EULER EQUATION WITH GEOMETRIC CLEBSCH VARIABLES
8.2 Hamiltonian Formulation for the Euler Equation
Here we present that the Euler equation is the symplectic gradient flow of the following Hamiltonian onM:
EEuler: M→R, EEuler [ψ] = Z
M 1 2η∧?η
where η = ψ∗ϑ is the unique solenoidal 1-form for the equivalence class [ψ]. Physically,EEuler is the total kinetic energy.
Theorem8.1. Under the symplectic gradient flow ofEEuler on (M,σM), the vor- ticity Clebsch map s =Π [ψ] satisfies
˙
s+Lus =0 (8.1)
whereu= η] is the velocity vector field.
Proof. Let us take the variation ofEEulerwith respecto to a variation ˚ψabout a representativeψ ∈[ψ] that expresses the solenoidal velocityη =ψ∗ϑ:
E˚Euler =Z
M
η˚∧?η. Now, sinceη = ψ∗ϑ, its variation is given by
η˚= d
ψ∗ιψ˚ϑ
+ψ∗ιψ˚ dϑ
= d ϑ
ψ˚ +ψ∗ιψ˚π∗σ
= d ϑ
ψ˚ +ψ∗π∗ιdπ(ψ)˚ σ
= d ϑ
ψ˚ +s∗ιs˚σ.
Therefore, using integration by parts we have E˚Euler =Z
M
d ϑ
ψ˚ ∧?η+s∗ιs˚σ∧?η
=− Z
M
ϑ ψ˚
∧ d? η
|{z}=0
+ιus∗ιs˚σ ?1
= Z
M
σ(s,˚ ds(u))?1
= σS(s,˚ ds(u)) C σM
[ψ˚]∗, sgradEEuler
That is, the symplectic gradient, sgradEEuler, satisfies dΠ
sgradEEuler
= ds(u)= Lus.
Now, suppose [ψ] evolves under the symplectic gradient flow [ψ˙]∗ =−sgradEEuler.
Then by applying dΠ to both sides of the equation, we obtain
˙
s= −Lus.
The evolution equation (8.1) for the vorticity Clebsch map can then be used to show that η satisfies the Euler Equation.
Theorem 8.2. Under the symplectic gradient flow of EEuler on (M,σM), the ve- locity1-formη satisfies
η˙+Luη = −dp for some exact1-form dp.
Proof. Along the time-dependent[ψ], letψbe a representative of[ψ]so that η =ψ∗ϑis the solenoidal1-form. Now take the time derivative ofη
η˙ = d
ψ∗ιψ˙ϑ
+ψ∗ιψ˙ dϑ
= d ϑ(ψ)˙
| {z }
C−p˜
+ψ∗ιψ˙π∗σ
=−dp˜+ψ∗π∗ιdπ(ψ)˙ σ
=−dp˜+s∗ιs˙σ.
By Theorem 8.1, s satisfies ˙s =−Lus =−ds(u). Therefore, η˙ =−dp˜−s∗ιds(u)σ
=−dp˜− ιus∗σ
=−dp˜− ιudη
=−dp˜+ d |u|2
| {z }
C−dp
−Luη.
8.3* Gauge Symmetries from Hamiltonian Group Actions on Σ
A physically observed fluid state is a solenoidal 1-form η, which is de- termined by the vorticity ω = dη and the harmonic component of η. As we represent these physical quantities using Clebsch variables, we have introduced additional gauge symmetries. For example, the vorticity is rep- resented as ω = s∗σ. If we post-compose s by a symplectomorphism τ: Σ → Σ, τ∗σ = σ, then (τ◦s)∗σ = s∗τ∗σ = s∗σ = ω still represents the same physically observed vorticity. When the domain M and the space Σ have non-trivial topology, invariance of vorticity itself may not be enough.
In this section, we discuss the gauge symmetries in the fluid system rep- resented by Clebsch variables with general topology, and their associated conserved quantities.
Hamiltonian Group Actions onΣ
We begin with the group Symp(Σ,σ), the group of symplectic diffeomor- phisms (i.e. symplectomorphisms) acting on Σ. That is, each element τ ∈ Symp(Σ,σ)is a smooth mapτ: Σ →Σthat satisfiesτ∗σ = σ. Letsymp(Σ,σ) be the Lie algebra associated to Symp(Σ,σ). Then the condition τ∗σ = σ forτ ∈Symp(Σ,σ) can be described infinitesimally by
LXξ σ =0 for all ξ ∈symp(Σ,σ)
whereXξ is the tangent vector field onΣgenerated by ξ,i.e.for eachp ∈Σ, Xξ(p) B ∂t∂
t=0exp(tξ)·p. Since0 = LXξ σ = dιXξσ+ ιXξ dσ = dιXξσ, one sees that ιXξσ is always closed.
In the case that ιXξσ is not only closed but also exact, the vector field Xξ is said to be Hamiltonian. The Lie algebra elements ξ ∈ symp(Σ,σ) that produce Hamiltonian vector fields Xξ form a subalgebra ham(Σ,σ) ⊂ symp(Σ,σ). And ham(Σ,σ) generates the group of Hamiltonian group ac- tions Ham(Σ,σ) ⊂ Symp(Σ,σ). See [Polterovich,2012] for more discussion on the groups Symp(Σ,σ) and Ham(Σ,σ).
Moment Map
For each ξ ∈ ham(Σ,σ), ιXξσ is exact. Therefore for each ξ there exists a function Uξ: Σ → R so that dUξ = ιXξσ. In fact, the values of Uξ can be chosen so that they are linear in ξ ∈ ham(Σ,σ). That is, there is a
ham∗(Σ,σ)-valued function
µΣ: Σ → ham∗(Σ,σ) that expresses Uξ as Uξ
p = D µΣ
p,ξE
, where h·,·i denotes the pairing be- tweenham∗(Σ,σ) andham(Σ,σ). In other words, µΣ satisfies
dD µΣ,ξE
= ιXξσ. (8.2) (In some context µΣ is further required to be equivariant under the co- adjoint actions of Ham(Σ,σ)onham∗(Σ,σ)). A functionµΣ: Σ →ham∗(Σ,σ) satisfying (8.2) is called a moment map.
The values of a moment map hµΣ,ξi, or simply Uξ, are the possible con- served quantities associated to ξ ∈ham(Σ,σ).
Noether’s Invariants
Although our dynamical system happens on function spaces Mand S, in order to make a simpler explanation of Noether’s invariants, let us consider dynamical systems on the finite dimensional space Σ. Suppose H: Σ → R is a Hamiltonian giving rise to the Hamiltonian flow on (Σ,σ)
˙
p=−sgradH p
for time-dependentponΣ. Here the symplectic gradient sgradH is defined by dH = σ
·, sgradH
. Now, suppose the Hamiltonian H is invariant under a Hamiltonian group action,i.e.there is ξ ∈ham(Σ,σ)so that
LXξ H =0.
Then under the Hamiltonian flow ˙p = −sgradH the value Uξ = D µΣ,ξE
is an integral of motion.
To see this, take the time derivative ofUξ. Using the fact that dUξ = ιXξσ we have
U˙ξ = dUξ p˙ =−dUξ
sgradH
= − ιXξσ
sgradH
=−σ
Xξ, sgradH
= dH Xξ
=LXξ H =0.
Therefore Uξ is conserved along the Hamiltonian flow of H whenever H has the symmetry LXξ H = 0.
Hamiltonian Group Actions onS
Our dynamical system for fluids happens on the symplectic manifold(M,σM), whose symplectic structure is given by (S,σS). We have seen that Hamil- tonian group actions on Σ have an associated moment map that repre- sents the candidates for conserved quantities. These notions on (Σ,σ) can be carried along to (S,σS). As a quick reminder, note that S consists of functions s mapping into Σ, and the symplectic structure σS is given by σS(s, ˚˙ s) =R
Mσ(s, ˚˙ s)?1.
Now, each Hamiltonian group action τ ∈ Ham(Σ,σ) gives rise to a sym- plectomorphism on (S,σS) by post-composition. Explicitly, let us denote j: Ham(Σ,σ) ,→ Symp(S,σS) defined by that for each τ ∈ Ham(Σ,σ), the action j(τ): S→Sis given by
j(τ)·s B τ◦s, s ∈S.
Here we check that j(τ) is indeed a symplectomorphism. For each ˙s, ˚s ∈ TsS, by chain rule the resulting variations mapped by the tangent map of the action j(τ) are
d j(τ)·
(s)˙ = dτ(s),˙ d j(τ)·
(s)˚ = dτ(s).˚ Therefore,
j(τ)·∗σS
(s, ˚˙ s) =σS
d j(τ)·
˙
s, d j(τ)·
˚ s
=Z
M
σ(dτ(s),˙ dτ(s))˚ ?1
= Z
M
σ(s, ˚˙ s)?1= σS(s, ˚˙ s).
Next, we show that elements of j(Ham(Σ,σ)) are not only symplectomor- phisms onSbut furthermore Hamiltonian actions. That is, j(Ham(Σ,σ)) ⊂ Ham(S,σS). We show it by directly constructing the associated moment map for j(Ham(Σ,σ)).
Let j∗(ham(Σ,σ)) ⊂ symp(S,σS) be the Lie algebra for j(Ham(Σ,σ)) ex- pressed by pushforwards. Let ⟪·,·⟫ be the pairing between j∗(ham(Σ,σ)) and its dual space.
Theorem8.3. The map