Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma

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1 Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma Mircea Neagu Abstract. In this paper, using Riemann-Lagrange geometrical methods, we construct a geometrical model on 1-jet spaces for the study of multitime relativistic magnetized non-viscous plasma, characterized by a given energy-stress momentum distinguished d-) tensor. In that arena, we give the conservation laws and the continuity equations for multi-time plasma. The partial differential equations of the stream sheets the equivalent of stream lines in the classical semi-riemannian geometrical approach of plasma) for multi-time plasma are also written. M.S.C. 2010: 53B21, 53B40, 53C80. Key words: generalized multi-time Lagrange space, energy-stress-momentum d- tensor of multi-time plasma, conservation laws, continuity equations, PDEs of stream sheets. 1 Introduction During that so-called the radiation epoch, in which photons are strongly coupled with the matter, the interactions between the various constituents of the Universal matter include radiation-plasma coupling, which is described by the plasma dynamics. Although it is not traditional to characterize the radiation epoch by the dominance of plasma interactions, however, it may be also called the plasma epoch please see [5]). This is because the electromagnetic interaction dominates all the four fundamental physical forces electrical, magnetic, gravitational and nuclear). In the present days, the Plasma Physics is an well established field of Theoretical Physics, although the formulation of magnetohydrodynamics in a curved space-time is a relatively new development please see Punsly [11]). The MHD processes in an isotropic space-time are intensively studied by a lot of physicists. For example, the MHD equations in an expanding Universe are investigated by Kleidis, Kuiroukidis, Papadopoulos and Vlahos in [5]. Considering the interaction of the gravitational waves with the plasma in the presence of a weak magnetic field, Papadopoulos also investigates the relativistic hydromagnetic equations [10]. The electromagnetic-gravitational dynamics into plasmas with pressure and viscosity is studied by Das, DeBenedictis, Differential Geometry - Dynamical Systems, Vol.19, 2017, pp c Balkan Society of Geometers, Geometry Balkan Press 2017.

2 88 Mircea Neagu Kloster and Tariq in the paper [2]. In their paper, the authors derive the relativistic Navier-Stokes equations that govern plasma. It is important to note that all preceding physical studies are done on an isotropic four-dimensional space-time, represented by a semi- pseudo-) Riemannian space with the signature +, +, +, ). Consequently, the Riemannian geometrical methods are used as a pattern over there. Geometrically speaking, using the Finslerian geometrical methods, the plasma dynamics was extended on non-isotropic space-times by V. Gîrţu and Ciubotariu in the paper [3]. More general, after the development of Lagrangian geometry on tangent bundle, due to Miron and Anastasiei [6], the generalized Lagrange geometrical objects describing the relativistic magnetized plasma were studied by M. Gîrţu, V. Gîrţu and Postolache in the paper [4]. According to Olver s opinion [9], we appreciate that the 1-jet fibre bundle is a basic object in the study of classical and quantum field theories. For such a reason, using as a pattern the Miron-Anastasiei s geometrical ideas [6], the author of this paper recently developed in the paper [7] that so-called the multi-time Riemann-Lagrange geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, gravitational and electromagnetic geometrical theories. We would like to point out that the geometrical construction on 1-jet spaces exposed in the article [7] was initiated by Asanov in [1] and further developed by the author of this paper. In this geometrical context, the aim of this paper is to create a multi-time extension on 1-jet spaces of the geometrical objects that characterize plasma in semi-riemannian and Lagrangian approaches. Thus, we introduce the energy-stress-momentum d-tensor of the multitime plasma and we give the geometrical-physical equations which govern it. 2 The semi-riemannian geometrical approach. Plasma in isotropic space-times Let SR n = M n, φ ij x)) be a semi-riemannian manifold, M n is an n-dimensional smooth manifold, whose coordinates are x = x i ) i=1,n, and φ ij x) ia a semi-riemannian metric having a constant signature. From a physical point of view, φ ij x) play the role of gravitational potentials. Note that, throughout this paper, the Latin letters run from 1 to n and the Einstein convention of summation is assumed. Let us consider the Christoffel symbols of the semi-riemannian metric φ ij, which are given by 2.1) γ i jk = φim 2 φjm x k + φ km x j φ ) jk x m. The Christoffel symbols 2.1) produce the Levi-Civita covariant derivative T ij... kl...;p = T ij... kl... x p + T mj... kl... γmp i + Tkl... im... γmp j +... T ij... ml... γm kp T ij... km... γm lp..., T = T ij... kl... x) x i x j dxk dx l... is an arbitrary tensor on M.

3 Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma89 Remark 2.1. The Levi-Civita covariant derivative has the metrical properties φ ij;k = φ ij ;k = 0. To define the energy-stress-momentum tensor T = T ij x)dx i dx j, that characterize the relativistic magnetized non-viscous plasma, we need the following geometrical objects [2], [4]: 1. the unit velocity-field of a test particle, given by U = u i x) x i,, if we denote u i = φ im u m, then we have u i u i = 1. Note that, physically speaking, if V = v i x) / x i) is the fluid space-like velocity vector, then we have u i v i = φrs v r v s ; 2. the 2-form of the electric field)-magnetic induction) is given by H = H ij x)dx i dx j and the 2-form of the electric induction)-magnetic field) is given by G = G ij x)dx i dx j. Note that, in physical applications, one takes H = G = F/ µ 0, F is the electromagnetic field and µ 0 is the electromagnetic permeability constant. 3. the Minkowski energy tensor of the electromagnetic field inside the plasma is given by the tensor E = E ij x)dx i dx j, whose components are E ij = 1 4 φ ijh rs G rs + φ rs H ir G js, G rs = φ rp φ sq G pq. In order to obey the relativistic Lorentz equation of motion for a charged test particle, the following Lorentz condition is required [2]: 2.2) E m i;mu i = 0, E m i = φ mp E pi. Obviously, using the notations H m r = φ mp H pr and G r i = φ rs G si, then we have E m i = 1 4 δm i H rs G rs H m r G r i, δ m i is the Kronecker symbol.

4 90 Mircea Neagu In this physical context, the components of the energy-stress-momentum tensor of plasma are defined by please see [2], [3], [4]) 2.3) T ij = ρ + p ) c 2 u i u j + pφ ij + E ij, c = const. is the speed of light, p = px) is the hydrostatic pressure and ρ = ρx) is the proper mass density of plasma. In the Riemannian framework of plasma, it is postulated that the following conservation laws for the components 2.3) are true: 2.4) T m i;m = 0, Ti m = φ mp T pi = ρ + p ) c 2 u m u i + pδi m + Ei m. By direct computations, the conservation equations 2.4) become 2.5) [ρ + p ) c 2 u m] u i + ρ + p ) ;m c 2 u m u i;m + p,i φ ir F r = 0, p,i = p/ x i and F r = φ rs Es;m m is the Lorentz force. Contracting the conservation equations 2.5) with u i and taking into account the Lorentz condition 2.2), we find the continuity equation of plasma, namely 2.6) [ρ + p ) c 2 u m] + p,m u m = 0, ;m we also used the equalities 0 = u i u i ;m = 1 2 ui u i),m = u i;mu i, the comma symbol,m representing the partial derivative / x m. Replacing the continuity law 2.6) into the conservation equations 2.5), we find the relativistic Euler equations for plasma, namely 2.7) ρ + p ) c 2 u i;m u m p,m u m u i δi m ) φ im F m = 0. If we put now u m = dx m /ds into Euler equations 2.7), we find out the equations of the stream lines of plasma, which are given by the second order DE system d 2 x k ] [γ ds 2 + krm c2 dx r dx m p + ρc 2 δk r p,m ds ds = c 2 [ F k p + ρc 2 φ km ] p,m, s is the natural parameter of the smooth curve c = x k s) ) k=1,n. 3 Generalized Lagrangian geometrical approach. Plasma in non-isotropic space-times Let GL n = M n, g ij x, y), N i j x, y)) be a generalized Lagrange space for more details, please see Miron and Anastasiei [6]). Let us consider that the tangent bundle

5 Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma91 T M, as smooth manifold of dimension 2n, has the local coordinates x i, y i). i=1,n Then, g ij x, y) is a metrical d-tensor on T M, which is symmetrical, non-degenerate and has a constant signature on T M\{0}. The local coefficients Nj i x, y) are the components of a nonlinear connection N on T M. The nonlinear connection N = ) Nj i produces on T M the following dual adapted bases of d-vectors and d-covectors: { } δ δx i, { y i X T M), dx i, δy i} X T M), δ δx i = x i N i m y m, δyi = dy i + N i mdx m. Note that the d-tensors on the tangent bundle T M behave like classical tensors. For example, on T M we have the global metrical d-tensor G = g ij dx i dx j + g ij δy i δy j, which is usually endowed with the physical meaning of non-isotropic gravitational potential. Following the geometrical ideas of Miron and Anastasiei from [6], the generalized Lagrange space GL n produces the Cartan canonical N-linear connection CΓN) = L i jk, C i jk), 3.1) L i jk = gim 2 C i jk = gim 2 δgjm δx k gjm y k + δg km δx j + g km y j δg ) jk δx m, g jk y m ). Further, the Cartan linear connection CΓN), given by 3.1), induces the horizontal h ) covariant derivative D ij... kl... p = δdij... kl... δx p and the vertical v ) covariant derivative kl... p = Dij... kl... y p D ij... is an arbitrary d-tensor on T M. + D mj... kl... Li mp + D im... kl... L j mp +... D ij... ml... Lm kp D ij... km... Lm lp... + D mj... kl... Ci mp + D im... kl... C j mp +... D ij... ml... Cm kp D ij... km... Cm lp..., D = D ij... δ kl... x, y) δx i y j dxk δy l... Remark 3.1. The Cartan covariant derivatives produced by CΓN) have the metrical properties g ij k = g ij k = 0, g ij k = g ij k = 0.

6 92 Mircea Neagu For the study of relativistic magnetized non-viscous plasma in the non-isotropic space-time GL n, one uses the following geometrical objects [4]: 1. the unit velocity-d-field of a test particle is given by U = u i x, y) y i,, if we use the notation ε 2 = g pq y p y q > 0, then we put u i = y i /ε. Obviously, we have u i u i = 1, u i = g im u m ; 2. the distinguished 2-form of the electric field)-magnetic induction) is given by H = H ij x, y)dx i dx j ; 3. the distinguished 2-form of the electric induction)-magnetic field) is given by G = G ij x, y)dx i dx j ; 4. the Minkowski energy d-tensor of the electromagnetic field inside the nonisotropic plasma is given by E = E ij x, y)dx i dx j + E ij x, y)δy i δy j. The Minkowski energy adapted components are defined by E ij = 1 4 g ijh rs G rs + g rs H ir G js, G rs = g rp g sq G pq, and they must verify the Lorentz conditions 3.2) E m i m ui = 0, E m i m u i = 0, E m i = g mp E pi. If we denote H m r = g mp H pr and G r i = grs G si, then we have E m i = 1 4 δm i H rs G rs H m r G r i. The energy-stress-momentum d-tensor, that characterize the relativistic magnetized non-viscous plasma in a non-isotropic space-time, is defined by the distinguished tensor [4] T = T ij x, y)dx i dx j + T ij x, y)δy i δy j, whose adapted components are 3.3) T ij = ρ + p c 2 ) u i u j + pg ij + E ij, c = constant, p = px, y) and ρ = ρx, y) have the similar physical meanings as in the semi-riemannian case. In the Lagrangian framework of plasma, one postulates that the following conservation laws for the components 3.3) are true [4]: 3.4) T m i m = 0, T m i m = 0,

7 Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma93 Ti m = g mp T pi = ρ + p ) c 2 u m u i + pδi m + Ei m. By direct computations, the conservation equations 3.4) become [ρ + p ) ] c 3.5) 2 u m u i + ρ + p ) m c 2 u m h u i m + p,,i g irf r = 0, [ρ + p ) ] c 2 u m m u i + ρ + p ) c 2 u m v u i m + p #i g irf r = 0, p,,i = δp/δx i, p #i = p/ y i and F h r = g rs Es m m F v r = g rs Es m m, respectively) is the horizontal vertical, respectively) Lorentz force. Contracting the conservation equations 3.5) with u i and taking into account the Lorentz conditions 3.2), we find the continuity equations of plasma in a non-isotropic medium: [ρ + p ) ] c 2 u m + p,,m u m = 0, 3.6) we also used the equalities m [ρ + p c 2 ) u m ] m + p #m u m = 0, 0 = u i u i m = 1 2 ui u i),,m = u i mu i, 0 = u i u i m = 1 ui u i) 2 #m = u i m u i, the symbols,,m and #m being the derivative operators δ/δx m and / y m. Replacing the continuity laws 3.6) into the conservation equations 3.5), we find the relativistic Euler equations for non-isotropic plasma, namely ρ + p ) 3.7) c 2 u i m u m p,,m u m u i δi m) g h imf m = 0, ρ + p ) c 2 u i m u m p #m u m u i δi m) g v imf m = 0. If we take now y m = dx m /dt, then we have u m = 1 ε 0 dx m dt = dxm ds, ε2 0 = g ij x, dx/dt) dxi dt dx j dt. Introducing this u m into Euler equations 3.7), we obtain the equations of the stream lines for non-isotropic plasma, which are given by the following second order DE systems: horizontal stream line DEs: d 2 x k ds N k m ε 0 N r m 2 [L krm c2 p + ρc 2 δk r p,,m dx m ds N p mg pr ε 0 dx r ds dx m ds g pq dx p dx q dx m dx k y r ds ds ds ds ; ] dx r ds dx k ds dx m ds = c 2 [ hf k p + ρc 2 g km p,,m] +

8 94 Mircea Neagu vertical stream line DEs: ] [C krm c2 dx r dx m p + ρc 2 δk r p #m ds ds = c 2 [ vf k p + ρc 2 g km p #m] g pq dx p dx q dx r dx k 2 y r ds ds ds ds. Remark 3.2. If the metrical d-tensor g ij x, y) is Finslerian one, that is we have g ij x, y) = 1 2 F 2 2 y i y j, F : T M R + is a Finslerian metric, then the DEs of stream lines of plasma in non-isotropic spaces reduce to horizontal stream line DEs: d 2 x k ] [L ds 2 + krm c2 dx r p + ρc 2 δk r p,,m ds + 2 [G k F 2 g pr G p dxr dx k ] ; ds ds vertical stream line DEs: p #m [ g mk dxm ds dx m ds = c 2 [ hf k p + ρc 2 g km p,,m] + dx k ] = F v k, ds, if the generalized Christoffel symbols of g ij x, y) are Γ i jkx, y) = gim 2 gjm x k + g km x j g ) jk x m, then we have G k = 1 2 Γk pqx, y)y p y q. 4 The Riemann-Lagrange geometrical approach. Multi-time plasma Let us consider that T p, h αβ t)) is a Riemannian manifold of dimension p, whose local coordinates are t α ) α=1,p. Suppose that the Christoffel symbols of the Riemannian metric h αβ t) are κ γ αβ t). Let J 1 T, M) be the 1-jet space it has the dimension p + n + pn) whose local coordinates are t α, x i, x i α). These transform by the rules t α = t α t β ) x i = x i x j ) x i α = xi x j t β t α xj β,

9 Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma95 det t α / t β) 0 and det x i / x j) 0. Note that, throughout this work, the Greek letters run from 1 to p, and the Latin ) letters run from 1 to n. Let GML n p = J 1 T, M), G α)β) i)j) = h αβ g ij be a multi-time generalized Lagrange space for more details, please see Neagu [7]), g ij t γ, x k, xγ) k is a metrical d- tensor on J 1 T, M), which is symmetrical, non-degenerate and has a constant signature. Let us consider that GML n p is endowed with a nonlinear connection having the form [7] ) Γ = M i) α)β = κγ αβ xi γ, N i) α)j. The nonlinear connection Γ produces on J 1 T, M) the following dual adapted bases of d-vectors and d-covectors: { } δ δt α, δ δx i, X J 1 { T, M)), dt α, dx i, δx i } α X J 1 T, M)), x i α δ δt α = t α + κγ αµx m γ x m µ, δ δx i = x i N m) µ)i x m µ, δx i α = dx i α κ γ αµx i γdt µ + N i) α)m dxm. Note that the d-tensors on the 1-jet space J 1 T, M) also behave like classical tensors. For example, on the 1-jet space J 1 T, M) we have the global metrical d-tensor G = h αβ dt α dt β + g ij dx i dx j + h αβ g ij δx i α δx j β, which may be endowed with the physical meaning of non-isotropic multi-time gravitational potential. It follows that G has the adapted components h αβ, for A = α, B = β g ij, for A = i, B = j G AB = h αβ g ij, for A = α) i), 0, otherwise. B =β) j) Following the geometrical ideas of Asanov [1] and Neagu [7], the preceding geometrical ingredients lead us to the Cartan canonical Γ-linear connection ) CΓ = κ γ αβ, Gk jγ, L i jk, C iγ) jk), 4.1) G k jγ = gkm 2 δg mj δt γ, Li jk = gim 2 C iγ) jk) = gim 2 δgjm + δg km δx j ) δx k g jm x k + g km γ x j g jk γ x m γ. δg ) jk δx m,

10 96 Mircea Neagu In the sequel, the Cartan linear connection CΓ, given by 4.1), induces the T - horizontal h T ) covariant derivative D αij)ν)... γkβ)l).../ε = δdαij)ν)... γkβ)l)... + D µij)ν)... γkβ)l)... κα µε + D αmj)ν)... γkβ)l)... G i mε + δt ε +D αim)ν)... γkβ)l)... Gj mε + D αij)µ)... γkβ)l)... κν µε +... D αij)ν)... µkβ)l)... κµ γε D αij)ν)... γmβ)l)... Gm kε D αij)ν)... γkµ)l)... κµ βε Dαij)ν)... γkβ)m)... Gm lε..., the M-horizontal h M ) covariant derivative D αij)ν)... γkβ)l)... p = δdαij)ν)... γkβ)l)... and the vertical v ) covariant derivative + D αmj)ν)... γkβ)l)... L i mp + D αim)ν)... γkβ)l)... Lj mp +... δx p D αij)ν)... γmβ)l)... Lm kp D αij)ν)... γkβ)m)... Lm lp... D αij)ν)... γkβ)l)... ε) p) = Dαij)ν)... γkβ)l)... x p ε + D αmj)ν)... γkβ)l)... C iε) mp) + Dαim)ν)... γkβ)l)... Cjε) mp) +... D αij)ν)... γmβ)l)... Cmε) kp) Dαij)ν)... γkβ)m)... Cmε) lp)..., D = D αij)ν)... γkβ)l)... tλ, x r, x r λ) δ δt α δ δx i x j β dt γ dx k δx l ν... is an arbitrary d-tensor on J 1 T, M). Remark 4.1. The Cartan covariant derivatives produced by CΓ have the metrical properties h αβ/γ = h αβ /γ = 0, h αβ k = h αβ k = 0, h αβ γ) k) = hαβ γ) k) = 0, g ij/γ = g ij /γ = 0, g ij k = g ij k = 0, g ij γ) k) = gij γ) k) = 0. For the study of the relativistic magnetized non-viscous plasma dynamics, in a Riemann-Lagrange geometrical multi-time approach, we use the following geometrical objects: 1. the unit multi-time velocity-d-field of a test particle is given by U = u i αt γ, x k, x k γ) x i, α, if we take ε 2 = h µν g pq x p µx q ν > 0, then we put u i α = x i α/ε. Obviously, we have h αβ u iα u i β = 1, u iα = g im u m α ;

11 Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma97 2. the distinguished multi-time 2-form of the electric field)-magnetic induction) is given by H = H ij t γ, x k, x k γ)dx i dx j ; 3. the distinguished multi-time 2-form of the electric induction)-magnetic field) is given by G = G ij t γ, x k, x k γ)dx i dx j ; 4. the multi-time Minkowski energy d-tensor of the electromagnetic field inside the multi-time plasma is given by E = E ij t γ, x k, x k γ)dx i dx j + h ην E ij t γ, x k, x k γ)δx i η δx j ν. The multi-time Minkowski energy adapted components are defined by the similar formulas E ij = 1 4 g ijh rs G rs + g rs H ir G js, G rs = g rp g sq G pq. Furthermore, we suppose that the multi-time Minkowski energy adapted components verify the multi-time Lorentz conditions 4.2) E m i m ui α = 0, E m i µ) m) ui µ = 0, E m i = g mp E pi. Obviously, if we use the notations H m r = g mp H pr and G r i = grs G si, we obtain E m i = 1 4 δm i H rs G rs H m r G r i. In our Riemann-Lagrange geometrical approach, the multi-time plasma is characterized by the energy-stress-momentum d-tensor defined by T = T ij t γ, x k, x k γ)dx i dx j + h ην T ij t γ, x k, x k γ)δx i η δx j ν, 4.3) T ij = ρ + p c 2 ) h αβ u iα u jβ + pg ij + E ij. The entities c = constant, p = pt γ, x k, x k γ) and ρ = ρt γ, x k, x k γ) have the multi-time extended physical meanings of their analogous entities from the semi-riemannian framework. Note that the adapted components of the energy-stress-momentum d- tensor T of multi-time plasma are given by T ij, for C = i, F = j 4.4) T CF = h ην T ij, for C = η) i), 0, otherwise. F =ν) j) In the multi-time Riemann-Lagrange framework of plasma, we postulate that the following multi-time conservation laws for the components 4.3) and 4.4) are true: { } 4.5) TA:M M = 0, A α, i, α) i),

12 98 Mircea Neagu the capital Latin letters A, M,... are indices of kind α, i or α) i), :M represents one of the local covariant derivatives h T, h M or v, and we have TA M = G MD T DA = 0, otherwise. Obviously, the d-tensor T m i T m i = g mp T pi = T m i, for A = i, M = m δ α µt m i, for A = α) i), M =µ) m) is given by the formula ρ + p c 2 ) h αβ u m α u iβ + pδ m i + E m i. The multi-time conservation laws 4.5) reduce to the multi-time conservation equations 4.6) T m i m = 0, T m i µ) m) = 0. By direct computations, the multi-time conservation equations 4.6) become 4.7) [ h αβ ρ + p ) ] c 2 u m α u iβ + ρ + p ) m c 2 h αβ u m h α u iβ m + p,,i g irf r = 0, ] µ) h αβ [ ρ + p c 2 ) u m α +p µ) m) #i) g irf rµ = 0, p,,i = δp/δx i, p µ) #i) v u iβ + ρ + p ) c 2 h αβ u m α u iβ µ) m) + = p/ xi µ and F h r = g rs Es m m is the multi-time horizontal Lorentz force; F v rµ = g rs Es m µ) m) is the multi-time vertical Lorentz d-tensor force. Contracting the multi-time conservation equations 4.7) with u i µ and taking into account the Lorentz conditions 4.2), we find the continuity equations of multi-time plasma, namely [ h αβ ρ + p ) ] c 2 u m α u iβ u i µ + ρ + p ) m c 2 h αβ u m α u iβ m u i µ + p,,m u m µ = 0, ] µ) h αβ [ ρ + p c 2 ) u m α m) If we take now x l η = x l / t η, then we have u iβ u i µ + ρ + p ) c 2 h αβ u m α u iβ µ) m) ui µ + p µ) #m) um µ = 0. u l η = xl η ε 0, ε 2 0 = h αβ t)g ij t γ, x k, x k γ)x i αx j β. Introducing this u l η into multi-time conservation equations 4.7), we obtain the second order PDEs of the stream sheets that characterize the multi-time plasma:

13 Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma99 horizontal stream sheet PDEs: [ h αβ ρ + p ) ] [ ] x m α c 2 x k β ε + ρ + p ) x k 0 m c 2 x m β α ε 0 [ ] hf = ε k 0 g km p,,m ; m = vertical stream sheet PDEs: [ h αβ ρ + p ) ] [ ] x m α µ) c 2 x k β ε + ρ + p ) x k 0 m) c 2 x m β α ε 0 [ ] vf = ε kµ 0 g km p µ) #m). µ) m) = Taking into account the local form of the h M and v covariant derivatives produced by the Cartan connection CΓ, the expressions of the PDEs of the stream sheets of multi-time plasma reduce to: horizontal stream sheet PDEs: { h αβ H m x m α x k β + 1 ρ + p ) [ ] ε 0 c 2 L k rmx r β N k) β)m x m α ρ + p ) [ ] } [ ] hf ε 0 c 2 L m rmx r α N m) α)m x k β = ε k 0 g km p,,m, H m = [ 1 ρ + p )] ε 0 c 2 + ρ + p ) [ ] 1,,m c 2 ; ε 0,,m vertical stream sheet PDEs: { h αβ V µ) m) xm α x k β + 1 ρ + p ) ] [n ε 0 c 2 δ αx µ k β + δ µ β xk α ε 0 ρ + p c 2 ) [ C kµ) mr) xm β n = dim M and 5 Conclusions V µ) m) = [ 1 ε 0 ρ + p c 2 ) ] µ) ] } [ ] vf + C mµ) rm) xk β x r α = ε kµ 0 g km p µ) #m), #m) + ρ + p ) [ ] µ) 1 c 2. ε 0 #m) The Riemann-Lagrange geometrical theory upon the Multi-Time Plasma Physics may be applied for a lot of interesting multi-time generalized Lagrange spaces with physical connotations [7]:

14 100 Mircea Neagu 5.1 The geometrical model GRGML n p for multi-time General Relativity and Electromagnetism This generalized multi-time Lagrange space is characterized by the fundamental metrical d-tensor G α)β) i)j) = h αβ t γ )e 2σtγ,x k,x k γ ) φ ij x k ) and the nonlinear connection Γ = M i) α)β = κµ αβ xi µ, N i) α)j = γi jmx m α 5.2 The geometrical model RGOGML n p for multi-time Relativistic Optics This generalized multi-time Lagrange space is characterized by the fundamental metrical d-tensor { [ ] } G α)β) i)j) = h αβ t γ ) φ ij x k 1 ) + 1 nt γ, x k, x k Y i Y j γ) and, again, by the nonlinear connection Γ, Y i = φ im x k )x m µ X µ t γ ). 5.3 The geometrical model EDML n p for multi-time Electrodynamics This generalized multi-time Lagrange space is characterized by the Lagrangian function for more details, please see [8]) L ED = h αβ t γ )φ ij x k )x i αx j β + U α) i) tγ, x k )x i α + Φt γ, x k ), which produces the fundamental metrical d-tensor ). i)j) = 1 2 L ED 2 x i α x j β G α)β) = h αβ φ ij and the nonlinear connection whose components are M i) α)β = κµ αβ xi µ and N i) α)j = γi jmx m α + h αµφ im µ) U m) 4 x j U µ) j) x m. 5.4 The geometrical model BSML n p for Bosonic Strings This is the multi-time Lagrange space corresponding to the multi-time Lagrangian function L BS = h αβ t γ )φ ij x k )x i αx j β. In this particular case, we have the fundamental metrical d-tensor i)j) = 1 2 L BS 2 x i α x j = h αβ φ ij β G α)β)

15 Riemann-Lagrange geometric dynamics for the multi-time magnetized non-viscous plasma101 and the canonical nonlinear connection Γ. Moreover, the Cartan canonical connection has the following simple form: ) C Γ = κ γ αβ, 0, γi jk, 0. It follows that, for the multi-time Lagrange space BSML n p, the PDEs of the stream sheets for multi-time plasma simplify as follows: horizontal stream sheet PDEs: h αβ H m x m α x k β = ε 0 [ hf k g km p,,m ] ; vertical stream sheet PDEs: { h αβ V µ) m) xm α x k β + 1 ρ + p ) [n ε 0 c 2 δ αx µ k β + δ µ β α] } xk = [ ] vf = ε kµ 0 g km p µ) #m). Open problem. Does there exist a real physical interpretations for our multi-time Riemann-Lagrange geometric dynamics of plasma? Acknowledgements. A version of this paper was presented at The X-th International Conference Differential Geometry and Dynamical Systems DGDS-2016), 28 August - 3 September 2016, Mangalia, Romania. References [1] G.S. Asanov, Jet extension of Finslerian gauge approach, Fortschritte der Physik 38, ), [2] A. Das, A. DeBenedictis, S. Kloster, N. Tariq, Relativistic particle, fluid and plasma mechanics coupled to gravity, arxiv:math-ph/ v1 2005). [3] V. Gîrţu, C. Ciubotariu, Finsler spaces associated with a general relativistic magnetized plasma, Tensor N.S. 56, ), [4] M. Gîrţu, V. Gîrţu, M. Postolache, Geodesic equations for magnetized plasma in GL-spaces, BSG Proceedings 3, Geometry Balkan Press, Bucharest 1999), [5] K. Kleidis, A. Kuiroukidis, D.B. Papadopoulos, L. Vlahos, Magnetohydrodynamics and plasma cosmology, arxiv:gr-qc/ v1 2005). [6] R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Academic Publishers, Dordrecht, [7] M. Neagu, Generalized metrical multi-time Lagrange geometry of physical fields, Forum Mathematicum ), [8] M. Neagu, The geometry of autonomous metrical multi-time Lagrange space of electrodynamics, International Journal of Mathematics and Mathematical Sciences 29, No ), 7-16.

16 102 Mircea Neagu [9] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, [10] D.B. Papadopoulos, Plasma waves driven by gravitational waves in an expanding universe, arxiv:gr-qc/ v1 2002). [11] B. Punsly, Black Hole Gravitohydromagnetics, Springer-Verlag, Berlin, Author s address: Mircea Neagu Transilvania University of Braşov, Department of Mathematics and Informatics, 50 Blvd. Iuliu Maniu, Braşov , Romania. mircea.neagu@unitbv.ro