Stability Subject to Dynamical Accessibility

Transcription

1 Stability Subject to Dynamical Accessibility P. J. Morrison Department of Physics and Institute for Fusion Studies The University of Texas at Austin morrison/ In Memory of Eliezer Hameiri NYU September 21, 2018

2 Hamilton s Equations Phase Space Coordinates: (q, p) = (q 1, q 2,..., q N ; p 1, p 2,..., p N ), Hamiltonian: H(q, p) s.t. phase space R Hamiltons Equations: ṗ i = H q i, qi = H p i, i = 1, 2,..., N Equivalent Form z = (q, p): ż α = J αβ c H z β =: {zα, H}, α, β = 1, 2,..., 2N (J αβ c ) = ( 0N I N I N 0 N ),

3 Geometry Dynamics takes place in phase space, Z (needn t be T Q), a differential manifold endowed with a closed, nondegenerate 2-form ω. A patch has canonical coordinates z = (q, p). Hamiltonian dynamics flow on symplectic manifold: i X ω = dh Poisson tensor (J c ) is bivector inverse of ω, defining the Poisson bracket {f, g} = df, J c (dg) = ω(x f, X g ) = f g z αjαβ c z β, α, β = 1, 2,... 2N Flows generated by Hamiltonian vector fields Z H = JdH, H a 0- form, dh a 1-form. Poisson bracket = commutator of Hamiltonian vector fields etc. Early refs.: Jost, Mackey, Souriau, Arnold, Abraham &Marsden

4 Noncanonical Hamiltonian Dynamics Sophus Lie (1890) Noncanonical Coordinates: ż a = J ab H z b = {za, H}, Poisson Bracket Properties: {f, g} = f z ajab (z) g z b, a, b = 1, 2,... M antisymmetry {f, g} = {g, f}, Jacobi identity {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0 G. Darboux: detj 0 = J J c Canonical Coordinates Sophus Lie: detj = 0 = Canonical Coordinates plus Casimirs J J d = 0 N I N 0 I N 0 N M 2N.

5 Flow on Poisson Manifold Definition. A Poisson manifold M is differentiable manifold with bracket {, } : C (M) C (M) C (M) st C (M) with {, } is a Lie algebra realization, i.e., is i) bilinear, ii) antisymmetric, iii) Jacobi, and iv) consider only Leibniz, i.e., acts as a derivation. Flows are integral curves of noncanonical Hamiltonian vector fields, Z H = JdH. Because of degeneracy, functions C st {f, C} = 0 for all f C (M). Called Casimir invariants (Lie s distinguished functions.)

6 Poisson Manifold M Cartoon Degeneracy in J Casimirs: {f, C} = 0 f : M R Lie-Darboux Foliation by Casimir (symplectic) leaves: Leaf vector fields, Z f = {z, f} = Jdf are tangent to leaves.

7 Lie-Poisson Brackets Matter models in Eulerian variables: where c ab c J ab = c ab c z c are the structure constants for some Lie algebra. Examples: 3-dimensional Bianchi algebras for free rigid body, Kida vortex, rattleback. (cf. Tokeida, Yoshida) Infinite-dimensional theories: Ideal fluid flow, MHD, shearflow, extended MHD, Vlasov-Maxwell, etc.

8 Lie-Poisson Geometry Lie Algebra: g, a vector space with [, ] : g g g, antisymmetric, bilinear, satisfies Jacobi identity Pairing: with g vector space dual to g, : g g R Lie-Poisson Bracket: {f, g} = z, [ f z, g ] z, z g, f z g

9 Magnetohydrodynamics Equations of Motion: Force Density Entropy Magnetic Field ρ v t ρ t s t B t = ρv v p + 1 c J B = (ρv) = v s = E = (v B) Energy: H = D d3 x ( ρ v 2 /2 + ρu(ρ, s) + B 2 /2 ) Thermodynamics: p = ρ 2 U ρ T = U s

10 Noncanonical Lie-Poisson Bracket (pjm & Greene 1980): {F, G} = D d3 x M i δf δm j δg x j δg δm i δm j + ρ δf δm δg δρ δg δm δf + σ δρ [ δf + B δm δg δb δg δm δf ] δb [ ( ) δf + +B δg ( ) δg δm δb δm δf x j δm i δf δm δg δσ δg δm δf δσ δf ], δb Dynamics: ρ t = {ρ, H}, s t = {s, H}, v t = {v, H}, and B t = {B, H}. Densities: M := ρv σ := ρs

11 Hamiltonian Description of Idea MHD Lagrangian fluid variable description is a continuum version of particle mechanics and, consequently, canonical, while Eulerian fluid variables are noncanonical variables. Lagrange to Euler is a reduction.

12 Eulerian Variable Description

13 Eulerian Variable Description Observables {s(r, t), ρ(r, t), v(r, t), B(r, t)} where r D R 3.

14 Eulerian Variable Description Observables {s(r, t)ρ(r, t), v(r, t), B(r, t)} constitue a Field Theory.

15 Lagrangian Variable Description Assume a continuum of fluid particles or fluid elements and follow them around. Lagrange Mécanique Analytique (1788) Newcomb (1962) MHD

16 Lagrangian Variable Description

17 Lagrangian Variable Description Dynamical canonical variables {q(a, t), π(a, t)}.

18 Stability in General Finite Systems General system: ż i = V i (z), i = 1, 2,..., M, Equilibrium point, z e, satisfies V (z e ) = 0. Definition: The equilibrium point z e is said to be stable if, for any neighborhood N of z e there exists a subneighborhood S N of z e such that if z(t = 0) S then z(t) N for all time t > 0. p H = const. S q N

19 Stability in Canonical Hamiltonian Systems Lagrange s Theorem: For separable Hamiltonians, H = p 2 /2 + V (q), an equilibrium point p e = 0 and q e a local minimum of V is stable. Converse? Dirichlet s Theorem: For general Hamiltonians, H(q, p), an equilibrium point H(q e, p e ) = 0 is stable if 2 H(z e )/ z i z j is definite. Note, H could an energy minimum or a maximum, e.g. for a localized vortex in fluid mechanics. Converse?

20 MHD δw Energy Principla Standard tool of MHD for stability. Bernstein et al.; Hain et al. Infinite-dimensional version of Lagrange s necessary and sufficient theorem. Flow in equilbrium considered by Frieman & Rotenberg. Sufficient condition is a version of Dirichlet s theorem.

21 Stability in Noncanonical Hamiltonian Systems What do Casimirs do? What does the cartoon geometry imply?

22 Stability in Noncanonical Hamiltonian Systems I. Energy-Casimir Stability: In noncanonical variables w ẇ α = {w α, H} = {w α, H + C} = J αβ (w) F w β = 0 Equilibrium variational principle: Linear Hamiltonian: δf = δ(h + C) = 0 δ 2 F = 2 F (w e ) w α wβ δwα δw β = H L definite stability Kruskal-Oberman (1958), Gardner, Arnold, Marsden, pjm, etc. II. Dynamically Accessible Stability: Constrained variations δw α = J αβ (w e )g β δ 2 F da definite stability pjm & Pfirsch (1989)

23 MHD Dynamically Stability Generator: G = x (g 1 M + g 2 σ + g 3 ρ + g 4 B) where M = ρv and σ = ρs. Then {χ, G} Energy Functional: with δ 2 H da [g] = δρ da = (ρg 1 ) δm da = ρ g 3 + ( M) g 1 + M g 1 + (M g 1 ) + σ g 2 + B ( g 4 ) δσ da = (σg 1 ) δb da = (B g 1 ) d 3 x ρ δv da g 1 v + v g δwla [g 1 ] δv da = (g 3 + g 1 v) g 1 ( v) + σ ρ g ρ B ( g 4) Hameiri: In certain relevant circumstances d 3 x ρ... 2 stabelizes!