# 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

• 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}

\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}

\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

• 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.