RECENT PROGRESSES IN FLUID DYNAMICS RESEARCH Proceedings of the Sixth International Conference on Fluid Mechanics AIP Conf. Proc. Vol 1376, 645-647 (2011) c
2011 American Institute of Physics 978-0-7354-0936-1/$30.00
Hamilton-Pontryagin Principle for Incompressible Ideal Fluids Hiroaki Yoshimura1 , Franc¸ois Gay-Balmaz2 1 Department 2 Laboratoire
of Applied Mechanics and Aerospace Engineering, Waseda University, Tokyo 169-8555, Japan de m´et´eorologie dynamique Ecole Normale Sup´erieure, 24 Rue Lhomond 75005 Paris, France
Email:
[email protected],
[email protected] Abstract We develop the Hamilton-Pontryagin principle for Lagrangians with advective parameters, which yields an implicit analogue of Euler-Poincar´e equations with advective parameters. Then, we derive the reduced HamiltonPontryagin principle and illustrate it with the example of incompressible ideal fluids, where the configuration space is given by the group of (volume preserving) diffeomorphisms. Incorporating pressure and momentum densities as Lagrange multipliers into the Hamilton-Pontryagin principle, we finally show that the dynamics of incompressible ideal fluids can be effectively formulated in the context of implicit Euler-Poincar´e equations. Key words: Hamilton-Pontryagin Principle, Implicit Euler-Poincar´e Equations, Advective Parameters
HAMILTON-PONTRYAGIN PRINCIPLE Let L be a right invariant Lagrangian on the tangent bundle T G of a Lie group G and let l : g → R be the reduced Lagrangian given by l := L|g, where we g = Te G ⊂ T G is the Lie algebra of G. Recall from [5] that the HamiltonPontryagin principle is given by the stationary condition of the action integral for curves (g(t), v(t), p(t)), t1 ≤ t ≤ t2 in T G ⊕ T ∗ G, namely, F(g, v, p) =
Z t2
[L(g(t), v(t)) + hp(t), g(t) ˙ − v(t)i] dt.
t1
Under the fixed point conditions for the curve g(t), namely, δ g(t1 ) = δ g(t2 ) = 0, it follows that the implicit EulerLagrange equations can be derived as p=
∂L , ∂v
g˙ = v,
p˙ =
∂L . ∂g
By left invariance, we have L(g, v) = l(η ) and the reduced Hamilton-Pontryagin principle is given by
δ
Z t2 t1
[l(η (t)) + hµ (t), ξ (t) − η (t)i] dt = 0,
(1)
where ξ = gg ˙ −1 , η = vg−1 and µ = pg−1 . By taking variations in Eq.(1) with δ ξ (t) = ζ˙ (t) − [ξ (t), ζ (t)], where ζ = (δ g)g−1 so that ζ (t) is an arbitrary curve in g satisfying ζ (t1 ) = ζ (t2 ) = 0, the stationary condition induces the implicit Euler-Poincar´e equations as
µ=
δl , δη
ξ = η,
µ˙ = −ad∗ξ µ .
IMPLICIT EULER-POINCARE´ EQUATIONS WITH ADVECTIVE PARAMETERS Following the approach of [4], we now construct the Hamilton-Pontryagin principle for Lagrangians with advective parameters. Consider a right representation of Lie group G on the vector space W and let G act on the right on 645
T G ×W ∗ as (vg , a)h = (vg h, ah). Assuming that L : T G ×W ∗ → R is right G-invariant, we get the reduced Lagrangian l : g ×V ∗ → R defined by l(vg g−1 , a0 g−1 ) = L(vg , a0 ). The reduced Hamilton-Pontryagin principle is given by
δ
Z t1 t0
[l(η (t), a(t)) + hµ (t), ξ (t) − η (t)i] dt = 0,
˙ −1 and a ∈ W ∗ are given by where a(0) = a0 , a(t) = a0 g(t)−1 , and the variations of ξ = gg
δ ξ = η˙ − [ξ , η ] and δ a = −η a It yields the implicit Euler-Poincar´e equations with advective parameters as
µ=
δl , δη
ξ = η,
µ˙ = −ad∗ξ µ +
δl ⋄a δa
(2)
In the above, the diamond operation ⋄ : W ×W ∗ → g∗ is defined as ⋄ : W ×W ∗ → g∗ ;
(v, w) 7→ v ⋄ w := ρv∗ (w),
where ρv : g → W ; ξ 7→ ρv (ξ ) := vξ is the Lie algebra action associated to the representation of G on W . Note that the relation a(t) = a0 g(t)−1 yields the advection equation a˙ + aξ = 0.
MOTION OF INCOMPRESSIBLE FLUIDS It is widely known that Euler equations for ideal fluids can be obtained from variational principles [1,2]. An extended variational principle for ideal continuum was developed by [3], in which one can incorporate Clebsch potentials into Hamilton’s variational principle. We follow the mathematical notations used in [4]. Let D be a bounded domain in R3 with smooth boundary ∂ D and let G = Diff(D) be the diffeomorphism group of D, which is the configuration space. Points in D represent the material or Lagrangian points of fluid particles, denoted by X ∈D and a fluid motion is described as ηt or as η (t), which is a time-dependent curve in Diff(D) giving an evolutional sequence of diffeomorphism from the reference configuration to the current configuration in D as x(X ,t) := ηt (X ) = η (t) · X ∈ D, where x is the current Eulerian spatial points of the path. The Lagrangian or material velocity of the system along the motion ηt is defined by taking the time derivative of the Lagrangian trajectory keeping the particle label X fixed
∂ ηt (X ) ∂ ∂ x(X ,t) := X ηt (X ) = , ∂t ∂t ∂t
V (X ,t) :=
while the Eulerian or spatial velocity of the system along the path x(X ,t) := ηt (X ) is defined by taking the time derivative of the path keeping the Eulerian point x fixed as v(x,t) := V (X ,t) :=
∂ ηt (X ) = V (ηt−1 (x),t), ∂t x
where v is the time dependent vector field on D and vt (x) := v(x,t). Thus, we have the following relationship between the Lagrangian material velocity (η , η˙ ) and the Eulerian spatial velocity v: v = η˙ ◦ η −1 ,
i.e.,
vt = Vt ◦ ηt−1 ,
where Vt (X ) := V (X ,t).
IMPLICIT EULER EQUATIONS Let us consider the (reduced) Lagrangian l : X(D) ×W ∗ → R with an advective parameter ρ ∈ W ∗ , which is given by l(u, ρ ) =
Z
D
1 1 ρ u·u d 3 x = hρ u, ui , 2 2 646
where u ∈ X(D) is the Eulerian velocity. Then, we construct the reduced Hamilton-Pontryagin principle in Eulerian coordinates as
δ
Z t2 Z t1
D
[l(u, ρ ) + h1 − ρ , pi + hΠ, v − ui] d 3 x dt = 0.
In the above, we incorporate the incompressibility condition for the mass density ρ and the second order condition for a velocity vector field u on D into the variational principle by using the Lagrange multipliers. Note that variations of v = η˙ ◦ η −1 and ρ are given by
δ v = w˙ − [v, w] and
δ (ρ d 3 x) = −£w (ρ d 3 x) = −∇ · (ρ w) d 3 x,
where w = δ η ◦ η −1 ∈ X(D) and ρ ⊗ d 3 x ∈ W ∗ = Den(D) denotes the mass density. Note that p ∈ W and Π ⊗ d 3 x ∈ X(D)∗ are Lagrange multipliers, each of which indicates pressure and momentum density. Under the boundary conditions
δ η (X ,ti ) = 0, for X ∈ D;
δ η (X ,t) · n(X ) = 0, for all t1 ≤ t ≤ t2 ,
The second equality being a consequence of the fact that a diffeomorphism η of D necessarily preserves the boundary ∂ D. The implicit Euler-Poincar´e equations with advective parameters given in equation Eq.(2) induce the following equations of motion:
δl = ρ u : momentum density, δu (ii) ρ = 1 : incompressibility constraint, (iii) v = u : second-order vector field, ∂Π 1 2 ∗ (iv) + adv Π = ρ ∇ |u| − p : equations of motion. ∂t 2 (i) Π =
It goes without saying that the above set of equations are equivalent with Euler equations of motion for incompressible ideal fluids. Note that ρ = 1 yields the incompressibility constraint from the advection equation ρ˙ + div(ρ u) = 0.
CONCLUSION The approach described here extends easily to all the fluid models with advected quantities described in [4] such as MHD or geophysical fluids.
Acknowledgements We thank Jerry Marsden for his helpful comments and suggestions. The research of H. Y. is partially supported by JSPS Grant-in-Aid 23560269, JST-CREST and Waseda University Grant for SR 2010A-606.
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