In this paper we consider the Hamiltonian formulation of the equations of
incompressible ideal fluid flow from the point of view of optimal control
theory. The equations are compared to the finite symmetric rigid body equations
analyzed earlier by the authors. We discuss various aspects of the Hamiltonian
structure of the Euler equations and show in particular that the optimal
control approach leads to a standard formulation of the Euler equations -- the
so-called impulse equations in their Lagrangian form. We discuss various other
aspects of the Euler equations from a pedagogical point of view. We show that
the Hamiltonian in the maximum principle is given by the pairing of the
Eulerian impulse density with the velocity. We provide a comparative discussion
of the flow equations in their Eulerian and Lagrangian form and describe how
these forms occur naturally in the context of optimal control. We demonstrate
that the extremal equations corresponding to the optimal control problem for
the flow have a natural canonical symplectic structure.Comment: 6 pages, no figures. To appear in Proceedings of the 39th IEEEE
  Conference on Decision and Contro

Bloch, A. M.

Crouch, P. E.

Holm, D. D.

Marsden, J. E.

English

arXiv

In this paper we consider the Hamiltonian formulation

of the equations of incompressible ideal fluid flow

from the point of view of optimal control theory. The

equations are compared to the finite symmetric rigid

body equations analyzed earlier by the authors. We

discuss various aspects of the Hamiltonian structure of

the Euler equations and show in particular that the optimal

control approach leads to a standard formulation

of the Euler equations – the so-called impulse equations

in their Lagrangian form. We discuss various other aspects

of the Euler equations from a pedagogical point

of view. We show that the Hamiltonian in the maximum

principle is given by the pairing of the Eulerian impulse density with the velocity. We provide a comparative

discussion of the flow equations in their Eulerian

and Lagrangian form and describe how these forms

occur naturally in the context of optimal control. We

demonstrate that the extremal equations corresponding

to the optimal control problem for the flow have a

natural canonical symplectic structure

Bloch, Anthony M.

Holm, Darryl D.

Crouch, Peter E.

Marsden, Jerrold E.

Caltech Authors - Main

AN OPTIMAL CONTROL FORMULATION FOR
INVISCID INCOMPRESSIBLE IDEAL FLUID FLOW
Proc. CDC, 39, 1273–1279
Anthony M. Bloch1
Department of Mathematics
University of Michigan
Ann Arbor MI 48109
abloch@math.lsa.umich.edu
Peter E. Crouch2
Center for Systems Science
and Engineering
Arizona State University
Tempe, AZ 85287
peter.crouch@asu.edu
Darryl D. Holm3
Mathematical Modeling and Analysis
Theoretical Division
Los Alamos National Laboratory
Los Alamos, NM 87545
dholm@lanl.gov
Jerrold E. Marsden4
Control and Dynamical Systems 107-81
California Institute of Technology
Pasadena, CA 91125
marsden@cds.caltech.edu
Contents
1 Introduction 1
2 Inviscid, Incompressible, Fluid Flow 2
3 Optimal Control Problem 3
4 Hamiltonian Structure of Extremals 5
5 Conclusions 5
6 References 6
Abstract
In this paper we consider the Hamiltonian formula-
tion of the equations of incompressible ideal ﬂuid ﬂow
from the point of view of optimal control theory. The
equations are compared to the ﬁnite symmetric rigid
body equations analyzed earlier by the authors. We
discuss various aspects of the Hamiltonian structure of
the Euler equations and show in particular that the op-
timal control approach leads to a standard formulation
of the Euler equations – the so-called impulse equations
in their Lagrangian form. We discuss various other as-
pects of the Euler equations from a pedagogical point
of view. We show that the Hamiltonian in the maxi-
mum principle is given by the pairing of the Eulerian
1Research partially supported by the NSF and AFOSR.
2Work supported in part by NSF and NATO.
3Work supported in part by DOE
4Research partially supported by NSF and AFOSR.
impulse density with the velocity. We provide a com-
parative discussion of the ﬂow equations in their Eule-
rian and Lagrangian form and describe how these forms
occur naturally in the context of optimal control. We
demonstrate that the extremal equations correspond-
ing to the optimal control problem for the ﬂow have a
natural canonical symplectic structure.
1 Introduction
In this paper we consider the Hamiltonian formula-
tion of the equations of incompressible ideal ﬂuid ﬂow
from the point of view of optimal control theory. Our
goal in this work is to compare the ﬂuid equation aris-
ing in this fashion with the symmetric generalized rigid
body equations derived in Bloch and Crouch [1996],
Bloch, Brockett and Crouch [1997] and Bloch, Crouch,
Marsden and Ratiu [1998]. In these papers we showed
that a natural control approach led to a form of the
rigid body equations on SO(n)× SO(n) rather than on
T ∗ SO(n). In contrast here we show that the optimal
control approach leads to a standard formulation of the
Euler equations – the so-called impulse equations in
their Lagrangian form. A nice survey of the impulse
equations in the various forms can be found in Russo
and Smereka [1999] (see also Kuz’min [1983], Osledets
[1989] and Maddocks and Pego [1995] for related work).
Lie-Poisson reduction is an important tool in rational-
izing many of these approaches (see e.g. Marsden and
Weinstein [1984]). In particular, the Hodge projection
to the divergence free vector ﬁelds, is a Poisson map
since it may be regarded as the dual of the natural inclu-
sion (this is a standard result; see Marsden and Ratiu
[1994]). Thus, the Hodge projection naturally takes the
unconstrained Poisson system to the constrained one.
Russo and Smereka and others concern themselves with
the numerical aspects of these equations, something we
do not consider here.
These impulse equations are not symmetric in the
same sense as the symmetric rigid body equations –
i.e. we do not obtain two symmetric equations evolving
on two copies of the diﬀeomorphism group, but it is
possible to get a more symmetric formulation which we
intend to discuss in a forthcoming publication and that
we mention brieﬂy in the conclusions.
In the remainder of the introduction we recall the
standard and symmetric rigid body equations.
We recall from Manakov [1976] and Ratiu [1980]
that the left invariant generalized rigid body equations
on SO(n) may be written as
Q˙ = QΩ
M˙ = [M,Ω] , (RBn)
where Q ∈ SO(n) denotes the conﬁguration space vari-
able (the attitude of the body), Ω = Q−1Q˙ ∈ so(n) is
the body angular velocity, and
M := J(Ω) = ΛΩ + ΩΛ ∈ so(n)
is the body angular momentum. Here J : so(n) →
so(n) is the symmetric (with respect to the above inner
product) positive deﬁnite operator deﬁned by
J(Ω) = ΛΩ + ΩΛ,
where Λ is a diagonal matrix satisfying Λi + Λj > 0
for all i = j. For n = 3 the elements of Λi are related
to the standard diagonal moment of inertia tensor I by
I1 = Λ2 + Λ3, I2 = Λ3 + Λ1, I3 = Λ1 + Λ2.
The equations M˙ = [M,Ω] are readily checked to
be the Euler-Poincare´ equations on so(n) for the La-
grangian
l(Ω) =
1
2
〈Ω, J(Ω)〉 .
The left invariant symmetric rigid body system is
given by the ﬁrst order equations
Q˙ = QΩ
P˙ = PΩ (1.1)
where Ω is regarded as a function of Q and P via the
equations
Ω := J−1(M) ∈ so(n) and M := QTP − PTQ.
These equations can be derived from the following
optimal control problem:
Deﬁnition 1 Let T > 0, Q0, QT ∈ SO(n) be given
and ﬁxed. Let the rigid body optimal control problem be
given by
min
U∈so(n)
1
4
∫ T
0
〈U, J(U)〉dt (1.2)
subject to the constraint on U that there be a curve
Q(t) ∈ SO(n) such that
Q˙ = QU Q(0) = Q0, Q(T ) = QT . (1.3)
Proposition 2 The rigid body optimal control problem
1 has extremal evolution equations (1.1) where P is the
costate vector given by the maximum principle.
The optimal controls in this case are given by
U = J−1(QTP − PTQ). (1.4)
In §3 we derive the impulse equations for ﬂuid ﬂow
from the optimal control point of view.
2 Inviscid, Incompressible, Fluid Flow
In this section we introduce the usual dynamics for
inviscid, incompressible ﬂuid ﬂow, impulse density and
the vorticity dynamics. The basic equations we con-
sider are:
∂v
∂t
+ (v · grad)v = − grad p; div v = 0 (2.1)
x ∈ Ω; v = v(x, t), p = p(x, t).
We assume, for simplicty only that the ﬂow is in all of
space or in a periodic box so we do not need to deal
with boundary conditions. This is not an essential re-
striction.
Here, v is the ﬂuid velocity and p is the pressure.
We introduce the impulse density z,
z = v + grad k. (2.2)
where k is an arbitrary scalar ﬁeld, k = k(x, t). No-
tice that the preceding equation gives the (Helmholtz)-
Hodge decomposition of z. In other words, the projec-
tion of z to v is the Hodge projection of z. We return to
this important remark in the conclusions to gain deeper
insight into what is going on with the calculations to
follow.
Take the time derivative of (2.2) to get
∂z
∂t
− v × curl z = grad Λ, div v = 0 (2.3)
where
Λ =
∂k
∂t
− p− 1
2
v · v;
Λ is called the gauge. Any choice of gauge is possible,
but to be concrete, we consider the “geometric gauge”
Λ = −v · z.
With this choice
∂z
∂t
+ (v · grad)z + (grad v)T z = 0, div v = 0
and k is now ﬁxed by the equation
dk
dt
= p− 1
2
v · v.
Let z =
∑
i zidxi(= z · dx) be the one form correspond-
ing to z. Then one can easily show that
Lvz = (z∗v + vT∗ z) · dx.
Thus the impulse density is governed by:
∂z
∂t
+ Lvz = 0, div v = 0. (2.4)
Hence by applying the exterior diﬀerential operator we
obtain
∂
∂t
dz + Lvdz = 0, div v = 0.
Lemma 3 w = curl z = curl v satisﬁes the vorticity
equation:
∂w
∂t
+ [v, w] = 0 (2.5)
Proof dz =
∑
i dzi ∧ dxi =
∑
ij( ˆcurlz)ijdxi ⊗ dxj ,
where aˆb = a× b. We may now compute:
Lvdz =
∑
ij
[
v, curlz
]∧
ij
dxi ⊗ dxj .
We now quickly review the two coordinate systems
associated with the ﬂuid system. We denote the La-
grange or material variables by Xi and the Euler or
spatial variables by xi, and set
xi = φi(X, t), 1 ≤ i ≤ 3.
We assume φ : Ω → Ω is a volume preserving diﬀeo-
morphism, with Jacobian equal to unity, |φ∗| = 1.
Let v(x, t) = spatial velocity, so that
∂xi
∂t
= vi(x, t).
Thus V (X, t) = v(φ(X, t)) is the material velocity.
Hence
∂φ
∂t
(X, t) = v(φ(X, t))
or
∂φ
∂t
= v ◦ φ. (2.6)
We note the “right invariance” of this system and its
evolution on the “Group” G = Diﬀvol(Ω) of volume
preserving diﬀeomorphisms of Ω. Setting
〈a, b〉 =
∫
R3
aT (x, t)b(x, t)dx,
and using |φ∗| = 1, we obtain the identity
〈a ◦ φ, b〉 = 〈a, b ◦ φ−1〉.
With this introduction we may introduce the total
vorticity equations:
∂φ
∂t
= v ◦ φ; ∂w
∂t
= [w, v] : div v = 0. (2.7)
It is interesting to compare these equations with the
right invariant Euler equations for the rigid body:
Q˙ = ΩQ; M˙ = [Ω,M ] (2.8)
[Ω,M ] = ΩM −MΩ
(
= [M,Ω] interpreted
as vector ﬁelds
)
.
These Euler equations are Hamiltonian on T ∗SO(3),
with the canonical symplectic structure. The equiva-
lent statements about (2.7) have been well studied, (see
references in the introduction). However, the derivation
of the symmetric version as in (1.1) provides our moti-
vation for this new study.
3 Optimal Control Problem
In this section we introduce an optimal control prob-
lem and discuss the corresponding extremals. The
problem can be posed as:
min
v(·)
1
2
∫ T
0
〈v, v〉dt
subject to:
div v = 0;
∂φ
∂t
= v ◦ φ (3.1)
and
φ(X, 0) = φ0(X), φ(X,T ) = φT (X) ﬁxed,
and, for ﬂow in all of space, suitable conditions at in-
ﬁnity.
This optimal control problem is of course identical
to the standard Hamilton principle for ideal ﬂuid me-
chanics. However our goal here is to analyze it from the
point of view of the Pontryagin maximum principle.
We solve this problem by introducing Lagrange mul-
tipliers and the cost
J(v, φ, π, k) =
∫ T
0
(〈
π, v ◦ φ− ∂φ
∂t
〉
− 1
2
〈v, v〉+ 〈k,div v〉
)
dt
The problem (3.1) may be recast as: min J , subject to
div v = 0, ∂φ∂t = v ◦ φ, and boundary conditions.
We may prove the following result:
Theorem 4 The extremals of problem (3.1) are given
by
∂π
∂t
=− (v∗ ◦ φ)Tπ, ∂φ
∂t
= v ◦ φ (3.2)
v =π ◦ φ−1 − grad k, div v = 0.
Sketch Proof
δJ =
∫ T
0
(〈π, δv ◦ φ+ (v∗ ◦ φ)δφ− ∂
∂t
δφ〉 − 〈v, δv〉
+〈k, div δv〉)dt
=
∫ T
0
(〈π, δv ◦ φ〉 − 〈v, δv〉+ 〈k, div δv〉)dt
+
∫ T
0
〈π, (v∗ ◦ φ)δφ− ∂
∂t
δφ〉dt
Noting that δv(∞, t) = δφ(x, 0) = δφ(x, T ) = 0, we
obtain
δJ =
∫ T
0
〈π ◦ φ−1 − v − grad k, δv〉dt
+
∫ T
0
〈(v∗ ◦ φ)Tπ + ∂π
∂t
, δφ〉dt.
The system (3.2) follows immediately.
We note that the system (3.2) should be interpreted
in terms of Lagrange and Euler variables in the form
∂π
∂t
(X, t) =−
(
∂v
∂x
(φ(X, t), t)
)T
π(X, t),
∂φ
∂t
(X, t) =v(φ(X, t)),
and
v(x, t) = π ◦ φ−1(x, t)− grad k(x, t).
We now study the Hamiltonian for the extremal
ﬂow. Employing the maximum principle we know that
the Hamiltonian corresponding to the problem (3.1) is
H(π, φ) = 〈π, v ◦ φ〉 − 1
2
〈v, v〉.
We introduce the vector potential for v
v = curlψ; divψ = 0,
(and ψ → 0 at inﬁnity in all of space). Thus
ω = curl v = curl curlψ = −∆ψ + grad divψ = −∆ψ,
so
ω = −∆ψ; ∆ = Laplacian; .
Thus ψ = Aω where A is an integral operator. From
these identities we may write:
H(π, φ) = 〈π ◦ φ−1, curlψ〉 − 1
2
〈v, curlψ〉
= 〈curlπ ◦ φ−1, ψ〉 − 1
2
〈curl v, ψ〉
But
v = π ◦ φ−1 − grad k
so
curl v = curlπ ◦ φ−1
and
H(π, φ) =
1
2
〈curlπ ◦ φ−1, ψ〉 (3.3)
=
1
2
〈π ◦ φ−1, v〉 (3.4)
=
1
2
〈ω,Aω〉 (3.5)
=
1
2
〈curlπ ◦ φ−1, A curlπ ◦ φ−1〉. (3.6)
We now compute along extremals (3.2)
∂
∂t
∑
i
πi(X, t)dφi(X, t) = 0
or
∂
∂t
∑
i
πidφi = 0.
Hence
∂
∂t
∑
i
dπi ∧ dφi = 0.
Thus the “canonical two form”
∑
i
dπi ∧ dφi =
∑
ijk
∂πi
∂Xj
∂φi
∂Xk
dXj ∧ dXk
is constant along extremals. We now have the following
critical result.
Lemma 5 Let z = π ◦ φ−1. Along extremals (3.2)
∂z
∂t
+ z∗v + vT∗ z = 0.
Thus z = z · dx satisﬁes
∂z
∂t
+ Lvz = 0, ∂φ
∂t
= v ◦ φ (3.7)
It follows that we have recovered the evolution of the
impulse density of the ﬂuid ﬂow, equation (2.4). Note
that H = 12 〈z, v〉, where −z · v = Λ is the geometric
gauge. The following result relates z to the canonical
two form.
Lemma 6 φ−1∗
∑
i dπi ∧ dφi =
∑
i dzi ∧ dxi = dz.
Substituting the relation
z = v + grad k
into the system (3.7) recovers the system (2.1) where
dk
dt
= p− 1
2
v.v. (3.8)
However k is not arbitrary, since it is determined by the
extremal system (3.2). In fact
div z = div π ◦ φ−1 = div grad k = ∆k.
Thus,
k =Adiv z = Adiv π ◦ φ−1. (3.9)
v =π ◦ φ−1 − grad Adiv π ◦ φ−1.
Hence the pressure p is also determined by the ﬂow.
Lemma 7 By augmenting the cost functional in the
optimal control problem (3.1) by a potential function
η
∫ T
0
(
1
2
〈v.v〉 −
∫
Ω
η ◦ φ
)
dt
the extremal ﬂow satisﬁes the system (2.1) with the
pressure determined by
dk
dt
= p− η − 1
2
v.v.
k and v are determined from (3.9).
Thus any pressure p may be obtained via a suitable
potential η.
4 Hamiltonian Structure of Extremals
We now brieﬂy explore the Hamiltonian nature of
(2.1) and the extremal equations (3.2). If u is a smooth
function of x and t, and h[u] is a function of x, t and
the jet of u, let
H[u] =
∫
R3
h[u]dx.
Deﬁne
δH[u] =
∫
R3
〈
δH(u)
δu
, δu
〉
dx.
We have the following result:
Theorem 8 For the Hamiltonian (3.6)
δH
δπ
(π, φ) = v ◦ φ; ∂H
∂φ
(π, φ) = (v∗ ◦ φ)Tπ.
Thus the extremal equations (3.2) may be written
as
∂π
∂t
= −δH
δφ
;
∂φ
∂t
=
δH
δπ
. (4.1)
These equations are canonical with respect to the nat-
ural symplectic form on L2(R3 : R3)× L2(R3 : R3)
ω((X1, Y1), (X2, Y2)) =
∫
R3
(Y2 ·X1 −X2 · Y1)dx
Thus we have expressed the extremal equations (3.2) in
terms of a canonical Hamiltonian system.
5 Conclusions
As described earlier, the Euler (impulse) equations
(3.2) are not quite in the symmetric form that we obtain
in the rigid body setting – i.e. we do not get a sym-
metric ﬂow on two copies of the diﬀeomorphism group.
However, it is possible to extend the analysis to this
setting by factoring π as
π = r ◦ ψ, (5.1)
where
∂ψ
∂t
= v ◦ ψ (5.2)
and ψ evolves on the diﬀeomorphim group. Thus we
do get symmetric equations for φ and ψ coupled to an
interesting radial equation for r. This also has an ana-
logue in the ﬁnite-dimensional setting – one allows P
to be in Gl(n) and considers the polar decomposition
P = RK where R is symmetric positive deﬁnite and
K lies in SO(n). We shall describe the details of this
analysis in a forthcoming publication.
In addressing these issues, a deeper understanding
of both the Hamiltonian and variational structure as
well as the geometry is needed. For example, we can
obtain more insight into some of the calculations done
in this paper as follows. Consider the Hodge projection
P : X → Xvol taking a vector ﬁeld z to its divergence
free part parallel to the boundary. As we have men-
tioned in the introduction, using the L2 pairing, this
map is a Poisson map. Taking the L2 kinetic energy as
the Hamiltonian on the unconstrained space X as well
as on the constrained space Xvol, we conclude that the
corresponding Hamiltonian systems with their Lie Pois-
son bracket structures are mapped one to the other (in-
cluding integral curves) by the Hodge projection. This
simple remark is, in fact, the essense of what is going
on in relaxing the divergence free constraints and in
relating the Hamiltonian structure in the formalism of
Osledets, Buttke, and Kusmin. We have, in fact, shown
some aspects of this remark in the above direct calcu-
lations. A deeper problem, to which we will return in
other work, is to carry this out in material representa-
tion, where one needs a nonlinear Hodge decomposition,
similar to the Moser decomposition (a diﬀeomorphism
group analogue of the polar decomposition) discussed
in Ebin and Marsden [1970]. Many of these issues are
addressed in work of Brenier; see, eg, Brenier [1999].
Acknowledgement: We would like to thank Peter
Smereka for useful conversations.
6 References
Benjamin T. Brooke [1984] Impulse, Flow Force and
Variational Principles, IMA Journal of Applied
Mathematics 32, 3-68.
Bloch, A.M., R. W. Brockett, and P.E. Crouch [1997]
Double bracket equations and geodesic ﬂows on
symmetric spaces. Comm. Math Phys 187,357-
373.
Bloch, A. M. and P. E. Crouch [1996] Optimal control
and geodesic ﬂows. Systems and Control Letters
28, 65-72.
Bloch, A. M., P. E. Crouch J.E. Marsden and T.S.
Ratiu [1998] Discrete rigid body dynamics and
optimal control Proceedings of the 37th CDC,
IEEE, 2249-2254.
Bloch, A.M., P.S. Krishnaprasad, J.E. Marsden and
T.S. Ratiu [1994] Dissipation induced instabili-
ties, Ann. Inst. H. Poincare´, Analyse Nonlineare
11, 37–90.
Brenier, Y. [1999] Minimal geodesics on groups of
volume-preserving maps and generalized solutions
of the Euler equations. Comm. Pure Appl. Math.
52, 411–452.
Buttke, T.F. [1993] Velocity methods: Lagrangian nu-
merical methods which preserve the Hamiltonian
structure of incompressible ﬂuid ﬂow. In Vortex
Flows and Related Numerical Methods (ed. J.T.
Beale, G.H. Cottet and S. Hueberson), Kluwer.
Buttke, T.F. and A.J. Chorin [1993], Turbulence cal-
culations in magentization variables, Appl. Nu-
mer. Maths. 12, 47.
Ebin, D.G. and J.E. Marsden [1970] Groups of diﬀeo-
morphisms and the motion of an incompressible
ﬂuid, Ann. Math. 92, 102-163.
Hamel, G. [1904] Die Lagrange-Eulerschen Gleichun-
gen der Mechanik, Z. fu¨r Mathematik u. Physik
50, 1-57.
Hamel, G. [1949] Theoretische Mechanik, Springer-
Verlag.
Holm, D.D. [1983] Magnetic tornadoes: three-
dimensional aﬃne motions in ideal magnetohy-
drodynamics, Physica D 8 (1983), 170-182.
Kuz’min, G.A. [1983] Ideal incompressible hydrody-
namics in terms of the vortex momentum density,
Phys. Let. 96A, 88-90.
Lamb, H. [1932] Hydrodynamics, 6th ed., Cambridge
University Press.
Maddocks, J.H. and R.L. Pego [1995] An uncon-
strained Hamiltonian formulation for incompress-
ible ﬂuid ﬂow Comm. Math. Phys. 170, 207-217.
Manakov, S.V. [1976] Note on the integration of
Euler’s equations of the dynamics of and n-
dimensional rigid body. Funct. Anal. and its
Appl. 10, 328–329.
Marsden, J.E., and T. Ratiu [1994] Introduction
to Mechanics and Symmetry, Texts in Applied
Math., 17, Springer-Verlag, 2nd edition, 1999.
Marsden, J.E. and A. Weinstein [1983] Coadjoint or-
bits, vortices, and Clebsch variables for incom-
pressible ﬂuids, Physica D 7, 305-323.
Osledets, V.I. [1989], On a new way of writing the
Navier Stokes equation: The Hamiltonian formal-
ism, Russ. Math. Surveys 44, 210-211.
Ratiu, T. [1980] The motion of the free n-dimensional
rigid body. Indiana U. Math. J., 29, 609-627.
Russo, G. and P. Smereka [1999] Impulse formulation
of the Euler equation: general properties and nu-
merical methods, J. Fluid Mechanics 391, 189-
209.
Serrin, J. [1959] in Mathematical Principles of Classi-
cal Fluid Mechanics, Vol. VIII/1 of Encyclopedia
of Physics, edited by S. Flu¨gge (Springer-Verlag,
Berlin), sections 14-15, pp. 125-263.
Smereka, P. [1996] A Vlasov description of the Euler
equation, Nonlinearity, 9, 1361.


An optimal control formulation for inviscid incompressible ideal fluid flow

A Vlasov description of the Euler equation,

An unconstrained Hamiltonian formulation for incompressible ﬂuid ﬂow

Brooke [1984] Impulse, Flow Force and Variational Principles,

Coadjoint orbits, vortices, and Clebsch variables for incompressible ﬂuids,

Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. f¨ ur Mathematik u.

Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. fu¨r Mathematik u.

Discrete rigid body dynamics and optimal control

Dissipation induced instabilities,

Double bracket equations and geodesic ﬂows on symmetric spaces.

Groups of diﬀeomorphisms and the motion of an incompressible ﬂuid,

Hydrodynamics, 6th ed.,

Ideal incompressible hydrodynamics in terms of the vortex momentum density,

Impulse formulation of the Euler equation: general properties and numerical methods,

Introduction to Mechanics and Symmetry,

Magnetic tornadoes: threedimensional aﬃne motions in ideal magnetohydrodynamics,

Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations.

Note on the integration of Euler’s equations of the dynamics of and ndimensional rigid body.

On a new way of writing the Navier Stokes equation: The Hamiltonian formalism,

Optimal control and geodesic ﬂows.

The motion of the free n-dimensional rigid body.

Theoretische Mechanik,

Turbulence calculations in magentization variables,

Velocity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible ﬂuid ﬂow.

https://authors.library.caltech.edu/20344/1/BlCrHoMa2000.pdf

An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid
  Flow

An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid Flow

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