In the Hamiltonian formalism of classical mechanics, a system with configuration space $Q$ is represented by a symplectic manifold $(T^*Q,\omega^\mathrm{can})$ called the phase space. The dynamics are then described by the flow of the Hamitlonian vector field of the Hamiltonian function $H\colon T^*Q \to\mathbb{R}$. If $\dim Q = n$, we have for $q\in Q$ that $T^*_qQ\cong\mathbb{R}^q$ and thus for $n>0$ such a system is non-compact, making some mathematics on it more difficult.

Reading about geometric quantization, I came across a classical system with a compact phase space, namely the classical spin particle with phase space $S_j^2$ (where the radius $j>0$ is the total angular momentum) and the canonical symplectic form $\omega_{\vec{J}}(\vec{v},\vec{w}) = j^{-2}\vec{J}\cdot(\vec{v}\times \vec{w})$. We interpret $\vec{J}\in S_j^2$ as the angular momentum, and in a magnetic field $\vec{B}$ we have $H(\vec{J}) = -\gamma\vec{J}\cdot\vec{B}$ so that the Hamiltonian vector field is given by $X_H(\vec{J}) = \gamma(\vec{J}\times \vec{B})$ describing Larmor precession. See also What precisely is a *classical* spin-1/2 particle?.

This made me curious whether there are any other mechanical systems that can be described in terms of a classical phase space (or, more generally, a phase space that is not just a cotangent bundle), or whether this is just a singular interesting case. It would also be nice to get some insight on why Hamilton's formalism also gives the correct answer in these more general phase spaces.


1 Answer 1


I could comment that a Hamiltonian system with compact symplectic phase space could appear after Lie-Poission reduction of a left (or right) invariant Hamiltonian defined on the cotangent bundle of a compact Lie group.

If $G$ is a compact Lie group, then $T^*G$ is a symplectic manifold. The left action $L_g \,:\, G \, \to \, G$ defined as $L_g(h) = g\,h$ induces a left action on the cotangent bundle $\big(L_{g^{-1}}\big)^*: \, T^*G \,\to \,T^*G $. The action has a moment map, $\big(R_{\pi(p)^{-1}}\big)^*_{\pi(p)} \,:\, T^*_{\pi(p)}G \,\to\, \mathfrak{g}^*$, where $R_g : G \to G$ is the right action $R_g(h) = hg^{-1}$. $\,\,$ If $H \,:\, T^*G \,\to \,\mathbb{R}$ is a left invariant Hamiltonain, i.e. $\big(L_{g^{-1}}\big)^* H = H$, the moment map will project the Hamiltonain vector field $X_H$ from a vector field on the whole $T^*G$ to a vector field on the dual Lie-algebra $\mathfrak{g}^*$ so that the resulting vector field is $$\frac{d}{dt}m \,=\, -\, \text{ad}^*\Big(dH(m)\Big)\, m$$ However, this latter vector field is actually tangent to each coadjoint orbit $O(m_0)^* \,=\,\text{Ad}(G)^*m_0$ of the group $G$, and since $G$ is compact, the coadjoint orbit is a compact submanifold of $\mathfrak{g}^*$. On top of that, there is a two form, $\omega_{m}\Big(\text{ad}^*(u)\,m,\, \text{ad}^*(v)\, m\Big) \, = \, m \cdot [u, v]$, which restricted to a coadjoint orbit $O(m_0)^*$ is in fact a symplectic form. As a result, the vector field $$\frac{d}{dt}m \,=\, -\, \text{ad}^*\Big(dH(m)\Big)\, m$$ restricted to a coadjoint orbit $O(m_0)^*$ is a Hamiltonain vector field with Hamiltonian $H(m)$, where $O(m_0)^*$ is a compact symplectic manifold.

So, as you can see, this is one possible mechanism that supplies examples of Hamiltonain vector fields on compact symplectic manifolds.

For example, in the case of a rigid body, freely rotating in space, the configuration space is the rotation group $\text{SO}(3)$, the original phase space is $T^*\text{SO}(3)$, and the Hamiltonian that describes the dynamics of the freely rotating body is left-invariant. After performing Lie-Poission reduction to this system, we switch from dynamics on $T^*\text{SO}(3)$ to dynamics on the dual Lie algebra $T_\text{Id}^*\text{SO}(3) = \text{so}(3)^* \cong \mathbb{R}^3$, which is the space of angular momenta, viewed in the body rotating frame, i.e. we switch from dynamics of $\text{SO}(3)-$rotations to dynamics of angular momenta. The reduced Hamiltonain on the space of angular momenta (i.e. the dual algebra) is $$H \,=\, \frac{1}{2}\, \vec{M}\cdot J\,\vec{M}$$ and $$\text{ad}^*\big(\vec{\Omega}\big) \,\vec{M} \,=\, \vec{\Omega}\,\times \,\vec{M}$$ As a result, one obtains Euler's equations of motion $$\frac{d}{dt} \vec{M} \,=\, -\,J\,\vec{M} \times \vec{M}$$
where $J$ is the inverse of the inertia tensor. In this case, $$\text{ad}^*\Big(dH(\vec{M})\Big) \,\vec{M}\,=\, J\vec{M} \times \vec{M}$$ If the initial angular momentum has magnitude $\|\vec{M}_0\| = c_0$, then all solutions lie on the sphere $$\|\vec{M}\|^2 = c_0^2$$ which is in fact the coadjoint orbit of the initial momentum $\vec{M}_0$ under the action of the rotation group $\text{SO}(3)$. Thus, the vector equation
$$\frac{d}{dt} \vec{M} \,=\, -\,J\,\vec{M} \times \vec{M}$$ gives rise to a vector field which is tangent to any sphere of type $$\|\vec{M}\|^2 = c_0^2$$ and that vector field is a Hamiltonian vector field on the symplectic manifold $\|\vec{M}\|^2 = c_0^2$.

Now, an upgrade of the latter classical example: add to the freely rotating body in space, a disk with center of mass coinciding with the center of mass of the body, and that disks rotating with constant angular velocity of magnitude $\omega_0$ and with axis the shortest inertial axis of the body (in body frame) spanned by the unit vector $\vec{i}$, then one obtains again a left invariant vector field on the group $\text{SO}(3)$ which again, by Lie-Poisson Reduction, reduces to the Hamiltonain vector field $$\frac{d}{dt} \vec{M} \,=\, -\,\Big(\,J\,\vec{M} - {I_0}{J_1} \omega_0\,\vec{i}\,\Big) \times \vec{M}$$ on a a two-dimensional sphere, which is a compact symplectic manifold, $$\|\vec{M}\,\|^2 \,=\, \vec{M}\cdot\vec{M} \,=\, c_0^2$$ with reduced Hamiltonain $$H = \big(\vec{M}\cdot J\,\vec{M}\big) - {I_0}{J_1} \omega_0\,\big(\vec{M} \cdot \vec{i}\big)$$ Here, $J$ is the inverse of the inertia tensor of the body with the disk as if being locked to the body, $J_1$ is the eigenvalue $J\,\vec{i} \,=\, J_1\, \vec{i}$ and $I_0$ is the scalar inertia moment of the disk around the axis spanned by $\vec{i}$. This particular system is used as a mathematical model of a satellite with a spinning disk rotating at a constant angular velocity, that makes the satellite's shortest inertial axis the one with the minimal energy. This latter fact, makes the shortest inertial axis, which is geometrically the longest axis of the satellite, the most stable one and thus the satellite's attitude is oriented along this axis and this attitude is stable. Without such rotating disk, the most stable inertial axis is the longest inertial axis, i.e. the shortest geometric axis.

Other examples of systems with compact symplectic phase spaces, arising as the result of Lie-Possion Reduction, could be added too, like for example classical Heisenberg models, where the reduction of the cartesian products of copies of the compact Lie group $\text{SU}(2)$ (or possibly $\text{SO}$(3) ) yield a compact symplectic manifold which is a product of 2D unit spheres $\|\vec{S}_i\| = 1$ and the dynamics is
$$\frac{d}{dt}\vec{S}_i \,=\, \Big(\sum_{j \text{ neighbour of } i}\,J_{ij}\,\vec{S}_j\,\,\Big) \times \vec{S}_i$$ with Hamiltonain $$H \,=\, \sum_{ij \text{ edge of lattice}}\,J_{ij}\,\vec{S}_i\cdot\vec{S}_j$$


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