Unimodularity and preservation of measures in nonholonomic mechanics

Transcription

1 Unimodularity and preservation of measures in nonholonomic mechanics Luis García-Naranjo (joint with Y. Fedorov and J.C. Marrero) Mathematics Department ITAM, Mexico City, MEXICO

2 ẋ = f (x), x M n, f smooth vector field on M n. Φ(t, x) Flow Φ(t, x) : = f (Φ(t, x)), Φ(0, x) =x. t M n orientable. ν(x) - non-vanishing differential n-form on M n. Look for a smooth invariant volume µ(x) ν(x). µ C (M n ), µ(x) > 0. µ(x) ν(x) = A Φ(t,A) µ(x) ν(x);

3 Liouville Equation d dt t=0 Φ(t,A) µ(x) ν(x) = A div ν (µf )(x) ν(x) Infinitesimal condition for measure preservation: div ν (µf )=0. In local coordinates PDE for µ. Existence of global solutions? Flow box theorem. ẋ 1 =1, ẋ 2 =0,...,ẋ n = 0.

4 Symplectic Hamiltonian systems M n is a symplectic manifold (n =2m). ω symplectic form. H C (M n ). ω(f, ) =dh( ) Then the measure ω m is preserved. In local canonical coordinates: q 0 I H = H, H = ṗ I 0 q, H. p div dq dp (f )= ω m = (const.) dq dp 2 H q i 2 H p i p i q i =0.

5 Homogeneous systems on vector spaces ẋ = f (x), x R n. f homogeneous of degree k N: i.e.f (λx) =λ k f (x). Kozlov 88: f preserves the smooth measure µ(x) dx iff it preserves the euclidean measure dx and µ(x) is a conserved quantity. Proof: Setµ(x) =e σ(x) div(e σ f )=0 e σ ( σ f +div(f )) = 0 σ Hence σ f =0anddiv(f ) = 0. f deg k = div(f ) deg k 1

6 Linear systems ẋ = Ax Measure preservation iff Trace(A) = 0.

7 Hydrodynamical Chaplygin Sleigh

8 Generic Dynamics of the Hydrodynamic Chaplygin Sleigh Interesting dynamics related to the non-existence of a preserved measure.

9 Mechanical Lie-Poisson systems Linear (almost) Poisson structure in R n : {F, G} =( F (x)) T π(x) G(x), (π(x)) αβ = C γ αβ x γ Skew-symmetry: C γ αβ = C γ βα Jacobi identity: C αδ C δ βγ + C γδ C δ αβ + C βδ C δ γα =0 R n is a Lie algebra: Hamiltonian system: [e α, e β ]=C γ αβ e γ Mechanical Hamiltonian: ẋ = π(x) H(x) = X H (x). Quadratic H(x) = 1 2 K αβ x α x β, K αβ symmetric positive definite

10 Mechanical Lie-Poisson systems Measure Preservation (Kozlov 88): ẋ α = C γ αβ K βδ x δ x γ div(x H (x)) = C γ αβ K βα x γ + Cαβ α K βδ x δ 0 There exists a smooth preserved measure iff C α αβ =0, β =1,...,n. If Jacobi identity holds: C α αβ =0, β =1,...,n Lie algebra is unimodular

11 Poisson structures and the modular vector field First suppose M n = R n. {F, G}(x) =( F (x)) T π(x) G(x) Skew-symmetry: π αβ = π βα Jacobi identity: π δα π βγ x δ Hamiltonian vector fields ẋ = π(x) H(x) :=X H (x); Take (euclidean) divergence + π δγ π αβ x δ div(x H (x)) = π αβ (x) H (x)+ x α x β + π δβ π γα x δ =0 ẋ α = π αβ (x) H x β (x) 0 2 H π αβ (x) (x) x α x β = M(x) H(x). modular vector field

12 More generally: {σ,h} div(e σ(x) X H (x)) = e σ(x) σ(x) X H (x) +div(x H (x)) = e σ(x) (M(x) X σ (x)) H(x) Definition: If M(x) is Hamiltonian = π is unimodular Unimodularity: Sufficient condition for the existence of an invariant measure. In certain cases (i.e. mechanical Lie-Poisson) this condition is also necessary. Remark: This definition of unimodularity and the consequences only depend on the skew-symmetry of π.

13 The modular class of a Poisson manifold If the Jacobi identity holds then the entries of M satisfy M γ π αβ x γ + π γα M β x γ π γβ M α x γ =0, L M π =0. M is a Poisson vector field. First Poisson cohomology group = {Vector fields that preserve π } {Hamiltonian vector fields} Representative of M is the modular class of π (Weinstein 96). Important objects in the study (topology, classification) of Poisson manifolds (Weinstein, Xu, Dufour, Grabowski, Lu, Evens,...) Unimodularity modular class is zero.

14 Summary: Unimodularity and invariant measures for (almost) Poisson Hamiltonian systems Remark: Discussion can be generalized to orientable (almost) Poisson manifolds. Unimodularity is an intrinsic global concept.

15 Kinetic mechanical equations of motion Phase space M n is a vector bundle over an orientable manifold Q. Local equations of motion: q i = ρ i α(q) H (q, p) p α ṗ α = ρ i α(q) H q i (q, p) C γ αβ (q)p H γ (q, p) p β H(q, p) = 1 2 K αβ (q)p α p β. K αβ (q) positive definite Skew-symmetry, Jacobi identity Candidates for invariant volumes are basic: µ(q) dq dp.

16 π(q, p) = 0 ρ(q) ρ(q) T C(q, p) ; C αβ (q, p) =C γ αβ (q)p γ Marrero 2010: Unimodularity Existence of invariant volume Unimodularity condition M = X σ for σ = σ(q). Locally: ρ i α q i (q)+c β βα (q) =ρi α(q) σ q i (q) for all α Difficult to verify!

17 Nonholonomic systems with symmetry Hamiltonian and constraints are invariant under the action of a Lie group. p a not compatible with group action p α divided into p A compatible with group action Technical details: rank condition, locality, connection. After reduction 0 ρ(q) 0 π(q, p) = ρ(q) T ; C C(q, p) αβ (q, p) =C γ αβ (q)p γ 0 H(q, p) = 1 2 K ab (q)p a p b + K AB (q)p A p B

18 Unimodularity condition M = X σ for σ = σ(q). Locally: ρ i a q i (q)+c β βa (q) =ρi a(q) σ q i (q) C β βa (q) =0 for all a for all A This unifies and generalizes all results existing in the literature Kozlov 88, Jovanović 98, Cantrijn et al 02, Zenkov, Bloch 03.

19 Rigid body with planar face rolling over sphere Necessary conditions for unimodularity: I 12 = I 23 = I 23 =0 (I 11 I 22 ) =0.

20 References Grabowski J. Modular classes of skew symmetric relations (2011), arxiv: Fedorov, Y. García-Naranjo L., Marrero J.C., Unimodularity and preservation of volumes in nonholonomic mechanics, In preparation.

21 References Grabowski J. Modular classes of skew symmetric relations (2011), arxiv: Fedorov, Y. García-Naranjo L., Marrero J.C., Unimodularity and preservation of volumes in nonholonomic mechanics, In preparation. Thanks