are no longer consistent with the equation when ∂t ∂x ∂x2 λ 6= 0, introducing numerical error, independently of the and we compare some classical schemes, namely the stan- step sizes. As for them, the invariantized schemes respect dard Euler FTCS, the Lax–Wendroff and the Crank– the physical property of the equation and provide quasi- Nicholson schemes, to their invariant versions. The sym- identical solutions when λ changes. metry transformations of (50) are: Another numerical test was carried out with FTCS • time translations: scheme. The exact solution is a self-similar solution under projections (54): g1 :(t, x, u) 7−→ (t + 1, x, u), (51) 1 x • space translations: u(t, x) = x − tanh( ) . (60) t 2ν g2 :(t, x, u) 7−→ (t, x + 2, u), (52) It corresponds to a viscous chock. The chock tends to be • scaling transformations: entropic when ν becomes closer to 0. Figure 5 show the numerical solutions obtained with the standard and the 23 3 −3 g3 :(t, x, u) 7−→ (te , xe , ue ), (53) invariantized FTCS schemes at t = 2s, with ν = 10−2, −2 −2 • projections: ∆t = 5.10 and ∆x = 5.10 . It shows that the solution given by the invariantized scheme remains close to the t x g4 :(t, x, u) 7−→ , , (1 − 4t)u + 4x , 1−4t 1−4t exact solution whereas the original FTCS scheme presents (54) a poor performance close to the chock location. 10 4 0.0 0.0 1.0 0.5 2.0 1.0 3.0 [6] D. Razafindralandy, A. Hamdouni, and N. A. Sayed, “Lie- 8 3 symmetry group and modeling in non-isothermal fluid mechan- 6

2 ics,” Physica A: Statistical Mechanics and its Applications, vol. 4 u 391, no. 20, pp. 4624 – 4636, 2012. 2 1 [7] M. Oberlack, “A uniﬁed approach for symmetries in plane par-

0 0 allel turbulent shear ﬂows,” Proceedings in Applied Mathematics −2 and Mechanics, vol. 427, pp. 299–328, 2001. −1

−4 −2 −1 0 1 2 3 4 5 −2 −1 0 1 2 3 X [8] P. Sagaut, Large eddy simulation for incompressible flows. An X 0.0 4 0.0 2 0.5 1.0 introduction. 3rd ed., ser. Scientific Computation. Springer, 1.0 2.0 3.0 2006. 1.5 3 [9] D. Razafindralandy and A. Hamdouni, “Invariant subgrid mod- 1 2 elling in large-eddy simulation of heat convection turbulence,” 0.5

1 Theoretical and Computational Fluid Dynamics, 2007. 0 [10] W. Zhang and Q. Chen, “Large eddy simulation of natural 0 −0.5 and mixed convection airﬂow indoors with two simple ﬁltered −1 −1 dynamic subgrid scale models,” Numerical Heat Transfer, Part −2 −1 0 1 2 3 −2 −1 0 1 2 3 4 5 X X 0.0 4 0.0 A: Applications, vol. 37, no. 5, pp. 447–463, 2000. 0.5 1.0 1.0 2.0 2 3.0 [11] J. Smagorinsky, “General circulation experiments with the 3 1.5 primitive equations,” Monthly Weather Review, vol. 91, no. 3,

1 2 pp. 99–164, 1963.

0.5 [12] P. Olver, “Geometric foundations of numerical algorithms and 1 0 symmetry,” Applicable Algebra in Engineering, Communication

0 −0.5 and Computing, vol. 11, no. 5, pp. 417–436, 2001.

−1 −1 [13] ——, “Moving frames,” Journal of Symbolic Computation,

−2 −1 0 1 2 3 −2 −1 0 1 2 3 4 5 X X vol. 36, no. 3-4, pp. 501–512, 2003. [14] M. Chhay and A. Hamdouni, “A new construction for invari- Fig. 4. Approximate solution, with λ = 0, λ = 0.5 and λ = 1, at ant numerical schemes using moving frames,” Comptes Rendus t = 1s. Left: standard schemes, right: invariantized schemes. Top: Mécanique, vol. 338, pp. 97–101, 2010. FTCS, middle: Lax-Wendroﬀ, bottom: Crank-Nicholson [15] M. Chhay, E. Hoarau, A. Hamdouni, and P. Sagaut, “Com- parison of some lie-symmetry-based integrators,” Journal of Computational Physics, vol. 230, pp. 2174–2188, 2011. Since the invariant scheme has the same invariance property as the solution under projections, it does not produce any artiﬁcial error. This shows the ability of invariantized scheme to respect the physics of the equation.

0.8 0.8 EXACT EXACT SYM FTCS 0.6 0.6

0.4 0.4

0.2 0.2 u 0 0

−0.2 −0.2

−0.4 −0.4

−0.6 −0.6

−0.8 −0.8 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 X X Fig. 5. Numerical solutions with FTCS (left) and invariantized FTCS (right) ν = 10−2. Self-similar case.

VI. Conclusion Throughout this communication, we showed the impor- tance of Lie-symmetry group theory in engineering science. We also presented ways to preserve symmetries when developing theoretical models and numerical schemes. References [1] P. Olver, Applications of Lie groups to differential equations, ser. Graduate texts in mathematics. Springer-Verlag, 1986. [2] N. Ibragimov, CRC handbook of Lie group analysis of differential equations. Vol 1: Symmetries, exact solutions and conservation laws. CRC Press, 1994. [3] N. A. Sayed, A. Hamdouni, E. Liberge, and D. Razafindralandy, “The symmetry group of the non-isothermal navier-stokes equations and turbulence modelling,” Symmetry, vol. 2, 2010. [4] B. Cantwell, “Similarity transformations for the two- dimensional, unsteady, stream-function equation,” Journal of Fluid Mechanics, vol. 85, pp. 257–271, 1978. [5] E. Nœther, “Invariante Variationsprobleme,” in Königliche Gesellschaft der Wissenschaften, 1918, pp. 235–257.