Symplectic geometry provides the natural mathematical framework for Hamiltonian mechanics. In geometric numerical integration, the primary objective of a symplectic integrator is to preserve the underlying symplectic structure of the continuous system at the discrete level.
This section formally introduces symplectic manifolds, symplectic forms, canonical coordinates, Darboux charts, and key structural identities. These concepts are foundational for Sections 2.2–2.3 and for all later Hamiltonian geometric integrators.
Let \(M\) be a smooth manifold of even dimension \(2m\). A 2-form on \(M\) is a smooth function assigning to each \(x\in M\) an alternating bilinear form
\[ \omega_x : T_xM \times T_xM \to \mathbb{R}, \qquad \omega_x(u,v) = -\omega_x(v,u). \]
A 2-form is called non-degenerate if
\[ \forall x\in M: \quad \omega_x(u,\cdot)=0 \implies u=0. \]
Non-degeneracy implies that the linear map
\[ T_xM \to T_x^*M, \qquad u \mapsto \omega_x(u,\cdot) \]
is an isomorphism; hence \(\dim M\) must be even.
The exterior derivative of a 2-form \(\omega\) is a 3-form \(d\omega\); the condition \(d\omega = 0\) is called closedness.
Definition 2.1 (Symplectic manifold). A pair \( (M, \omega) \) is called a symplectic manifold if:
A symplectic manifold is necessarily even-dimensional, but unlike Riemannian manifolds it has no preferred notion of length or angle. It is purely “area-based”.
\[ \omega_0 = \sum_{i=1}^m dq_i \wedge dp_i = dq_1\wedge dp_1 + \cdots + dq_m\wedge dp_m. \]
In matrix form, writing \( z=(q,p) = (q_1,\dots,q_m,p_1,\dots,p_m) \), \(\omega_0(u,v) = u^\top \Omega v\), where
\[ \Omega = \begin{pmatrix} 0 & I_m\\ -I_m & 0 \end{pmatrix}. \]
This will be the prototype for numerical symplecticity in later chapters.
Theorem 2.1 (Darboux, 1882). Let \( (M,\omega) \) be a symplectic manifold. For every point \(x_0\in M\), there exists a neighbourhood \(U\) and local coordinates
\[ (q_1,\dots,q_m,p_1,\dots,p_m): U \to \mathbb{R}^{2m} \]
such that, in these coordinates,
\[ \omega = \sum_{i=1}^m dq_i \wedge dp_i. \]
Thus all symplectic manifolds are locally “canonical”, unlike Riemannian manifolds which have curvature invariants.
Darboux's theorem is crucial because it implies that symplecticity is a coordinate-free property. Numerical symplectic integrators should therefore preserve the symplectic form in any coordinate chart.
Definition 2.2 (Symplectic diffeomorphism). A diffeomorphism \( \Psi : M \to M \) is symplectic if
\[ \Psi^* \omega = \omega, \]
or equivalently, for all \(x\in M\),
\[ D\Psi(x)^\top\, \omega_{\Psi(x)}\, D\Psi(x) = \omega_x. \]
A symplectic diffeomorphism is traditionally called a canonical transformation in classical mechanics.
Proposition 2.1. Let \(X_H\) be the Hamiltonian vector field defined by \(i_{X_H}\omega = dH\). Then its flow \( \varphi_H^t \) satisfies
\[ (\varphi_H^t)^*\omega = \omega. \]
Proof sketch. Consider the Lie derivative:
\[ \frac{d}{dt} (\varphi_H^t)^*\omega = (\varphi_H^t)^*(\mathcal{L}_{X_H}\omega) = (\varphi_H^t)^*(d(i_{X_H}\omega) + i_{X_H}d\omega) = (\varphi_H^t)^*(ddH + i_{X_H} 0) = 0. \]
Thus, \((\varphi_H^t)^*\omega\) is constant in \(t\), equal to \(\omega\) at \(t=0\).
This property is the backbone of the long-time qualitative accuracy of symplectic integrators.
The distinguishing feature of symplectic maps is that they preserve oriented 2-dimensional areas defined by the symplectic form, not necessarily Euclidean areas.
The interactive figure below demonstrates how a linear map behaves when it is:
Notice that:
In this section we introduced:
These ideas are the backbone of symplectic integration. Section 2.2 will introduce Hamiltonian vector fields, Poisson brackets, and the deep relation between \(\omega\), Hamilton's equations, and structure preservation.
Hamiltonian mechanics is naturally expressed on a symplectic manifold \((M,\omega)\) by associating to each Hamiltonian function \(H : M \to \mathbb{R}\) a unique vector field \(X_H\) via the symplectic form. In canonical coordinates, this yields Hamilton’s equations. In this section we:
Let \((M,\omega)\) be a symplectic manifold as defined in §2.1. For each smooth Hamiltonian \(H : M \to \mathbb{R}\), we define the associated Hamiltonian vector field \(X_H\) by:
\[ i_{X_H}\,\omega = dH, \]
where \(i_{X_H}\) denotes interior contraction with the vector field \(X_H\). At each point \(x\in M\),
\[ \omega_x(X_H(x), \cdot) = dH_x(\cdot). \]
Because \(\omega_x\) is non-degenerate, this relation uniquely determines \(X_H(x)\) for each \(x\), giving a globally defined vector field \(X_H\).
The map
\[ \flat_\omega : T_xM \to T_x^*M, \qquad v \mapsto \omega_x(v,\cdot) \]
is an isomorphism (“musical isomorphism”). Then
\[ X_H(x) = \big(\flat_\omega^{-1}\big)(dH_x), \]
i.e. \(X_H\) is the vector field obtained by “raising the index” of the differential \(dH\) using the symplectic form. The resulting flow \(\varphi_H^t\) is the Hamiltonian evolution of the system with Hamiltonian \(H\).
On the standard symplectic vector space \(M = \mathbb{R}^{2m}\) with coordinates \(z = (q,p) = (q_1,\dots,q_m,p_1,\dots,p_m)\) and symplectic form
\[ \omega_0 = \sum_{i=1}^m dq_i \wedge dp_i, \]
the Hamiltonian vector field has the familiar coordinate expression given by Hamilton’s equations.
Writing \(X_H(z) = (\dot{q},\dot{p})\), the condition \(i_{X_H}\omega_0 = dH\) yields
\[ \dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}, \qquad i=1,\dots,m. \]
In matrix notation, with
\[ \Omega = \begin{pmatrix} 0 & I_m\\ -I_m & 0 \end{pmatrix}, \qquad z = \begin{pmatrix}q\\p\end{pmatrix}, \]
we can write the Hamiltonian system as
\[ \dot{z} = X_H(z) = \Omega \nabla H(z). \]
The flow \(\varphi_H^t\) generated by this ODE is symplectic, as shown in §2.1.4.
A fundamental property of Hamiltonian flows is that the Hamiltonian itself is a first integral of its own flow.
Proposition 2.2 (Conservation of energy). Let \(H : M \to \mathbb{R}\) be a Hamiltonian and \(X_H\) the associated Hamiltonian vector field. Then, along the flow \(\varphi_H^t\),
\[ \frac{d}{dt} H(\varphi_H^t(x)) = 0. \]
Proof. We have
\[ \frac{d}{dt} H(\varphi_H^t(x)) = dH_{\varphi_H^t(x)}\!\big(\dot{\varphi}_H^t(x)\big) = dH_{\varphi_H^t(x)}\!\big(X_H(\varphi_H^t(x))\big). \]
Using the defining relation \(i_{X_H}\omega = dH\), we get
\[ dH(X_H) = \omega(X_H, X_H) = 0 \]
because \(\omega\) is skew-symmetric. Hence the derivative vanishes identically. \(\square\)
Thus Hamiltonian trajectories lie on level sets of \(H\). In canonical coordinates this means:
\[ \nabla H(z) \cdot X_H(z) = 0, \]
i.e. the vector field is everywhere tangent to the level surfaces of \(H\).
On a symplectic manifold \((M,\omega)\), the assignment \(H \mapsto X_H\) can be used to define a Lie algebra structure on smooth functions via the Poisson bracket.
Definition 2.3 (Poisson bracket). For \(F,G \in C^\infty(M)\), their Poisson bracket is
\[ \{F,G\} := \omega(X_F,X_G) = dF(X_G) = -\,dG(X_F). \]
In canonical coordinates on \(\mathbb{R}^{2m}\), this reduces to the familiar formula
\[ \{F,G\} = \sum_{i=1}^m \left( \frac{\partial F}{\partial q_i} \frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i} \frac{\partial G}{\partial q_i} \right). \]
The Poisson bracket has the following properties:
In particular, the Jacobi identity means that \((C^\infty(M),\{\cdot,\cdot\})\) is a Lie algebra. The map
\[ C^\infty(M) \to \mathfrak{X}(M), \qquad H \mapsto X_H \]
is a Lie algebra homomorphism up to sign:
\[ [X_F,X_G] = X_{\{F,G\}}. \]
This identity is pivotal in the analysis of splitting and composition methods, where commutators of vector fields appear in the BCH expansion (see §1.3.4).
Let \(H\) be the Hamiltonian of the system. A function \(I : M\to\mathbb{R}\) is a constant of motion (first integral) if
\[ \{I,H\} = 0. \]
To see this, note that the time derivative of \(I\) along the Hamiltonian flow is
\[ \frac{d}{dt} I(\varphi_H^t(x)) = \{I,H\}(\varphi_H^t(x)). \]
Thus \(\frac{d}{dt} I(\varphi_H^t(x)) = 0\) iff \(\{I,H\} = 0\) along the trajectory.
When we design geometric integrators, we are often interested in methods that preserve certain Poisson commuting quantities, at least approximately.
For the harmonic oscillator with Hamiltonian
\[ H(q,p) = \tfrac{1}{2}(p^2 + q^2), \]
we have
\[ \frac{\partial H}{\partial p} = p, \qquad \frac{\partial H}{\partial q} = q, \]
thus
\[ \dot{q} = p, \qquad \dot{p} = -q. \]
The flow is a rotation in phase space; energy level sets are concentric circles.
Consider a particle in \(\mathbb{R}^3\) with position \(x\) and momentum \(p\), Hamiltonian
\[ H(x,p) = \frac{1}{2m} |p|^2 + V(|x|), \]
where \(V\) is a radial potential. The angular momentum
\[ L = x \times p \]
is a constant of motion, which can be verified by checking \(\{L_i,H\}=0\), or geometrically via rotational symmetry and Noether’s theorem. This is a basic example where geometric structure (rotational invariance) leads to conserved quantities.
| Hamiltonian \(H\) | Phase space | Hamiltonian vector field \(X_H\) | Conserved quantity |
|---|---|---|---|
| \(\tfrac{1}{2}(p^2 + q^2)\) | \(\mathbb{R}^2\) | \((p,-q)\) | Energy \(H\) |
| \(\tfrac{1}{2m}|p|^2 + V(|x|)\) | \(\mathbb{R}^3\times\mathbb{R}^3\) | \((p/m,\,-\nabla V)\) | Energy \(H\), angular momentum \(L\) |
The small interactive demo below evaluates the Poisson bracket \(\{F,G\}\) at a grid of points for some simple choices of functions \(F,G\) in the plane. It highlights (in colour) where the bracket vanishes.
In this example, \(\{H,q\} = -p\), so the bracket vanishes along the horizontal axis \(p=0\) and changes sign across it. This simple picture connects the algebraic definition with a geometric field on phase space.
We have:
In the next section (2.3), we develop the role of symmetry in Hamiltonian systems, introduce momentum maps, and state a version of Noether’s theorem in the symplectic setting, preparing the ground for symmetry-preserving geometric integrators.
The structure of Hamiltonian systems is deeply connected with symmetry. Continuous symmetries give rise to conserved quantities, and these conserved quantities in turn constrain the geometry of the system’s flow. This section formalises:
Let \(G\) be a Lie group with Lie algebra \(\mathfrak{g}\). A smooth (left) group action on a symplectic manifold \((M,\omega)\) is a map
\[ \Phi: G \times M \to M, \qquad \Phi(g,x) = g\cdot x, \]
satisfying:
The action is called symplectic if each map
\[ \Phi_g : M \to M, \qquad x\mapsto g\cdot x, \]
is a symplectic diffeomorphism:
\[ \Phi_g^* \omega = \omega. \]
For each \(\xi \in \mathfrak{g}\) we define the infinitesimal generator of the action:
\[ \xi_M(x) := \left.\frac{d}{dt}\right|_{t=0} \exp(t\xi)\cdot x \in T_xM. \]
Because the action is symplectic, each \(\xi_M\) satisfies:
\[ \mathcal{L}_{\xi_M}\omega = 0. \]
The vector field \(\xi_M\) is symplectic but not necessarily Hamiltonian. If there exists a smooth function \(J_\xi\) such that
\[ i_{\xi_M}\omega = dJ_\xi, \]
then the symmetry is called Hamiltonian.
For a Hamiltonian group action, the map
\[ J : M \to \mathfrak{g}^*, \]
is called the momentum map if
\[ \langle J(x), \xi\rangle = J_\xi(x), \qquad dJ_\xi = i_{\xi_M}\omega, \]
for all \(\xi\in\mathfrak{g}\).
The momentum map collects all conserved quantities associated with the symmetry.
A momentum map is called equivariant if:
\[ J(g\cdot x) = \mathrm{Ad}^*_{g} \, J(x), \]
where \(\mathrm{Ad}^*\) is the coadjoint action. Equivariance leads to powerful structural results but is not strictly required for basic conservation laws.
Theorem 2.3 (Noether). Let \(G\) act on a symplectic manifold \((M,\omega)\) by symplectic diffeomorphisms. Suppose the Hamiltonian \(H : M\to\mathbb{R}\) is invariant under the action:
\[ H(g\cdot x) = H(x),\qquad \forall g\in G. \]
Then for each \(\xi\in\mathfrak{g}\), the corresponding momentum map component \(J_\xi = \langle J,\xi\rangle\) satisfies:
\[ \{J_\xi, H\} = 0. \]
That is, symmetry implies a conserved quantity.
If \(H\) is invariant under the flow of \(\xi_M\), then:
\[ \mathcal{L}_{\xi_M} H = 0. \]
But
\[ \mathcal{L}_{\xi_M} H = dH(\xi_M) = \omega(X_H,\xi_M) = \{J_\xi, H\}. \]
Hence the Poisson bracket vanishes, giving conservation of \(J_\xi\).
On \(M=\mathbb{R}^{2m}\) with coordinates \((q,p)\), the group \(G=\mathbb{R}^m\) acts by translations:
\[ v \cdot (q,p) = (q+v,p). \]
The infinitesimal generator corresponding to direction \(e_i\in\mathbb{R}^m\) is simply
\[ (e_i)_M = \frac{\partial}{\partial q_i}. \]
Solving \(i_{(e_i)_M}\omega = dJ_{e_i}\) gives
\[ J_{e_i} = p_i. \]
Thus the momentum map is \(J(q,p) = p\), i.e. linear momentum.
For \(G = SO(3)\) acting on phase space \(M = T^*\mathbb{R}^3 \cong \mathbb{R}^3 \times \mathbb{R}^3\) by rotations:
\[ R\cdot(x,p) = (Rx, Rp), \]
the Lie algebra \(\mathfrak{so}(3)\) identifies with \(\mathbb{R}^3\), with infinitesimal action:
\[ \xi_M = (\xi \times x,\, \xi \times p). \]
Then the momentum map is
\[ J(x,p) = x\times p = L, \]
i.e. the angular momentum vector.
The JavaScript demo below shows the vector field for the Hamiltonian \[ H(x,y,p_x,p_y) = \frac{1}{2}(p_x^2+p_y^2) + V(r), \qquad r=\sqrt{x^2+y^2}, \] with \(V(r) = \tfrac{1}{2}r^2\) for simplicity. The motion occurs on circles and angular momentum \(L = x p_y - y p_x\) is conserved.
In this section, we introduced:
This prepares the ground for Chapters 3–4: Symplectic integrators (leapfrog, splitting, variational) must preserve symplectic form, and **symmetry-preserving methods** preserve momentum maps. Both ideas are essential in long-time stability analysis.
Symplectic manifolds model unconstrained Hamiltonian mechanics. But many important systems are constrained, reduced by symmetry, or live on spaces whose geometry is not symplectic — e.g. a rigid body’s angular momentum space \(\mathfrak{so}(3)^*\), plasma models, Euler equations, Lie–Poisson systems, and systems obtained after symmetry reduction.
Poisson manifolds provide the natural geometric framework for all of these. The theory is a strict generalisation of symplectic geometry:
A Poisson bracket on a smooth manifold \(M\) is a bilinear map
\[ \{\,\cdot\,,\,\cdot\,\}: C^\infty(M)\times C^\infty(M) \to C^\infty(M) \]
satisfying:
\[ \{f,\{g,h\}\} + \{g,\{h,f\}\} + \{h,\{f,g\}\} = 0. \]
A manifold equipped with such a bracket is a Poisson manifold \((M,\{\cdot,\cdot\})\).
A Poisson bracket is encoded by a bivector field \(\pi \in \Gamma(\Lambda^2 TM)\) such that:
\[ \{f,g\} = \pi(df,dg). \]
The Jacobi identity is equivalent to:
\[ [\pi,\pi]_{\mathrm{SN}} = 0, \]
where \([\cdot,\cdot]_{\mathrm{SN}}\) is the Schouten–Nijenhuis bracket.
In canonical coordinates \((q,p) \in \mathbb{R}^{2m}\),
\[ \pi = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix}. \]
This reproduces the standard symplectic bracket. In a general Poisson manifold, the rank of \(\pi(x)\) may vary with \(x\).
Given \(H\in C^\infty(M)\), the Hamiltonian vector field is:
\[ X_H(f) = \{f,H\},\qquad \forall f\in C^\infty(M). \]
Equivalently,
\[ X_H = \pi^\sharp(dH), \]
where \(\pi^\sharp: T^*M\to TM\) is contraction with \(\pi\):
\[ \pi^\sharp(\alpha) = \pi(\alpha,\cdot). \]
Thus, Poisson geometry generalises symplectic dynamics to allow degeneracy.
If \(\pi(x)\) has rank \(2k(x)\), we define the distribution
\[ \mathcal{D}_x = \pi^\sharp(T_x^* M)\subset T_xM. \]
The Stefan–Sussmann theorem implies that this integrable distribution defines a foliation of \(M\) into immersed submanifolds symplectic leaves \(S\subset M\), on which:
Poisson manifolds are therefore “stratified symplectic manifolds”.
A smooth map \(F: (M,\pi_M)\to (N,\pi_N)\) is a Poisson map if:
\[ \{f\circ F, g\circ F\}_M = \{f,g\}_N \circ F \qquad \forall f,g\in C^\infty(N). \]
Example: symplectic maps are Poisson maps with respect to the canonical bracket.
In geometric numerical integration, a Poisson integrator satisfies:
\[ \{f,g\}\circ \Phi_h = \{f\circ \Phi_h,\, g\circ \Phi_h\}, \]
which generalises structure preservation when the system is not symplectic globally.
A fundamental class of Poisson manifolds arises from Lie algebras. If \(G\) is a Lie group with Lie algebra \(\mathfrak{g}\), then the dual space \(\mathfrak{g}^*\) carries a natural Poisson structure:
\[ \{F,G\}(\mu) = \langle \mu,\; [dF(\mu), dG(\mu)] \rangle, \]
where the bracket is the Lie bracket on \(\mathfrak{g}\).
Take \(\mathfrak{g} = \mathfrak{so}(3)\cong\mathbb{R}^3\) with cross product. Then the Lie–Poisson equations are Euler’s equations:
\[ \dot{L} = L \times \nabla H(L), \]
where \(L\in\mathbb{R}^3\) is angular momentum.
Symplectic leaves are spheres \(||L||=\mathrm{const}\).
The following JavaScript simulation shows a simple Lie–Poisson flow on the unit sphere:
\[ \dot{L} = L \times (A L), \]
where \(A=\mathrm{diag}(a_1,a_2,a_3)\) models anisotropic rigid body inertia.
In this section we introduced the Poisson generalisation of Hamiltonian mechanics:
These ideas underpin geometric numerical integration for systems that are not globally symplectic—particularly Lie–Poisson integration, splitting on noncanonical brackets, and reduced systems.
One of the most profound results in symplectic geometry is that it admits no local invariants. Unlike Riemannian geometry—where curvature, torsion, and other invariants distinguish local geometries—every symplectic structure looks locally identical to the standard canonical form.
This is the content of Darboux’s theorem. It implies that Hamilton’s equations always admit canonical coordinates \((q,p)\) locally, regardless of global geometry, topology, or constraints.
Theorem 2.5 (Darboux). Let \((M,\omega)\) be a symplectic manifold of dimension \(2m\). For every point \(x\in M\), there exists a coordinate chart
\[ (U; q_1,\dots,q_m, p_1,\dots,p_m) \]
such that on \(U\):
\[ \omega = \sum_{i=1}^m dq_i \wedge dp_i. \]
In other words, all symplectic manifolds are locally canonical.
No metric, no curvature, no Christoffel symbols: symplectic geometry has no local invariants.
Darboux’s theorem tells us that:
The importance for numerical integration is immense:
The classical proof uses Moser’s method. The idea is to connect the given symplectic form \(\omega\) to the canonical form \(\omega_0\) smoothly, and construct a diffeomorphism that carries one to the other.
Thus, \(\varphi_1\) gives the desired canonical coordinates.
In Darboux coordinates, the Poisson tensor becomes:
\[ \pi = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}, \qquad \{q_i,p_j\} = \delta_{ij},\quad \{q_i,q_j\}=0,\quad \{p_i,p_j\}=0. \]
Hence, Hamilton’s equations take their standard form:
\[ \dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}. \]
The appearance of these equations is not tied to special coordinates: every Hamiltonian system is locally canonical.
Darboux’s theorem implies:
The theorem is therefore foundational for the entire numerical theory in later chapters.
Darboux’s theorem has no analogue on Poisson manifolds:
However:
In previous sections we saw:
We now treat Lie–Poisson reduction — one of the deepest links between symmetry, geometry, and Hamiltonian mechanics. It explains why systems such as:
evolve on dual Lie algebras and preserve the coadjoint orbit structure.
This section provides the foundation for Lie–Poisson integrators, which appear later in geometric numerical integration: algorithms that preserve Casimirs, coadjoint orbits, and Lie–Poisson brackets exactly.
Suppose a Hamiltonian system has a symmetry group \(G\). By Noether’s theorem:
\[ \text{symmetry} \;\Rightarrow\; \text{momentum map} \;\Rightarrow\; \text{conserved quantity}. \]
Reduction asks the opposite question:
Can we eliminate redundant degrees of freedom introduced by symmetry?
For instance, the rigid body rotation group \(SO(3)\) acts freely on the configuration space. By reducing the symmetry, the true dynamics live not on the tangent bundle of the group but on \(\mathfrak{so}(3)^*\cong\mathbb{R}^3\).
Let \(G\) be a Lie group with Lie algebra \(\mathfrak{g}\). The adjoint action of \(G\) on \(\mathfrak{g}\) is:
\[ \mathrm{Ad}_g(X) := g X g^{-1}. \]
Dualising gives the coadjoint action:
\[ \mathrm{Ad}^*_g : \mathfrak{g}^* \to \mathfrak{g}^*, \qquad \langle \mathrm{Ad}_g^*\mu, X\rangle = \langle \mu, \mathrm{Ad}_{g^{-1}}X\rangle. \]
The orbits of this action,
\[ \mathcal{O}_\mu = \{\mathrm{Ad}_g^*\mu : g\in G\}, \]
are called coadjoint orbits.
Crucially:
Coadjoint orbits are naturally symplectic manifolds.
Each coadjoint orbit \(\mathcal{O}_\mu\subset \mathfrak{g}^*\) carries a canonical symplectic form defined as follows.
\[ \omega_\mu(\xi_{\mathfrak{g}^*},\eta_{\mathfrak{g}^*}) = \langle \mu,\;[\xi,\eta]\rangle. \]
Here \(\xi_{\mathfrak{g}^*}\) is the infinitesimal generator of the coadjoint action associated with \(\xi\in\mathfrak{g}\):
\[ \xi_{\mathfrak{g}^*}(\mu) = \mathrm{ad}_\xi^*\mu. \]
Thus each coadjoint orbit is a symplectic leaf of the Lie–Poisson bracket of Section 2.4.
For \(\mathfrak{g}^*\) the Poisson bracket of functions \(F,G\) is:
\[ \{F,G\}(\mu) = \langle \mu,\; [dF(\mu), dG(\mu)] \rangle. \]
This provides a direct bridge between symmetry reduction and Poisson geometry.
Consider a Hamiltonian \(H:\mathfrak{g}^*\to\mathbb{R}\). The Hamiltonian vector field on the Poisson manifold \(\mathfrak{g}^*\) is:
\[ \dot{\mu} = \mathrm{ad}^*_{\nabla H(\mu)} \mu. \]
This is the Lie–Poisson equation.
For \(\mathfrak{so}(3)^*\cong\mathbb{R}^3\):
\[ \dot{L} = L \times \Omega, \qquad \Omega = \nabla H(L) = (I^{-1}L). \]
This yields Euler’s equations of rigid body rotation — a Hamiltonian flow on the sphere \(|L|=\mathrm{const}\).
A Casimir on a Poisson manifold satisfies:
\[ \{C,F\} = 0\qquad\forall F. \]
On a Lie–Poisson manifold \(\mathfrak{g}^*\), Casimirs label the symplectic leaves.
\[ C(L) = \|L\|^2 \]
is a Casimir. It defines spherical leaves:
\[ \mathcal{O}_r = \{L\in\mathbb{R}^3 : \|L\|= r\}. \]
A Lie–Poisson integrator must preserve Casimirs exactly — a primary requirement in geometric numerical integration.
The following JavaScript simulation shows motion on a coadjoint orbit for the \(\mathfrak{so}(3)^*\) example:
In this section we developed the geometric machinery of reduced Hamiltonian systems:
This geometric background is essential for numerical methods that preserve: