Tutorial on Euler-Poincaré Reduction

Introduction

development of structure-preserving discretisation schemes based on discretisation of action principles (variational integrators)
important theories in plasma physics can be derived from action principles, e.g., magnetohydrodynamics, Vlasov-Poisson, Vlasov-Maxwell, gyrokinetics
known action principles are not suited for discretisation
- use of Lagrangian labeling (material frame)
- constrained variations (e.g., Lin constraints)
- based on field representations unsuited for numerical simulation (e.g., Clebsch coordinates)
Vlasov: lots of choices, but none is suited for our means
- Low’58, Sugama’00, Sugama’14: Lagrangian labeling
- Brizard’00: constrained variations, 8D phasespace
- Larsson’92’93, Fla’94: generating functions
- Ye/Morrison’92: Clebsch potentials
- Pfirsch’84: Hamilton-Jacobi
formal Lagrangians (Kraus'14): work well, very flexible, but unphysical

Euler-Poincaré Reduction

use symmetries of the Lagrangian to simplify the dynamics of the system
- the original dynamics takes place on a Lie group
- the reduced dynamics takes place on the dual of the corresponding Lie algebra
plasma physics: use particle relabeling symmetry to transform Lagrangian action principles into Eulerian action principles
- the original Lagrangian is defined on the tangent bundle of the group of diffeomorphisms on the configuration space (e.g., maps which transport particles from their initial position $x_{0}$ to their current position $x_{t}$)
- the reduced Lagrangian is defined on the corresponding Lie algebra
- constraints follow in a natural and systematic way
entirely based on variational principles with symmetries
- no requirements on nondegeneracy of the Lagrangian or the availability of canonical coordinates
- systematic proof of conservation laws with the Kelvin-Noether theorem
- equivalent to Hamiltonian formulation if Legendre transform is invertible
- systematic derivation of Lie-Poisson brackets
the geometric structure underlying the theory aids understanding and thereby the development of novel numerical methods

Some Geometry

Hamilton's Action Principle

action: functional of a curve $q(t)$

\[\begin{aligned} S [q] = \int \limits_{0}^{T} L \big( q(t), \dot{q} (t) \big) \, dt \end{aligned}\]

Hamilton’s principle of stationary action: among all possible trajectories $q$, nature chooses the one that makes the action integral $S$ stationary
variation and integration by parts (endpoints fixed: $\delta q(0) = \delta q(T) = 0$)

\[\begin{aligned} \delta S = \int \limits_{0}^{T} \left[ \dfrac{\partial L}{\partial q} \cdot \delta q + \dfrac{\partial L}{\partial \dot{q}} \cdot \delta \dot{q} \right] dt = \int \limits_{0}^{T} \left[ \dfrac{\partial L}{\partial q} - \dfrac{d}{dt} \left( \dfrac{\partial L}{\partial \dot{q}} \right) \right] \cdot \delta q \, dt \end{aligned}\]

the variation of the action has to vanish for all $\delta q$, thus the integrand has to vanish, and we get the Euler-Lagrange equations

\[\begin{aligned} \dfrac{\partial L}{\partial q} (q, \dot{q}) - \dfrac{d}{dt} \left( \dfrac{\partial L}{\partial \dot{q}} (q, \dot{q}) \right) = 0 \end{aligned}\]

Euler-Poincaré Theory and Geometry

Euler-Poincaré theory has a rich geometric structure.
We need to understand this structure in order to develop new structure- preserving numerical methods.
What are tangent spaces, vector fields, Lie groups, Lie algebras, symmetries, quotient spaces, and how do they interact?
Benefit: applicable to many problems from plasma physics, e.g., incompressible fluids, compressible fluids, magnetohydrodynamics, kinetics, gyrokinetics.

Tangent Space

a point $v \in \rsp^{n}$ is frequently pictured as an arrow from $0$ to $v$

often we want to picture the same arrow starting at a different point $q \in \rsp^{n}$

we describe the arrow from $q$ to $q+v$ by the pair $(q, v)$, often denoted by $v_{q}$
(the vector $v$ at point $q$)
the set of all such pairs is just $\rsp^{n} \times \rsp^{n}$, which we will denote by $T\rsp^{n}$
$T\rsp^{n}$ is called the tangent bundle of $\rsp^{n}$, its elements are called tangent vectors
the vector $v$ at point $q$ is given by the pair $(q, v)$, but there are many more vectors at $q$

the set of all vectors based at the point $q$ is the tangent space to $\rsp^{n}$ at $q$, denoted by $T_{q} \rsp^{n}$ and given by $\rsp^{n}$
the tangent space $T_{q} \rsp^{n}$ corresponds to the derivatives of all possible parametrised curves in $\rsp^{n}$ passing through $q$
the tangent bundle $T\rsp^{n}$ of the space $\rsp^{n}$ is given by the disjoint union of the tangent spaces to $\rsp^{n}$ at all points $q \in \rsp^{n}$

\[\begin{aligned} T\rsp^{n} = \bigcup \limits_{q \in \rsp^{n}} T_{q} \rsp^{n} = \rsp^{n} \times \rsp^{n} \end{aligned}\]

this construction generalises straight forwardly to general manifolds $M$

consider a smooth curve $q : \rsp \rightarrow M$ depending on the parameter $t$
the velocity $v_{q} = (q(t), \dot{q} (t))$ at time $t$ is tangent to the curve $q(t)$
the velocity vectors at different times belong to different tangent spaces,

\[\begin{aligned} \big( q(t_{1}), \dot{q} (t_{1}) \big) \in T_{q(t_{1})} M \neq T_{q(t_{2})} M \ni \big( q(t_{2}), \dot{q} (t_{2}) \big) \end{aligned}\]

both $\big( q(t_{1}), \dot{q} (t_{1}) \big)$ and $\big( q(t_{2}), \dot{q} (t_{2}) \big)$ belong to the same tangent bundle $TM$

\[\begin{aligned} \big( q(t), \dot{q} (t) \big) \in TM \hspace{1em} \text{for all $t$} \end{aligned}\]

Example:
The tangent bundle $TS^{1}$ of the unit circle $S^{1}$ may be visualised as the union of the circle with a one-dimensional vector space of line vectors attached to each point of the circle.

Vector Fields

Definition: Vector Field A vector field $X$ on $M$ is a map

\[\begin{aligned} X : M &\rightarrow TM \hspace{3em} \text{such that} \hspace{3em} X_{m} \in T_{m} M \hspace{1em} \text{for all} \hspace{1em} m \in M \end{aligned}\]

A vector field can be visualised as an arrow attached to each point of $M$, chosen to be tangent to $M$ and to vary continuously from point to point.
A smooth vector field is one that is a smooth map from $M$ to $TM$.
The set of smooth vector fields on $M$ are denoted by $\mfrak{X}(M)$.
Example: Consider a fluid flowing in physical space. To each point $x$, we associate the fluid velocity $u(x)$. We expect $u(x)$ to be a tangent vector, which however depends on $x$ and therefore is a vector field.
Definition: Tangent Lift and Push-Forward The tangent lift of a function $f : M \rightarrow M$ is a function $Tf : TM \rightarrow TM$ given by

\[\begin{aligned} (x,v) \mapsto Tf (x,v) = \big( f(x) , Df(x) \cdot v \big) \hspace{1em} \text{with the Jacobian} \hspace{1em} (Df)^{j}_{i} = \partial f^{j} / \partial x^{i} . \end{aligned}\]

The push-forward of a vector field $X$ on $M$ by $f$ is given by the restriction of $Tf$,

\[\begin{aligned} f_{*} X \big( f(x) \big) = T_{x} f \big( X(x) \big) = Tf \cdot X(x) \hspace{1em} \text{for all} \hspace{1em} x \in M . \end{aligned}\]

Lie Groups and Lie Algebras

Informal Definition: Groups A group $G$, acting on a set $M$, ist a set of transformations from $M$ to $M$, s.th. (1) $G$ includes the identity transformation $e$, defined by $e \cdot m = m$ for all $m \in M$, (2) $G$ is closed, i.e., $g_{1}, g_{2} \in G$ implies $g_{1} \circ g_{2} \in G$, (3) for each $g \in G$ there exists $g^{-1} \in G$, such that $g^{-1} \circ g = e$.
Informal Definition: Lie Groups A Lie group $G$ is a group which is also a smooth manifold, i.e., (1) the composition $\circ$ is a smooth map $\circ : G \times G \rightarrow G$, (2) inversion is a smooth map $G \rightarrow G$.
Examples:

- The Euclidean space $\rsp^{n}$ is a group with vector addition as group operation. - The group $SO(n)$ of all orthogonal $n \times n$ matrices with determinant $1$. - The group of diffeomorphisms $\diff(M)$ on a smooth manifold $M$.

Definition: Diffeomorphism A diffeomorphism from a manifold $M$ to another manifold $N$ is a smooth bijective map (one-to-one and onto) $f : M \rightarrow N$ that has a smooth inverse.
Definition: Lie Algebra and Lie Bracket A Lie algebra $\la{g}$ is a vector space endowed with a bracket $[ \cdot , \cdot ] : \la{g} \times \la{g} \rightarrow \la{g}$, which (1) is linear, $[a \xi, \eta] = a \, [\xi, \eta]$ for $a \in \rsp$ and $\xi, \eta \in \la{g}$, (2) is anti-symmetric, $[\xi, \eta] = - [\eta, \xi]$ for $\xi, \eta \in \la{g}$, (3) satisfies the Jacobi identity, $[ [\xi, \eta], \chi ] + [ [\eta, \chi], \xi ] + [ [\chi, \xi], \eta ] = 0$ for $\xi, \eta, \chi \in \la{g}$.
Examples:

- The vector space $\rsp^{3}$ is a Lie algebra when endowed with a bracket given by

\[\begin{aligned} [x, y] = x \times y . \end{aligned}\]

- The space of skew-symmetric $n \times n$ matrices is a Lie algebra with a bracket given by the commutator

\[\begin{aligned} [A, B] = AB - BA . \end{aligned}\]

- The space of all smooth vector fields $\mfrak{X} (M)$ on a smooth manifold $M$ is a Lie algebra with the Jacobi-Lie bracket

\[\begin{aligned} [X,Y] = (X \cdot \nabla) Y - (Y \cdot \nabla) X . \end{aligned}\]

Group Actions and Symmetry

Definition: Left and Right Action
The left action $\phi$ of a Lie group $G$ on a smooth manifold $M$ is a smooth map $G \times M \rightarrow M$, often written as $\phi (g,m) = g \cdot m$ or just $\phi (g,m) = gm$, such that (1) $\phi (e, m) = m$ for all $m \in M$, (2) $\phi (h, \phi (g, m)) = \phi (hg, m) = hg \cdot m$ for all $g, h \in G$ and all $m \in M$.
The right action of a group $G$ on a smooth manifold $M$ is a smooth map $M \times G \rightarrow M$, often written as $\phi (g,m) = m \cdot g$ or just $\phi (g,m) = mg$, such that (1) $\phi (e, m) = m$ for all $m \in M$, (2) $\phi (h, \phi (g, m)) = \phi (gh, m) = m \cdot gh$ for all $g, h \in G$ and all $m \in M$.
For every $g \in G$ the map $\phi_{g} : M \rightarrow M$ given by $\phi_{g} (m) = \phi (g,m)$ is a diffeomorphism with inverse $\phi_{g}^{-1} (m) = \phi_{g^{-1}} (m)$.
Informal Definition: Symmetry
An object defined on $M$ is invariant (symmetric) with respect to a group $G$ if it is unchanged by the action $\phi$ of elements of $G$.
Example:
A function $f : M \rightarrow \rsp$ is invariant under the action $\phi$ of a Lie group $G$ if

\[\begin{aligned} (f \circ \phi_{g}) (m) = f(g \cdot m) = f(m) \hspace{1em} \text{for all} \hspace{1em} g \in G . \end{aligned}\]

Noether Theorem:
If the Lagrangian $L : TM \rightarrow \rsp$ is invariant under the action of a Lie group $G$, then there exists a corresponding conservation law of the equations of motion.
Given a group action $\phi_{g}$ on $M$ we need to determine the corresponding action on the whole tangent bundle $TM$.
Definition: Tangent Lift of a Group Action
If $\phi : G \times M \rightarrow M$ is a (left or right) group action so that $\phi_{g} : M \rightarrow M$ for all $g \in G$, then the tangent lift $T \phi_{g} : TM \rightarrow TM$ is the corresponding action on the tangent bundle $TM$ given by

\[\begin{aligned} (x,v) \mapsto T_{x} \phi_{g} (x,v) = \big( \phi_{g} (x), T_{x} \phi_{g} (v) \big) . \end{aligned}\]

Invariance Condition:

\[\begin{aligned} \big( L \circ T \phi_{g} \big) \big( q (t), \dot{q} (t) \big) = L \big( \phi_{g} (q (t)), T_{q(t)} \phi_{g} ( \dot{q} (t) ) \big) = L \big( q (t), \dot{q} (t) \big) \end{aligned}\]

In shorthand notation (implying an appropriate interpretation of $\cdot$), this is

\[\begin{aligned} g \cdot L \big( q (t), \dot{q} (t) \big) = L \big( g \cdot q (t), g \cdot \dot{q} (t) \big) = L \big( q (t), \dot{q} (t) \big) \end{aligned}\]

Action of a Lie Group on Itself

Definition: Action of a Lie Group on Itself Left multiplication is a map $\mathrm{L}_{g} : G \rightarrow G$ given by $h \mapsto gh = \mathrm{L}_{g} (h)$. Right multiplication is a map $\mathrm{R}_{g} : G \rightarrow G$ given by $h \mapsto hg = \mathrm{R}_{g} (h)$.
From that the composition rules follow as

\[\begin{aligned} \mathrm{L}_{g_{1}} \circ \mathrm{L}_{g_{2}} &= \mathrm{L}_{g_{1} g_{2}} & &\text{and}& \mathrm{R}_{g_{1}} \circ \mathrm{R}_{g_{2}} &= \mathrm{R}_{g_{2} g_{1}} . & \end{aligned}\]

Definition: Left Invariant Vector Field (I) A vector field $X$ on a Lie group $G$ is called left invariant if

\[\begin{aligned} \mathrm{L}_{g*} X = T \mathrm{L}_{g} \circ X \circ L_{g}^{-1} = X \hspace{1em} \text{for all} \hspace{1em} g \in G . \end{aligned}\]

Definition: Left Invariant Vector Field (II) A vector field $X$ on a Lie group $G$ is called left invariant if

\[\begin{aligned} T_{h} \mathrm{L}_{g}(X(h)) = X(\mathrm{L}_{g}(h)) = X(gh) \hspace{1em} \text{for all} \hspace{1em} g, h \in G . \end{aligned}\]

The set of left invariant vector fields on $G$ is denoted $\mfrak{X}_{L} (G) \subset \mfrak{X} (G)$.
The tangent map $T_{e} \mathrm{L}_{g}$ shifts vectors based at $e$ to vectors based at $g \in G$.
By applying this operation to some vector $\xi \in T_{e} G$ for every $g \in G$ we define a smooth vector field $X_{\xi}$ on $G$.
Definition: Left Extension The left extension of any $\xi \in T_{e} G$ is the vector field $X_{\xi}$ given by

\[\begin{aligned} X_{\xi} (g) := T_{e} \mathrm{L}_{g} (\xi) \hspace{1em} \text{for all} \hspace{1em} g \in G . \end{aligned}\]

A vector field on $G$ is left invariant iff it equals $X_{\xi}$ for some $\xi \in T_{e} G$, i.e.,

\[\begin{aligned} \mfrak{X}_{L} (G) = \big\{ X_{\xi} \; \big\vert \; \xi \in T_{e} G \big\} . \end{aligned}\]

The Lie algebra $\la{g}$ of $G$ is the vector space of left invariant vector fields on $G$. It can be identified with the fibre of $TG$ at the identity, $\la{g} = T_{e} G$.

Orbits and Quotients

Definition: Orbit The $G$-orbit of $m$ is the subset of $M$ that results from the (left or right) action of all elements of $G$ on $m$,

\[\begin{aligned} \orb (m) = \big\{ g \cdot m \; \big\vert \; g \in G \big\} \subset M . \end{aligned}\]

Definition: Quotient (Orbit Space) Two elements $m_{1}, m_{2} \in M$ are said to be equivalent, denoted by $m_{1} \sim_{G} m_{2}$, if they belong to the same orbit, that is if there exists some $g \in G$ such that $m_{1} = g m_{2}$ and hence $m_{1} \in \orb(m_{2})$.
The quotient of $M$ by $G$ is the set of equivalence classes (the set of orbits),

\[\begin{aligned} M/G := \big\{ \orb(m) \; \big\vert \; m \in M \big\} . \end{aligned}\]

The quotient can be understood as a parametrisation of the orbits.
Example: Translational Symmetry

- Consider translations by vectors in the Euclidean space $\rsp^{3}$. The group is $G=\rsp$ acting on $M=\rsp^{3}$ by $g \cdot m = m + g v$ with $v$ some unit vector in $M$. - The orbits are lines in direction of $v$, given by

\[\begin{aligned} \orb(m) = \big\{ m + g v \; \big\vert \; g \in \rsp \big\} . \end{aligned}\]

- The quotient space is a plane (e.g., the one perpendicular to $v$), given by

\[\begin{aligned} M/G = \rsp^{3} / \rsp \simeq \rsp^{2} . \end{aligned}\]

Example: Rotational Symmetry

- Consider rotations in the two dimensional Euclidean space. The group is $G=SO(2)$ acting on $M=\rsp^{2}$ by matrix-vector multiplication. - The orbits are circles about the origin,

\[\begin{aligned} \orb(m) = \bigg\{ \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} \begin{pmatrix} m_{1} \\ m_{2} \end{pmatrix} \; \bigg\vert \; g \in SO(2) \bigg\} . \end{aligned}\]

- The quotient space corresponds to the positive real numbers

\[\begin{aligned} M/G = \rsp^{2} / SO(2) \simeq [0, + \infty) . \end{aligned}\]

Example: Left-Invariant Vector Fields

- Consider a vector $X_{g} = (g, \dot{g})$ on a Lie group $G$, s.th. $X_{g} \in T_{g} G$. - Let $G$ act on $TG$ by left multiplication, that is $h \cdot X_{g} = T_{g} \mathrm{L}_{h} (X_{g})$. - The orbits are left invariant vector fields

\[\begin{aligned} \orb(X_{g}) = \big\{ X_{hg} = T_{g} \mathrm{L}_{h} (X_{g}) \; \big\vert \; h \in G \big\} . \end{aligned}\]

- One element of this orbit is $X_{e} = (g^{-1} g, g^{-1} \dot{g}) = (e, \xi)$, so that we can write

\[\begin{aligned} \orb(X_{g}) = \orb(X_{e}) = \big\{ X_{h} = T_{e} \mathrm{L}_{h} (X_{e}) \; \big\vert \; h \in G \big\} . \end{aligned}\]

- The quotient corresponds to the Lie algebra $\mfrak{g}$ of the Lie group $G$,

\[\begin{aligned} TG/G \simeq T_{e} G \simeq \mfrak{g} , \end{aligned}\]

with $X_{e} = (e, \xi) \in T_{e} G$ and $\xi \in \mfrak{g}$.

Action of a Lie Algebra

Definition: Adjoint and coAdjoint Operator For each $\xi \in \la{g}$ we define the map $\ad_{\xi} : \la{g} \rightarrow \la{g}$ by

\[\begin{aligned} \ad_{\xi} (\eta) = [\xi, \eta] . \end{aligned}\]

The dual map $\ad_{\xi}^{*} : \la{g}^{*} \rightarrow \la{g}^{*}$ is defined by the condition

\[\begin{aligned} \bracket{\alpha, \ad_{\xi} (\eta)} = \bracket{\ad_{\xi}^{*} (\alpha), \eta} \end{aligned}\]

for given $\alpha \in \la{g}^{*}$ and $\eta \in \la{g}$, where $\bracket{ \cdot , \cdot } : \mfrak{g}^{*} \times \mfrak{g} \rightarrow \rsp$ is the duality pairing between $\mfrak{g}^{*}$ and $\mfrak{g}$, and $\mfrak{g}^{*}$ is the dual vector space of the Lie algebra $\mfrak{g}$, i.e., the space of linear functionals on $\mfrak{g}$.

Example:

- Consider the adjoint operator on the Lie algebra $\mfrak{so}(3)$, which can be identified with the vector space $\rsp^{3}$ equipped with the cross product as Lie bracket,

\[\begin{aligned} \ad_{\xi} \eta = \xi \times \eta . \end{aligned}\]

- For $\alpha \in \rsp^{3}$, we find

\[\begin{aligned} \bracket{ \alpha , \ad_{\xi} \eta } = \bracket{ \alpha , \xi \times \eta } = \bracket{ - \xi \times \alpha , \eta } \hspace{1em} \text{so that} \hspace{1em} \ad_{\xi}^{*} (\alpha) = - \xi \times \alpha . \end{aligned}\]

Basic Euler-Poincaré Reduction

Lagrangian reduction by symmetry: assuming that the Lagrangian

\[\begin{aligned} L : TQ \rightarrow \rsp \end{aligned}\]

is invariant under the action of a group $G$,

\[\begin{aligned} g \cdot L (q, \dot{q}) = L (g q, g \dot{q}) = L (q, \dot{q}) , \end{aligned}\]

there exists a well defined object $l$ on the reduced space $TQ/G$ which describes the same dynamics

Euler-Poincaré reduction considers the case of a $G$-invariant Lagrangian defined on $TG$
if the Lagrangian $L : TG \rightarrow \rsp$ is invariant under left-action of the group $G$, then $L(g, \dot{g})$ is constant on the corresponding orbits of $X_{g} = (g, \dot{g})$
as $\orb(X_{g})$ contains $X_{e}$ we can reduce the Lagrangian to a function of $\xi$ only,

\[\begin{aligned} X_{e} = T_{g} \mathrm{L}_{g^{-1}} (X_{g}) = (g^{-1} g, g^{-1} \dot{g}) = (e, g^{-1} \dot{g}) = (e, \xi) \end{aligned}\]

the reduced Lagrangian is a function on $TG/G \simeq \la{g}$, so that the equations of motion are defined on the dual $\la{g}^{*}$ of the Lie algebra $\la{g}$ of $G$
these equations are called the Euler-Poincaré equations
if the Lagrangian $L : TG \rightarrow \rsp$ is invariant under left-action of the group $G$,

\[\begin{aligned} L(hg, h\dot{g}) = L(g, \dot{g}) \hspace{1em} \text{for all $g, h \in G$} \end{aligned}\]

then this is in particular true for $h = g^{-1}$

this allows us to define the reduced Lagrangian $l : \la{g} \rightarrow \rsp$ by

\[\begin{aligned} l(\xi) := L(e, \xi) = L (g^{-1} g, g^{-1} \dot{g}) = L (g, \dot{g}) \end{aligned}\]

with

\[\begin{aligned} \xi (t) = g^{-1} (t) \, \dot{g} (t) \in \la{g} \simeq TG/G \end{aligned}\]

If $\xi (t)$ extremises the reduced action $s = \int_{0}^{T} l (\xi) \, dt$ with respect to constrained variations $\delta \xi = \dot{\eta} + [\eta, \xi]$, then $\xi (t)$ satisfies the Euler-Poincaré equation

\[\begin{aligned} \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} = \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} . \end{aligned}\]

Let $g(t)$ be a curve in $G$ and set $\tfrac{dg}{dt} = \dot{g} \in T_{g(t)} G$ and $\xi (t) = g^{-1} (t) \, \dot{g} (t) \in \la{g}$.
If $g$ extremises the action $S = \int_{0}^{T} L(g, \dot{g}) \, dt$ for an arbitrary variation $\delta g$ with fixed end points, then $g$ satisfies the Euler-Lagrange equations

\[\begin{aligned} \dfrac{d}{dt} \dfrac{\partial L}{\partial \dot{g}} - \dfrac{\partial L}{\partial g} = 0 . \end{aligned}\]

Show that $\xi (t)$ extremises the reduced action

\[\begin{aligned} s = \int_{0}^{T} l (\xi) \, dt \end{aligned}\]

with respect to constrained variations

\[\begin{aligned} \delta \xi = \dot{\eta} + [\xi, \eta] \end{aligned}\]

for an arbitrary curve $\eta(t) \in \la{g}$ which vanishes at the endpoints.

Show that $\xi (t)$ satisfies the Euler-Poincaré equation

\[\begin{aligned} \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} = \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} . \end{aligned}\]

Let $\delta g(t) = \tfrac{d}{d\eps} \big\vert_{\eps=0} g_{\epsilon} (t)$ be a variation of $g(t)$ induced by the deformation $g_{\eps} (t)$ where $g_{0} (t) = g(t)$. Fixed endpoints means $\delta g(0) = \delta g(T) = 0$.
We start by calculating the variations in $\xi$ induced by the variations in $g$.
If we define $\eta = g^{-1} \, \delta g$, we see that

\[\begin{aligned} \delta \xi = \dfrac{\partial}{\partial \eps} (g^{-1} \dot{g}) \bigg\vert_{\eps=0} = - g^{-1} \delta g \, g^{-1} \dot{g} + g^{-1} \dfrac{\partial^{2} g_{\eps}}{\partial t \, \partial \eps} \bigg\vert_{\eps=0} = - \eta \xi + g^{-1} \dfrac{\partial^{2} g_{\eps}}{\partial t \, \partial \eps} \bigg\vert_{\eps=0} , \end{aligned}\]

and similarly

\[\begin{aligned} \dot{\eta} = \dfrac{\partial}{\partial t} ( g^{-1} \delta g ) \bigg\vert_{\eps=0} = - g^{-1} \dot{g} \, g^{-1} \delta g + g^{-1} \dfrac{\partial^{2} g_{\eps}}{\partial t \, \partial \eps} \bigg\vert_{\eps=0} = - \xi \eta + g^{-1} \dfrac{\partial^{2} g_{\eps}}{\partial t \, \partial \eps} \bigg\vert_{\eps=0} . \end{aligned}\]

As a result we see that $\delta \xi - \dot{\eta} = \xi \eta - \eta \xi$ or in other words $\delta \xi = \dot{\eta} + [\xi, \eta]$.
If this is the case, we immediately see that

\[\begin{aligned} \delta \int L (g, \dot{g}) \, dt = \delta \int L ( g^{-1} g, g^{-1} \dot{g} ) \, dt = \delta \int l (\xi) \, dt , \end{aligned}\]

where the last integral is extremised w.r.t. variations of $\xi$ of the desired form.

Assume that $\xi (t)$ extremises the reduced action $s = \int_{0}^{T} l (\xi) \, dt$ with respect to variations $\delta \xi = \dot{\eta} + [\xi, \eta]$, then

\[\begin{aligned} \delta \int l (\xi) \, dt = \int \bracket{ \dfrac{\delta l}{\delta \xi} , \, \dot{\eta} + [\xi, \eta] } dt . \end{aligned}\]

Using integration by parts on the time derivative of $\eta$ and by definition of $\ad_{\xi}^{*}$,

\[\begin{aligned} \delta \int l (\xi) \, dt &= \int \bigg[ \bracket{\dfrac{\delta l}{\delta \xi} , \, \dot{\eta} } + \bracket{ \dfrac{\delta l}{\delta \xi} , \, \ad_{\xi} (\eta) } \bigg] dt \\ &= \int \bracket{ - \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} + \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} , \, \eta } dt . \end{aligned}\]

Since $\eta$ is arbitrary, the integrand has to vanish, which implies the Euler-Poincaré equations,

\[\begin{aligned} \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} = \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} . \end{aligned}\]

Summary: For a given Lagrangian $L : TG \rightarrow \rsp$ that is invariant under left action of $G$, basic Euler-Poincaré reduction consists of (1) determining the reduced Lagrangian $l(\xi) = g^{-1} \cdot L(g, \dot{g})$ with $\xi = g^{-1} \dot{g}$, (2) identifying the Lie algebra $\mfrak{g} \simeq T_{e} G$ and the corresponding Lie bracket $[\eta, \xi]$, (3) computing the coadjoint operator $\ad_{\xi}^{*}$ from $\ad_{\xi}= [ \cdot , \xi]$.
Remark: For right $G$-invariant Lagrangians, the variations take the form $\delta \xi = \dot{\eta} + [\eta, \xi]$ and the corresponding Euler-Poincaré equation is

\[\begin{aligned} \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} = - \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} \hspace{3em} \text{with} \hspace{3em} \xi = \dot{g} (t) \, g^{-1} (t) . \end{aligned}\]

The Rigid Body

Consider a rigid body rotating about its center of mass in $\rsp^{3}$ with angular velocity $\omega \in \rsp^{3}$ and angular momentum $\Pi = \identity \omega \in \rsp^{3}$, where $\identity$ is the inertia tensor.
The rigid body equations of motion,

\[\begin{aligned} \dot{\Pi} = \Pi \times ( \identity^{-1} \Pi ) , \end{aligned}\]

are the Euler-Lagrange equations for a left invariant Lagrangian on the special orthogonal group $SO(3)$, that is the orthogonal $3 \times 3$ matrices with determinant $1$.

Consider a point $x_{0}$ at time $t=0$, called reference configuration or label.
Its position at time $t$ is given by $x(t) = R(t) x_{0}$ where $R(t)$ is a proper rotation of $\rsp^{3}$, that is $R(t)$ is an element of the three dimensional Lie group $SO(3)$.
The configuration of the body is given by the rotation matrix $R(t) \in SO(3)$ acting on the reference configuration, so that the transformation group $SO(3)$ serves as the configuration space of the rigid body.
The rotation matrices $R(t)$ do not live in a vector space or a surface in $\rsp^{3}$ but in differentiable manifold which is also a Lie group.
The Lagrangian of the rigid body, $L : TSO(3) \rightarrow \rsp$, is given by the kinetic energy,

\[\begin{aligned} L(R, \dot{R}) &= \dfrac{1}{2} \int_{B} \rho(x_{0}) \, \bnorm{\dot{x}}^{2} dx_{0} \\ &= \dfrac{1}{2} \int_{B} \rho(x_{0}) \, \bnorm{\dot{R} x_{0}}^{2} dx_{0} \\ &= \dfrac{1}{2} \sum \limits_{i,j,k=1}^{3} \int_{B} \rho(x_{0}) \, x_{0}^{i} x_{0}^{j} \, \dot{R}_{ik} \dot{R}_{jk} \, dx_{0} \\ &= \dfrac{1}{2} \sum \limits_{i,j,k=1}^{3} \dot{R}_{ik} \dot{R}_{jk} \int_{B} \rho(x_{0}) \, x_{0}^{i} x_{0}^{j} \, dx_{0} \\ &= \dfrac{1}{2} \trace \big( \dot{R}^{T} \identity \dot{R} \big ) , \end{aligned}\]

with the mass distribution $\rho(x)$ and the inertia tensor $\identity$, given by

\[\begin{aligned} \identity^{ij} = \int_{B} \rho(x) \, x^{i} x^{j} \, dx . \end{aligned}\]

The Lagrangian $L : TSO(3) \rightarrow \rsp$ is left $SO(3)$ invariant (rotational symmetry),

\[\begin{aligned} L(R, \dot{R}) = L (R^{-1} R, R^{-1} \dot{R}) = L (\unity, R^{T} \dot{R}) , \end{aligned}\]

as $R^{-1} = R^{T}$ is orthogonal (and thus preserves length), such that

\[\begin{aligned} \bnorm{\dot{R} x_{0}}^{2} = \big( \dot{R} x_{0} \big) \cdot \big( \dot{R} x_{0} \big) = R^{-1} \big( \dot{R} x_{0} \big) \cdot R^{-1} \big( \dot{R} x_{0} \big) = \bnorm{R^{-1} \dot{R} x_{0}}^{2} \end{aligned}\]

For orthogonal matrices we have

\[\begin{aligned} R^{T} R = \unity . \end{aligned}\]

Taking the time derivatives gives

\[\begin{aligned} \dfrac{d}{dt} \big( R^{T} R \big) = \dot{R}^{T} R + R^{T} \dot{R} = 0 , \end{aligned}\]

which says that $R^{T} \dot{R}$ is an anti-symmetric (skew symmetric) matrix,

\[\begin{aligned} R^{T} \dot{R} = - \dot{R}^{T} R = - (R^{T} \dot{R})^{T} . \end{aligned}\]

This shows that the Lie algebra $\la{so}(3)$ of the Lie group $SO(3)$ consists of the vector space of anti-symmetric $3 \times 3$ matrices.
We can use the basis

\[\begin{aligned} e_{1} &= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix} , & e_{2} &= \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} , & e_{3} &= \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} , \end{aligned}\]

and the hat map

\[\begin{aligned} \hat{\omega} = R^{T} \dot{R} = \begin{pmatrix} \hphantom{-} 0\hphantom{^{2}} & - \omega^{3} & \hphantom{-} \omega^{2} \\ \hphantom{-} \omega^{3} & \hphantom{-} 0\hphantom{^{2}} & - \omega^{1} \\ - \omega^{2} & \hphantom{-} \omega^{1} & \hphantom{-} 0\hphantom{^{2}} \end{pmatrix} , \end{aligned}\]

to identify anti-symmetric matrices $\hat{\omega} \in \la{so}(3)$ with vectors $\omega \in \rsp^{3}$.

Applying $\hat{\omega}$ to a vector $\xi \in \rsp^{3}$ amounts to the cross product of $\omega$ and $\xi$,

\[\begin{aligned} \hat{\omega} \xi = \omega \times \xi . \end{aligned}\]

The hat map identifies the cross product of two vectors $\omega$ and $\xi$ in $\rsp^{3}$ with the commutator of two anti-symmetric $3 \times 3$ matrices $\hat{\omega}$ and $\hat{\xi}$,

\[\begin{aligned} (\omega \times \xi )^{\hat{}} = \big[ \hat{\omega} , \hat{\xi} \big] . \end{aligned}\]

This shows that the Lie algebra $\la{so}(3)$ is isomorphic to $\rsp^{3}$ using

\[\begin{aligned} \hat{} \, : \big( \rsp^{3}, \times \big) \rightarrow \big( \la{so}(3), [ \cdot , \cdot ] \big) . \end{aligned}\]

The reduced Lagrangian $l : \rsp^{3} \rightarrow \rsp$ of the rigid body is defined by

\[\begin{aligned} l(\omega) = L(\unity, \hat{\omega}) = \dfrac{1}{2} \omega^{T} \identity \omega , \end{aligned}\]

so that

\[\begin{aligned} \dfrac{\delta l}{\delta \omega} = \omega^{T} \identity . \end{aligned}\]

The adjoint operator is

\[\begin{aligned} \ad_{\omega} \xi = \hat{\omega} \hat{\xi} - \hat{\xi} \hat{\omega} = \omega \times \xi = \hat{\omega} \xi , \end{aligned}\]

so that for $\alpha \in \rsp^{3}$ we find

\[\begin{aligned} \bracket{ \alpha , \ad_{\omega} \xi } = \bracket{ \hat{\omega}^{T} \alpha , \xi } = \bracket{ - \hat{\omega} \alpha , \xi } . \end{aligned}\]

Therefore, the Euler-Poincaré equations are

\[\begin{aligned} \identity \dot{\omega} = - \omega \times (\identity \omega) . \end{aligned}\]

Letting $\Pi = \identity \omega$, we get

\[\begin{aligned} \dot{\Pi} = \Pi \times ( \identity^{-1} \Pi ) . \end{aligned}\]

Incompressible Fluids

The Euler equations for an ideal incompressible fluid,

\[\begin{aligned} \dfrac{\partial u}{\partial t} + u \cdot \nabla u + \dfrac{1}{\rho} \nabla p = 0 , \end{aligned}\]

are the Euler-Poincaré equations for a right invariant Lagrangian on $\diff_{\mathrm{Vol}} (M)$, the group of volume preserving diffeomorphisms on $M$.

$u$ is a velocity field, $p$ is the pressure and $\rho$ is the density, assumed to be $1$
Consider an incompressible fluid filling a region $\Omega$ in $\rsp^{3}$ free of external forces.
Given a reference configuration (label) $x_{0} \in \Omega$ at time $t=0$, the trajectory of a fluid particle is determined by a smooth invertible map $\varphi_{t}$ which takes $x_{0}$ to the current position $x(t)$,

\[\begin{aligned} x(t) = \varphi_{t} (x_{0}) . \end{aligned}\]

For fixed $t$, the map $\varphi_{t}$ is a diffeomorphism of $\Omega$, and since the fluid is incompressible, $\varphi_{t}$ even is a volume preserving diffeomorphism,

\[\begin{aligned} \varphi_{t} \in \diff_{\mathrm{Vol}} (\Omega) . \end{aligned}\]

The configuration space is a transformation group, namely the infinite- dimensional Lie group of volume preserving diffeomorphisms of $\Omega$.
The Lagrangian of an incompressible fluid is a map

\[\begin{aligned} L : T\diff_{\mathrm{Vol}} (\Omega) \rightarrow \diff_{\mathrm{Vol}} (\Omega) , \end{aligned}\]

given by the kinetic energy of the fluid in the Lagrangian frame,

\[\begin{aligned} L (\varphi, \dot{\varphi}) = \dfrac{1}{2} \int \limits_{\Omega} \bnorm{\dot{\varphi} (x)}^{2} dx . \end{aligned}\]

$L$ is right $\diff_{\mathrm{Vol}} (\Omega)$ invariant since for any $\phi \in \diff_{\mathrm{Vol}} (\Omega)$, we find

\[\begin{aligned} L (\varphi \circ \phi, \dot{\varphi} \circ \phi) &= \dfrac{1}{2} \int \limits_{\Omega} \bnorm{\dot{\varphi} (\phi(x))}^{2} dx \\ &= \dfrac{1}{2} \int \limits_{\phi(\Omega)} \bnorm{\dot{\varphi} (x)}^{2} \det (D \phi^{-1}) \, dx \\ &= \dfrac{1}{2} \int \limits_{\Omega} \bnorm{\dot{\varphi} (x)}^{2} dx = L (\varphi, \dot{\varphi}) . \end{aligned}\]

Therefore, we can identify $L$ with a function on the Lie algebra of $\diff_{\mathrm{Vol}} (\Omega)$.
The Lie algebra of $\diff_{\mathrm{Vol}} (\Omega)$ is the set of divergence free vector fields $\mfrak{X}_{\div} (\Omega)$ equipped with the standard Lie bracket of vector fields.
Let $\varphi_{t}$ be a one-parameter group of volume preserving diffeomorphisms and let $X$ be the vector field $\tfrac{d}{dt} \big\vert_{t=0} \varphi_{t}$. Then for an arbitrary open set $B \subset \Omega$, we have

\[\begin{aligned} \int_{B} dx = \int_{\varphi_{t} (B)} dx \hspace{3em} \text{for all $t$} . \end{aligned}\]

Taking $B$ to an infinitesimal box and taking the time derivative, we get

\[\begin{aligned} 0 = \dfrac{d}{dt} \int_{\varphi_{t} (B)} dx = \dfrac{d}{dt} \int_{B} J^{t} dx = \int_{B} (\nabla \cdot X) \, J^{t} \, dx = \int_{\varphi_{t} (B)} (\nabla \cdot X) \, dx . \end{aligned}\]

This means the following three statements are equivalent (see Chorin & Marsden, 1993), (i) the fluid is incompressible, (ii) $\div X = 0$, (iii) $J^{t}=1$.
Since $\varphi_{t}$ is arbitrary, we find that the Lie algebra of $\diff_{\mathrm{Vol}} (\Omega)$ is a subset of $\mfrak{X}_{\div} (\Omega)$.
By reversing the arguments given, we find that the flow is a volume preserving map and so $\mfrak{X}_{\div} (\Omega)$ is a subset of the Lie algebra of $\diff_{\mathrm{Vol}} (\Omega)$.
Let $u, v \in \mfrak{X}_{\div} (\Omega)$ and define the reduced Lagrangian to be

\[\begin{aligned} l(u) = \dfrac{1}{2} \int \limits_{\Omega} \bnorm{u}^{2} dx = L (e, u) = L (\varphi \varphi^{-1} , \dot{\varphi} \varphi^{-1}) = L (\varphi , \dot{\varphi}) . \end{aligned}\]

Computing the functional derivative of $l$ w.r.t. $u$, we find

\[\begin{aligned} \bracket{ \dfrac{\delta l}{\delta u} , \delta u } = \dfrac{\partial}{\partial \eps} \bigg\vert_{\eps=0} l(u + \eps \delta u) = \int \limits_{\Omega} u \cdot \delta u \, dx , \end{aligned}\]

so that

\[\begin{aligned} \bracket{ \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta u} , v } = \int \limits_{\Omega} u_{t} \cdot v \, dx . \end{aligned}\]

Let $u, v \in \mfrak{X}_{\div} (\Omega)$ and define the reduced Lagrangian to be

\[\begin{aligned} l(u) = \dfrac{1}{2} \int \limits_{\Omega} \bnorm{u}^{2} dx . \end{aligned}\]

Additionally, we know that

\[\begin{aligned} \ad_{u} v = [v,u] = (v \cdot \nabla) u - (u \cdot \nabla) v, \end{aligned}\]

so that the coadjoint operator is determined by

\[\begin{aligned} \bracket{ \ad_{u}^{*} \dfrac{\delta l}{\delta u} , v } &= \bracket{ \dfrac{\delta l}{\delta u} , \, \ad_{u} v } \\ &= \bracket{ u , \, (v \cdot \nabla) u - (u \cdot \nabla) v } \\ &= \bracket{ \tfrac{1}{2} \nabla ( u \cdot u ) + (u \cdot \nabla) u + u \, (\nabla \cdot u) , \, v } \\ &= \bracket{ \tfrac{1}{2} \nabla ( u \cdot u ) + (u \cdot \nabla) u , \, v } , \end{aligned}\]

where we used that $u$ is divergence free.

Summarising, the variation of the reduced action leads to

\[\begin{aligned} \delta \int l(u) \, dt &= \int \bracket{ \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} + \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} , \, v } \, dt \\ &= \int \bracket{ u_{t} + \tfrac{1}{2} \nabla ( u \cdot u ) + (u \cdot \nabla) u , \, v } \, dt . \end{aligned}\]

As $v \in \mfrak{X}_{\div} (\Omega)$ but otherwise arbitrary, this vanishes if the term in the brackets is the gradient of a function (as can be seen by integration by parts),

\[\begin{aligned} u_{t} + (u \cdot \nabla) u + \tfrac{1}{2} \nabla \norm{u}^{2} = \nabla q . \end{aligned}\]

Setting

\[\begin{aligned} q = \tfrac{1}{2} \norm{u}^{2} - p , \end{aligned}\]

we get the incompressible Euler equation,

\[\begin{aligned} u_{t} + (u \cdot \nabla) u + \nabla p = 0 . \end{aligned}\]

With the Helmholtz decomposition theorem, one can show that $q$ and thus the pressure $p$ is uniquely determined.

Euler-Poincaré Reduction with Advected Parameters

To treat compressible fluids and kinetic theories, we need to extend the theory to the case when the system includes advected parameters $a_{0}$.
The advected Eulerian parameters are assumed to belong to $V^{*}$ (the dual of the vector space $V$) and are defined to be those variables which are Lie transported by the flow of the Eulerian velocity field.
Such parameters tend to break the symmetry of the Lagrangian $L_{a_{0}} : TG \rightarrow \rsp$.
The symmetry breaking part can often be identified as a dual vector in $V^{*}$, which allows us to rewrite the Lagrangian as

\[\begin{aligned} L &: TG \times V^{*} \rightarrow \rsp & & \text{with} & L (g, \dot{g}, a_{0}) &= L_{a_{0}} (g, \dot{g}) , \end{aligned}\]

in such a way that the symmetry is restored, i.e.,

\[\begin{aligned} g^{-1} \cdot L (g, \dot{g}, a_{0}) = L (g^{-1} g, g^{-1} \dot{g}, g^{-1} a_{0}) = L (g, \dot{g}, a_{0}) . \end{aligned}\]

For Vlasov-Maxwell and magnetohydrodynamics we need to further generalise the theory to Lagrangians $L : TG \times V^{*} \times TQ$ with the electromagnetic fields living on $TQ$.
If $G$ acts in a trivial way on $TQ$, we obtain the Euler-Poincaré equations for the fluid part and the usual Euler-Lagrange equations for the electromagnetic part.
We assume $L : TG \times V^{*} \rightarrow \rsp$ to be invariant under left or right action of $G$.
Upon fixing $a_{0} \in V^{*}$, we define the parameter dependend function $L_{a_{0}} : TG \rightarrow \rsp$ by

\[\begin{aligned} L_{a_{0}} (g, \dot{g}) := L(g, \dot{g}, a_{0}) . \end{aligned}\]

The reduced Lagrangian $l : \la{g} \times V^{*} \rightarrow \rsp$ is defined by

\[\begin{aligned} l (\xi, a) = L (e, \xi, a) = L (g^{-1} g, g^{-1} \dot{g}, g^{-1} a_{0}) = L (g, \dot{g}, a_{0}) , \end{aligned}\]

where $\xi(t) = g^{-1} (t) \, \dot{g} (t)$ and $a(t) = g^{-1} (t) \, a_{0}$.

If $\xi (t)$ and $a(t)$ extremise the reduced action $s = \int_{0}^{T} l (\xi, a) \, dt$ with respect to constrained variations $\delta \xi = \dot{\eta} + [\eta, \xi]$ and $\delta a = - \eta a$, then $\xi (t)$ and $a(t)$ satisfy the Euler-Poincaré equation

\[\begin{aligned} \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} &= \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} + \dfrac{\delta l}{\delta a} \diamond a & & \text{and} & \dfrac{\partial a}{\partial t} &= - \xi a . & \end{aligned}\]

If $g$ extremises the action $S = \int_{0}^{T} L_{a_{0}} (g, \dot{g}) \, dt$ for an arbitrary variation $\delta g$ with fixed end points, then $g$ satisfies the Euler-Lagrange equations

\[\begin{aligned} \dfrac{d}{dt} \dfrac{\partial L_{a_{0}}}{\partial \dot{g}} - \dfrac{\partial L_{a_{0}}}{\partial g} = 0 . \end{aligned}\]

$\xi (t)$ extremises the reduced action $s = \int_{0}^{T} l (\xi, a) \, dt$ with respect to variations

\[\begin{aligned} \delta \xi &= \dot{\eta} + [\xi, \eta] , \\ \delta a &= - \eta a , \end{aligned}\]

for an arbitrary curve $\eta(t) \in \la{g}$ which vanishes at the endpoints.

$\xi (t)$ satisfies the Euler-Poincaré equations on $\la{g} \times V^{*}$, defined by

\[\begin{aligned} \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} &= \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} + \dfrac{\delta l}{\delta a} \diamond a , \\ \dfrac{\partial a}{\partial t} &= - \xi a , \end{aligned}\]

where $\diamond : V \times V^{*} \rightarrow \la{g}^{*}$ is a bilinear operator, defined by

\[\begin{aligned} \bracket{ \alpha \diamond a , \eta }_{\la{g}^{*} \times \la{g}} &= - \bracket{ \eta a , \alpha }_{V^{*} \times V} & \text{for all $\alpha \in V$, $a \in V^{*}$, and $\eta \in \la{g}$} . \end{aligned}\]

Summary: For a given Lagrangian $L : TG \times V^{*} \rightarrow \rsp$ that is invariant under left action of $G$, basic Euler-Poincaré reduction consists of

(1) determining the reduced Lagrangian $l(\xi, a) = g^{-1} \cdot L(g, \dot{g}, a_{0})$ with $\xi = g^{-1} \dot{g}$,

(2) identifying the Lie algebra $\mfrak{g} \simeq T_{e} G$ and the corresponding Lie bracket $[\eta, \xi]$,

(3) computing the coadjoint operator $\ad_{\xi}^{*}$ from $\ad_{\xi}= [ \cdot , \xi]$,

(4) computing the diamond operator $\diamond$ from the left action of $\eta$ on elements of $V^{*}$.
Remark: For right $G$-invariant Lagrangians, the variations take the form

\[\begin{aligned} \delta \xi &= \dot{\eta} + [\xi, \eta] , \\ \delta a &= - a \eta , \end{aligned}\]

and the corresponding Euler-Poincaré equations are

\[\begin{aligned} \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta \xi} &= - \ad_{\xi}^{*} \dfrac{\delta l}{\delta \xi} + \dfrac{\delta l}{\delta a} \diamond a , \\ \dfrac{\partial a}{\partial t} &= - a \xi , \end{aligned}\]

where $\xi = \dot{g} (t) \, g^{-1} (t)$, $a(t) = a_{0} g^{-1} (t)$, and $\diamond : V \times V^{*} \rightarrow \la{g}^{*}$ is defined by

\[\begin{aligned} \bracket{ \alpha \diamond a , \eta }_{\la{g}^{*} \times \la{g}} &= - \bracket{ a \eta , \alpha }_{V^{*} \times V} & \text{for all $\alpha \in V$, $a \in V^{*}$, and $\eta \in \la{g}$} . \end{aligned}\]

The Vlasov-Maxwell System

Low’s Lagrangian for Vlasov-Maxwell in Lagrangian labeling is given by

\[\begin{aligned} L_{f_{0}} &= \sum \limits_{s} \int dz_{0} \, f_{0} (z_{0}) \, \bigg[ \bigg( \dfrac{e}{c} A(x) + mv \bigg) \cdot \dot{x} - \dfrac{1}{2} m v^{2} - e \phi (x) \bigg] \\ &+ \dfrac{1}{8\pi} \int dx \, \bigg[ \abs{ - \nabla \phi - \dfrac{\partial A}{\partial t} }^{2} - \abs{ \nabla \times A }^{2} \bigg] . \end{aligned}\]

The advected parameter corresponds to the distribution function $f$.
The coordinates $z=(x,v)$ evolve under the action of the diffeomorphism group $\diff(\rsp^{6})$ by $z(t) = \varphi_{t} (z_{0})$ with $\varphi_{t} \in \diff(\rsp^{6})$.
The particle relabeling map $\varphi$ acts on $f$ from the right as $f (z) = (f_{0} \circ \varphi_{t}^{-1}) (z)$ so that $f (z (z_{0}, t)) = f_{0} (z_{0})$ with $f_{0}$ the initial distribution function.
$\varphi$ does not act on the potentials $\phi$ and $A$ in the electromagnetic part of the Lagrangian, since electromagnetic dynamics must be independent of particle relabeling.
The phasespace velocity in the Lagrangian frame is $\dot{z} = \dot{\varphi}_{t} (z_{0})$, that is the rate of change of $z$ at the position $z$.
The Eulerian phasespace velocity is $u(z) = (\dot{\varphi}_{t} \circ \varphi_{t}^{-1}) (z)$, as we first have to take $z$ back to $z_{0}$ with $\varphi_{t}^{-1}$ before we can evaluate $\dot{\varphi}_{t}$ at $z_{0}$.
Low’s Lagrangian for Vlasov-Maxwell in Lagrangian labeling is given by

\[\begin{aligned} L_{f_{0}} &= \sum \limits_{s} \int dz_{0} \, f_{0} (z_{0}) \, \bigg[ \bigg( \dfrac{e}{c} A(t,x) + mv \bigg) \cdot \dot{x} - \dfrac{1}{2} m v^{2} - e \phi (t,x) \bigg] \\ &+ \dfrac{1}{8\pi} \int dx \, \bigg[ \abs{ - \nabla \phi (t,x) - \dfrac{\partial A (t,x)}{\partial t} }^{2} - \abs{ \nabla \times A (t,x) }^{2} \bigg] . \end{aligned}\]

The Euler-Lagrange equations for $z = (x,v)$ yield Newton’s equations for particle trajectories which together with $f(z(z_{0},t)) = f(z_{0})$ describe the evolution of the distribution function.
Maxwell’s equation correspond to the Euler-Lagrange equations for $(\phi, A)$.
The Lagrangian is invariant under particle relabeling, i.e.,

\[\begin{aligned} \varphi_{t}^{-1} \cdot L (\varphi_{t}, \dot{\varphi}_{t}, \phi, \dot{\phi}, A, \dot{A}, f_{0}) &= L (\varphi_{t} \varphi_{t}^{-1}, \dot{\varphi}_{t} \varphi_{t}^{-1}, \phi, \dot{\phi}, A, \dot{A}, f_{0} \varphi_{t}^{-1}) \\ &= l(u, \phi, \dot{\phi}, A, \dot{A}, f) , \end{aligned}\]

with the Eulerian phasespace velocity $u(z) = \dot{\varphi}_{t} \circ \varphi_{t}^{-1}$.

The reduced Lagrangian is given by

\[\begin{aligned} l &= \sum \limits_{s} \int dz \, f (t,x,v) \, \bigg[ \bigg( \dfrac{e}{c} A(t,x) + mv \bigg) \cdot u_{x} (t,x,v) - \dfrac{1}{2} m v^{2} - e \phi (t,x) \bigg] \\ &+ \dfrac{1}{8\pi} \int dx \, \bigg[ \abs{ - \nabla \phi (t,x) - \dfrac{\partial A (t,x)}{\partial t} }^{2} - \abs{ \nabla \times A (t,x) }^{2} \bigg] . \end{aligned}\]

Extremising the reduced action with constrained variations

\[\begin{aligned} \delta u &= \dot{w} + [w,u] = \dot{w} + (w \cdot \nabla_{z}) u - (u \cdot \nabla_{z}) w , & \delta f &= - \nabla_{z} \cdot (fw) , \end{aligned}\]

for some vector field $w$, leads to the Euler-Poincaré equations

\[\begin{aligned} \dfrac{\partial}{\partial t} \dfrac{\delta l}{\delta u} &= - \ad_{u}^{*} \dfrac{\delta l}{\delta u} + \dfrac{\delta l}{\delta f} \diamond f , & \dfrac{\partial f}{\partial t} &= - \nabla_{z} \cdot (fu) , \end{aligned}\]

given by

\[\begin{aligned} u_{x} &= v , & u_{v} &= E + \dfrac{1}{c} \, v \times B , & \dfrac{\partial f}{\partial t} + \nabla_{z} \cdot (fu) &= 0 . \end{aligned}\]

Maxwell’s equation are obtained as the usual Euler-Lagrange equations from variations of the action with respect to $(\phi, A)$.

Summary and Outlook

Application to Gyrokinetics

Squire’13 apply an analogous procedure to a gyrokinetic action principle in Lagrangian labeling in order to obtain an Eulerian gyrokinetic action principle on 6D phasespace
gyrokinetic reduction in the particle frame is well understood (Lie transforms)
starting point: any consistent gyrokinetic particle Lagrangian (Poincaré-Cartan form) together with the corresponding electromagnetic field Lagrangian
the Euler-Poincaré reduction procedure provides the constrained variations

Some Questions

Extension of Squire’13 to account for dynamic background fields à la Sugama’14?
Can Brizard’00 be obtained by applying Euler-Poincaré reduction to some spacetime-covariant action principle in Lagrangian labeling?

The Hamiltonian Side

How to pass from an Euler-Poincaré formulation to a Poisson bracket formulation?

regular Lagrangian: invertible Legendre transform (Holm’98, Marsden’84)
- use the Legendre transform to pass from a parametrised Lagrangian description to a parametrised Hamiltonian description
- transform parameters into dynamical variables
- identify the Poisson bracket of the system
Vlasov-Maxwell: degenerate Lagrangian, Legendre transform not invertible
Dirac constraints (Cendra’98, Squire’13)
- use a generalised Legendre transform from the parametrised Lagrangian to a parametrised Hamiltonian
- use Dirac’s theory to enforce the constraints associated with the degeneracy of the Lagrangian
- leads to Morrison-Marsden-Weinstein bracket
Poisson reduction of Peierls brackets (ongoing work of Josh Burby, PPPL)
- eliminate the parameters in the Lagrangian by introducing Lagrange multipliers
- identify the Peierls bracket of the system using the boundary symplectic form
- observe that the brackets and the Hamiltonian are invariant under the action of a semidirect product
- use this invariance to perform Poisson reduction to identify the Lie-Poisson bracket of the system and to eliminate the Lagrange multipliers

Discretisation of Euler-Poincaré Action Principles

incompressible fluids (Pavlov’09’11, Gawlik’11)
- approximate the (infinite-dimensional) group of volume-preserving diffeomorphisms by stochastic matrices $\Omega$
- stochastic matrices form a Lie group with the Lie algebra of $\Omega$-antisymmetric matrices $A$, correspond to discrete vector fields, for which $A^{T} \Omega + \Omega A = 0$
- after identifying discrete scalar fields, discrete one-forms, and a discrete pairing of elements of the Lie algebra to elements of the dual algebra, the discrete Euler-Poincaré equations follow along the lines of the continuous derivation
- discrete Kelvin-Noether theorem assures adherence to conservation laws
Vlasov-Maxwell
- the dynamics takes place on the group of symplectomorphisms (diffeomorphisms on phasespace which preserve the symplectic structure)
- in 1D1V: identical to volume-preserving diffeomorphisms
open questions
- how to obtain higher order discretisations
- breaking of symplecticity due to nonholonomic constraint

Summary

Euler-Poincaré reduction is a particular Lagrangian reduction method applicable to systems whose configuration space is a Lie group and whose Lagrangian is invariant under the action of this group.
Euler-Poincaré reduction is applicable to many problems from plasma physics like fluid, kinetic and gyrokinetic theories, where it provides a systematic method to derive Eulerian action principles from Lagrangian action principles by symmetry reduction.
A generalised Legendre transform allows for the transition to the Hamiltonian side and the derivation of Lie-Poisson brackets.
Knowledge of the underlying geometry is essential in devising new numerical methods which preserve the structure of the equations.

References

Geometry and Mechanics
- Michael Spivak. A Comprehensive Introduction to Differential Geometry Volume 1. Publish or Perish, 1999.
- John M. Lee. Introduction to Smooth Manifolds. Springer, 2012.
- Jerrold E. Marsden, Tudor S. Ratiu. Introduction to Mechanics and Symmetry. Springer, 2002.
- Darryl D. Holm, Tanya Schmah, Cristina Stoica. Mechanics and Symmetry. Oxford University Press, 2009.
- Darryl D. Holm. Geometric Mechanics Part I & II. Imperial College Press, 2011.
- Charles Cruickshank, Mark Ransley, Cesare Tronci. Lecture Notes on Geometric Mechanics. 2011.
Euler-Poincaré Reduction
- Henri Poincaré. Sur une forme nouvelle des équations de la mécanique. CR Acad. Sci, Volume 132, pages 369-371, 1901.
- Vladimir Arnold. Sur la géométrie différentielle des groupes de lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. In Annales de l’institut Fourier, Volume 16, pages 319-361. Institut Fourier, 1966.
- Vladimir I. Arnold, Boris A. Khesin. Topological Methods in Hydrodynamics. Springer, 1998.
- Darryl D. Holm, Jerrold E. Marsden, Tudor S. Ratiu. The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories. Advances in Mathematics, Volume 137, pages 1-81, 1998.
- Hernán Cendra, Darryl D. Holm, Mark J. W. Hoyle, Jerrold E. Marsden. The Maxwell–Vlasov Equations in Euler–Poincaré Form. Journal of Mathematical Physics, Volume 39, pages 3138-3157, 1998.
- Jonathan Squire, Hong Qin, William M. Tang, Christel Chandre, 2013, The Hamiltonian Structure and Euler-Poincaré Formulation of the Vlasov-Maxwell and Gyrokinetic Systems, Physics of Plasmas, 20 022501.
- Henry Jacobs. A Crash Course in the Euler–Poincaré Equation, 2011.
- Henry Jacobs. A Crash Course in Euler–Poincaré Reduction, 2013.
Discretisation Techniques
- Dmitry Pavlov. Structure-Preserving Discretization of Incompressible Fluids. PhD thesis, California Institute of Technology, 2009.
- Dmitry Pavlov, Patrick Mullen, Yiying Tong, Eva Kanso, Jerrold E. Marsden, Mathieu Desbrun, 2011, Structure-preserving discretization of incompressible fluids, Physica D: Nonlinear Phenomena, Volume 240, pages 443-458.
- Evan S. Gawlik, Patrick Mullen, Dmitry Pavlov, Jerrold E. Marsden, Mathieu Desbrun, 2011, Geometric, variational discretization of continuum theories, Physica D: Nonlinear Phenomena, Volume 240, pages 1724-1760.