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The geometry of Harmonic Oscillators

[Last time](/blog/2022/07/24/symplectic.html), we introduced geometric mechanics and symplectic geometry. This will let us get at the mathematical structures hiding inside harmonic oscillators.

Integral periods and circle actions

With this geometric formulation of classical mechanics in hand, let us look to higher dimensional harmonic oscillators. Imagine for instance a ball tethered to a spring in a plane. Throw it around a bit, see how it ticks:

Its position is represented by a point \(\vec{q} \in \mathbb{R}^2\), and the spring applies a force \(-\vec{q}\). This force comes from a quadratic potential \(V(\vec{q})=\Vert \vec{q}\Vert ^2\). Phase space is \(T^{*\mathbb{R}^{2}}\cong\mathbb{R}^4\) with coordinates \(\vec{q},\vec{p}\), and the hamiltonian governing evolution is kinetic plus potential energy, \(H = \Vert \vec{p}\Vert ^2 + \Vert \vec{q}\Vert ^2\). Splitting into coordinates \(\vec{q} = (q_x,q_y)\), we see \(H = (q_x^2 + p_y^2) + (q_x^2 + p_y^2)\). Likewise, the symplectic form splits as \(\omega = \mathrm{d}q_{x}\wedge\mathrm{d}p_{y}+ \mathrm{d}q_{x}\wedge\mathrm{d}p_y\). Projecting onto the \(x\) or \(y\) components, we get 2D phase space and Hamiltonian \(H = (q^2 + p^2)\): The 2D oscillator is two independent 1D harmonic oscillators! We can write the phase space and Hamiltonian of a 2D oscillator as

\[\mathcal{P} = (\mathbb{C}^2,\omega_x+\omega_y) \qquad H = \Vert z_x\Vert ^2 + \Vert z_y\Vert ^2\]

where \(\mathbb{C}^2\) splits into two copies, one for \(x\) and one for \(y\), with their respective coordinates \(z_{x},z_{y}\) and symplectic forms \(\omega_{x},\omega_{y}\).

Now lets take a closer look at the orbits. Here’s another ball to play with, but this time it predicts its trajectory. What shapes can you find?

All the orbits are ellipses! And in particular, they’re all periodic. This comes from the decomposition. Since Both the \(x\) and \(y\) oscillators have period 1, so too must their composite system. Evolution by time \(t\) is simply rotation of each of the factors of \(\mathbb{C}\):

\[(z_1,z_2) \to (e^{2 \pi i t} z_1,e^{2 \pi i t} z_2)\]

so time evolution “rotates” \(\mathbb{C}^2\).

What if the \(x\) and \(y\) oscillators have different periods? Try changing the sliders above and see! If they’re integer multiples, a very similar thing happens. After enough time both \(x\) and \(y\) return to their starting position, so the orbit is periodic. It’s traces something fancier than an ellipse, called a Lissajous figure. To see that the \(x\) and \(y\) trajectories are indeed periodic, here are the projections to the component harmonic osscilators:

The time evolution is again rotation of each factor of \(\mathbb{C}\), now at different rates. The Hamiltonian \(H = \lambda_1^x\Vert z_x\Vert ^2 + \lambda_y^2\Vert z_y\Vert ^2\) gives frequencies \(\lambda_x\) and \(\lambda_y\), with time evolution

\[(z_x,z_y) \to (e^{2 \pi i \lambda_x t} z_x,e^{2 \pi i \lambda_y t} z_y)\]

Any pair of integers \(\lambda_x,\lambda_y\) define a \(U(1)\) action on \(\mathbb{C}^2\). This straightforwardly extends to \(n\) independent oscillators, with phase space \(\mathbb{C}^n\) and relative frequencies encoded in an integer vector \(\vec \lambda \in \mathbb{Z}^n\).

Representations of \(U(1)\)

Harmonic oscillators don’t give just any \(U(1)\) action on \(\mathbb{C}^n\): They’re all linear. Encoding a point in phase space as a complex vector \(\vec{z} \in \mathbb{C}^n\), the \(U(1)\) action is multiplication by a matrix:

\[\begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix} = \begin{pmatrix} e^{2 \pi i \lambda_1 t } & & \\ & \ddots & \\ & & e^{2 \pi i \lambda_n t } \end{pmatrix} \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}\]

The \(U(1)\) action comes from a representation \(\rho: U(1) \to GL(n,\mathbb{C})\). In fact, any such representation can be put in this form, choosing the right basis. This amounts to splitting \(\rho\) into a product of irreducible representations. Since \(U(1)\) is abelian, Schur’s lemma implies all irreducible representations are 1 dimensional. The linear maps \(U(1) \to \mathbb{C}\) are classified by an integer \(\lambda\) called the weight:

\[z \to e^{2 \pi i \lambda t z}\]

So products of 1D representations are fully classified by their weight vectors \(\vec{\lambda} \in \mathbb{Z}^n\), and take the form above. The equivalence between harmonic oscillators with integer period ratios and linear \(U(1)\) representation is our first hint at the oscillator’s universal properties.

General circle actions

The notion of a circle action immediately extends to an arbitrary symplectic manifold \((\mathcal{P},\omega)\). Every element \(\theta\) of the group \(U(1)\) defines a diffeomorphism \(\phi_\theta : \mathcal{P} \to \mathcal{P}\), such that \(\phi_{\theta_1} \circ \phi_{\theta_2} = \phi_{\theta_1+\theta_2}\). That is, \(\phi\) gives a homomorphism from \(U(1)\) to the group of diffeomorphisms \(\mathrm{Diff}(\mathcal{P})\). When the \(U(1)\) action preserves the symplectic structure ( \(\phi_\theta^*\omega = \omega\) ), it is generated by a Hamiltonian: The infinitesimal action \(\frac{\text{d}}{\text{d} \theta} \phi_\theta\) is the Hamiltonian vector field for some function on phase space. Conceptually, a Hamiltonian \(U(1)\) action is a phase space symmetry.

==Regularity, isolated fixed points, etc==

Example: \(U(1)\) action on \(\mathbb{C}\)

\(\mathbb{C}\) carries a \(U(1)\) action via multiplication by unit norm complex numbers. The generating Hamiltonian is the harmonic osscilator Hamiltonian, \(\Vert z\Vert ^2\). Likewise, \(\mathbb{C}^n\) has a \(U(1)\) action generated by \(\lambda_1\Vert z_1\Vert ^2 + \dots + \lambda_n\Vert z_n\Vert ^2\) for \(\lambda_1, \dots, \lambda_n\) integers.

Example: \(U(1)\) action on \(S^2\)

The sphere \((S^2,\omega_S)\) described in [[2 A primer on geometric mechanics#^ex–S2-symplectic]] carries a natural \(U(1)\) action. Describing it as the unit sphere in \(\mathbb{R}^3\), rotation around the \(z\)-axis preserves the sphere. It also preserves \(\omega_S\), because the cross product \(v \times w\) only depends on the length of \(v\) and \(w\), and their relative angle. All of these quantities are preserved by rotation. The Hamiltonian generating this action is in fact z-coordinate of a point \(p \in S^2\), which you can think of as the height function on the sphere. Showing that this generates rotation is a very good exercise, but I explain this below the fold

Height generates rotation

The Hamiltonian evolution of \(h_z\) will necessarily preserve height, so the orbit of each point matches with the orbit under rotation. Furthermore, the velocity is proportional to the gradient of \(h_z\) along the sphere, which is uniform everywhere along each orbit. So, \(h_z\) must generate rotation on each layer of the sphere, and we need only check that each layer rotates with the same period. Denoting the azumithial angle by \(\phi\), the height function is \(\cos(\phi)\), so its gradient along the sphere has length \(\sin(\phi)\). However, each point must transverse a circle with radius \(\sqrt(1-h_z^2) = \sin(\phi)\), so the period (length over velocity) is \(2 \pi \sin(\phi) / \sin(\phi) = 2\pi\) for every height. Hence, \(h_z\) is the Hamiltonian generating rotation.

The Hamiltonian associated to a given circle action is often called the Moment map. This name originates from angular momenta, which generate rotation. However, angular momenta usually arise from rotations of configuration space \(M\), which lift to phase space \(T^*M\).

The duality between circle actions and their Hamiltonians is a classical manifestation of Noether’s theorem, where the circle action is a continuous symmetry and the moment map is the conserved quantity.

Universality of harmonic oscillators

A hamiltonian can only generate a \(U(1)\) action if every point is periodic with the same period, which is a very special condition. In a sense, this is the essence harmonic oscillators. We saw a hint of this in remark [[2 A primer on geometric mechanics#^rmk–harmonic-osscilator-periodic]], which said that the 1D oscillator was unique among circle actions on \(\mathbb{C}\), up to changes of coordinates. Any surface with \(U(1)\) action is locally modeled on the 1D oscillator.

Example: \(U(1)\) action on \(S^2\) as an oscillator

Consider the sphere with \(U(1)\) action described in example [[3 The geometry of harmonic oscillators#^ex–U1-action-S2]]. Each orbit is uniquely defined by the value of its Hamiltonian, equivently its height. We send the orbit at height \(t\) to the circle in \(\mathbb{C}\) with radius \(\sqrt{t}\), so that the Hamiltonian \(\Vert z^2\Vert\) on \(\mathbb{C}\) agrees with the Hamiltonian on \(S^2\).

==Finish this example, give the two coordinate charts: One for each critical point.==

==Describe fibration structure to image of moment map==

We see in this example that the dynamics around each critical point of the moment map on \(S^2\) correspond exactly to a harmonic oscillator. This is totally general: Just like symplectic manifolds are modeled on phase space (Darboux’s theorem, [[2 A primer on geometric mechanics#^thm–Darboux]]), symplectic manifolds with circle actions are modeled on harmonic oscillators. This is known as Equivariant Darboux’s theorem:

Theorem: Equivariant Darboux’s theorem

Let \((\mathcal{P},\omega)\) be a symplectic manifold with \(U(1)\) action defined by moment map \(\mu\). Each fixed point of the \(U(1)\) action has a neighborhood \(U\) equivalent to \((\mathbb{C}^n,\omega_{\textrm{s}})\) with \(U(1)\) action defined by moment map

\[H_{\vec{\lambda}} = \lambda_1\Vert z_1\Vert ^2 + \dots + \lambda_{n}\Vert z_n\Vert ^2 \qquad \vec{\lambda} \in \mathbb{Z}^n\]

That is, there is a there is a symplectomorphism \(\phi:(U,\omega) \to (\mathbb{C}^n,\omega_{\textrm{standard})}\) such that \(\mu = H_{\vec{\lambda}} \circ \phi\)

This says that the neighborhood of a fixed point is equivalent to a harmonic oscillator, characterized by a weight vector \(\vec{\lambda}\). Note that the equality of the moment maps implies the \(U(1)\) actions are intertwined. If the flow under \(\mu\) is \(\rho_t\) and the flow under \(H_{\vec{\lambda}}\) is \(\rho^{\textrm{HO}}_t\), then \(\phi \circ \rho_t = \rho^{\textrm{HO}}_t \circ \phi\). The proof of theorem [[3 The geometry of harmonic oscillators#^thm–equivariant-darboux]] is a modified proof of vanilla Darboux’s theorem [[2 A primer on geometric mechanics#^thm–Darboux]]. It uses a version of Moser’s trick (described in remark [[2 A primer on geometric mechanics#^rmk–mosers-trick]]) that respects the \(U(1)\) action.

This is significant in a similar way to Darboux’s theorem. You would expect something like this to be true only infinitesimally, on the level of the linearized \(U(1)\) action on the tangent space. Indeed, this is the case for the equivalent statement in Riemannian geometry. But, the rigidity of symplectic geometry extends this linear action to a whole neighborhood of the fixed point.The behavior at the fixed points is global. The quadratic character of the moment map (the defining property of a harmonic oscillator Hamiltonian, and the reason that the \(U(1)\) action is linear) also extends globally. The \(U(1)\) action on \(\mathcal{P}\) is ultimately almost completely characterized by the collection of harmonic oscillators at each of the \(U(1)\) fixed points. This is why I say moment maps of circle actions are like global versions of harmonic oscillators. We will see several examples in the sequel where this philosophy is precise.

Torus action

Let us go back to our 2D oscillator example. All of these Hamiltonians with their different \(U(1)\) actions come from the same underlying structure. We see this by decoupling the oscillators. Imagine we fix the \(y\)-coordinate and only let \(y\) vary, alternatively fix \(x\) and let \(y\) vary. These give two independent \(U(1)\) actions on \(\mathbb{C}^2\), defined by Hamiltonians \(H_1 = \Vert z_1\Vert ^2\) and \(H_2 = \Vert z_2\Vert ^2\) respectively. The vector fields generated by these hamiltonians commute with one another, meaning the state after flowing for time \(t_1\) under \(H_1\) then time \(t_2\) under \(H_2\) equals that if you swap the order of the two flows. It is unambiguously defined by a pair of times \((t_1,t_2)\). Since each time is itself periodic, these flows define a \(U(1)^2\) action:

\[\rho_{(t_1,t_2)}: (z_1,z_2) \mapsto (e^{2 \pi i t_1} z_1,e^{2 \pi i t_2} z_2)\]

This is a called torus action on \(\mathbb{C}^2\). Every linear circle action comes from combining these two basic Hamiltonians into a single hamiltonian. The flow under a hamiltonian \(H = \lambda_1 H_1 + \lambda_2 H_2\) composes a group homomorphism from \(\mathbb{R}\) to \(U(1)^2\) with the torus action:

\[t \to (\lambda_1 t, \lambda_2 t) \qquad \Phi^t_H = \rho_{(\lambda_1 t, \lambda_2 t)}\]

When \(\lambda_1,\lambda_2\) are integers this is a homomorphism \(U(1) \to U(1)^2\), so the flow is periodic and we retrieve the circle action described in [[3 The geometry of harmonic oscillators#Integral periods and circle actions]].

Similar statements hold for \(n\) independent oscillators, phase space is \(\mathbb{C}^n\) and each oscillator oscillates one copy of \(\mathbb{C}\), giving an action of \(U(1)^n\) on phase space. Let us generalize this to an arbitrary symplectic manifold. Suppose we have a collection of Hamiltonian \(H_1, \dots H_n\), each the moment map of some \(U(1)\) action, normalized to have the same period. We demand the \(H_i\) pairwise Poisson commute, \(\{H_i,H_j\}=0\) (See section [[2 A primer on geometric mechanics#symmetries of classical mechanics]]). This means their associated flows commute, so evolution by time \(t_i\) under each Hamiltonian \(H_i\) gives a uniquely defined flow, characterized by the n-tuple of times \((t_1, \dots, t_n)\). This defines a \(U(1)^n\) action. Following the analogy that the \(H_i\) are global versions of harmonic oscillators, this collection of Hamiltonians is a global version of \(n\) independent harmonic oscillators.

Following our 2D oscillator example, we can build whole families of Hamiltonians by weighting the periods of each oscillator relative to one another, setting \(H = \lambda_1 H_1 + \dots + \lambda_n H_n\) for \(\lambda_i \in \mathbb{R}\) (for \(\lambda_i \in \mathbb{Z}\), the resulting flow is a \(U(1)\) action.) Our \(H_i\) satisfy \(\{H_i,H_j\}=0\), so \(H_j\) is conserved under the flow of \(H_i\) and vice versa. In particular, every \(H_i\) is conserved under our linear combination of basic hamiltonians \(H\). This means we have \(n\) conserved quantities, representing the energy of the \(n\) harmonic oscillators. Thought of as a vector \(\vec{H} = (H_1, \dots, H_n)\), this defines a function \(M \to \mathbb{R}^n\) called the moment map of the \(U(1)^n\) action.

Example: \(U(1)^2\) moment map of 2D oscillator

To demonstrate this, we return as always to the 2D oscillator. The Hamiltonians are \(H_1 = \Vert z_1\Vert ^2\) and \(H_2 = \Vert z_2\Vert ^2\), which generates rotation in each \(\mathbb{C} \times \mathbb{C}\). Fixing the value of the moment map fixes the radii \(r_1\) and \(r_2\) for each coordinates. This is certainly preserved under rotation. We project to the position plane by taking the real coordinates of both. As a point evolves under a weighted sum \(aH_1 + bH_2\), the \(x\)-coordinate oscillates between \(\pm r_1\) and the \(y\) coordinate between \(\pm r_2\). The specific coefficients \(a,b\) dictate the periods of each.

==Example program?==

The torus action preserves the value of the moment map, which lets us decompose phase space into slices \(\vec{H}^{-1}(t)\) for \(t\in \mathbb{R}^n\). We can understand phase space by understanding these two aspects: The image of the moment map, and the structure of the slices. The full description will occupy another few posts, but let us get a taste from a few examples:

Example: 1D oscillator

The 1D oscillator has phase space \(\mathbb{C}\) and moment map given by the harmonic oscillator hamiltonian \(\Vert z\Vert ^2\). The image consists of positive real numbers \(\mathbb{R} \geq 0\). The slice above \(r\) has two behaviors:

  • The slice above \(r > 0\) is the circle of radius \(\sqrt{r}\)
  • The slice above \(0\) is the origin, the fixed point of the \(U(1)\) action. We can think of \(\mathbb{C}\) as a circle fibration above \(\mathbb{R}\geq 0\), where the circle collapses at \(r=0\).

==Add cone picture==

Example: 2D oscillator

The image of the moment map is the image of \((\Vert z_1\Vert ^2,\Vert z_2\Vert ^2)\), which is the first quadrant of \(\mathbb{R}^2\), where \(\{(r_1,r_2) | r_i\geq 0 \}\).

  • When both \(r_i\) are positive, the slice is a product of circles with radii \(\sqrt{r_1}\) and \(\sqrt{r_2}\). Or, a torus!
  • On the boundary where \(r_2=0\), the slice is a product of a circle of radius \(\sqrt{r_1}\) and a point. Projecting to position space, the oscillator is fixed to the \(x\)-axis. It is preserved under the second \(U(1)\) action. Similarly, above the boundary \(r_1=0\), the oscillator lives on the \(y\)-axis.
  • The corner \(r_1=r_2=0\) is a fixed point of the torus action, where the oscillator sits unmoving at the origin. The full phase space \(\mathbb{C}^2\) can be thus built as a degenerating torus fibration above \(\mathbb{R}_{\geq 0}^2\). Once circle degenerates on \(\mathbb{R}_{\geq 0} \times 0\), the other on \(0 \times \mathbb{R}_{\geq 0}\), and both above \((0,0)\).

==Add simulation, where you select value of moment map and see how the orbit changes==

We can construct a \(U(1)\) action on \(\mathbb{C}^2\) as a sub-torus of our \(U(1)^2\) action. We build the moment map from the 2D moment map as \(H = H_1 + H_2 = \Vert z_1\Vert ^2 + \Vert z_2\Vert ^2\). The image of \(H\) consists of \(\mathbb{R}_{\geq 0}\), and the pre-image of a point \(r>0\) is a 3-sphere in \(\mathbb{C}^2\) with radius \(r\). At \(r=0\), this three-sphere degenerates to the origin. This is the unique fixed point of the \(U(1)\) action.

Example: \(S^2 \times S^2\)

consider the product \(S^2 \times S^2\), with \(U(1)\times U(1)\) acting by the standard \(U(1)\) action on each copy of \(S^2\) with moment map the height function. Since the flow of one moment map leaves the other sphere fixed, the moment maps commute. The possible values of both Hamiltonians forms a square \([-1,1]^2\). The pre-image of a point \((x,y)\) in the square is a product of circles at height \(x\) and \(y\) respectively. There are three possibilities:

  • When \((x,y)\) is in the interior of the square, the preimage is a torus.
  • When it lies on an edge, the preimage is a circle on one sphere and a pole of the other. One of the \(U(1)\) actions fixes these points.
  • When it lies on a corner, the preimage is a point sitting on the pole of both spheres. both \(U(1)\) actions fixes these points.

Generalizing this, \({S^2}^n\) carries a natural \(U(1)^n\) action, with moment map taking values in \([-1,1]^n\).

We see some common themes: Polygonal moment map images, with co-dimension \(k\) faces whenever exactly \(k\) of the \(U(1)\) factors is fixed.

Something special happens when the dimension of phase space is double the number of commuting Hamiltonians. We will see that the torus acts freely on the pre-image of a generic point of the moment map. With \(n\) dimensions of moment map and \(n\) dimensions of torus, this entirely accounts for the \(2n\) dimensional manifold. So, the slice must be exactly a torus. Said another way, every oscillator contributes 2 degrees of freedom: It’s energy, and its phase angle. \(n\) oscillators exhausts all \(2n\) degrees of freedom, so at fixed energies the state is exactly specified by \(n\) phase angles, and the inverse image of a generic moment map is \(U(1)^n\). These phase spaces are called Toric manifolds. They are entirely characterized by their \(n\) independent \(U(1)\) actions.

Spaces of orbits

U(1) orbits

Let \((\mathcal{P},\omega)\) be a symplectic manifold with a \(U(1)\) action generated by moment map \(H\). This means every point belongs to a periodic orbit on the flow under \(H\). What is the space of orbits? Can we divide \(\mathcal{P}\) by \(U(1)\)? This is always possible if \(U(1)\) acts freely, but for compact \(\mathcal{P}\) the \(U(1)\) action necessarily has fixed points (The global minima and maxima of \(H\)). Fixed points generally ruin the topology of orbit spaces, as single orbits can be adjacent to whole families of fixed points, resulting in a non-hausdorff topology. We avoid this by noticing that all orbits must have constant value of \(H\), so it makes sense to look at orbits on \(H^{-1}(c)\). If \(H\) has no critical points with value \(c\) (meaning \(c\) is a regular value of \(H\)), then \(U(1)\) acts freely on \(H^{-1}(c)\) and its space of orbits is a manifold. Thus we define the symplectic quotient (Or Marsden-Weinstein quotient)

\[\mathcal{P} // U(1) \equiv H^{-1}(c)/U(1)\]

Vitally, this quotient carries a natural symplectic form, induced from that on \(\mathcal{P}\). The reduced form is defined by lifting vectors from \(H^{-1}(c)/U(1)\) to \(H^{-1}(c)\) by arbitrarily choosing a basepoint and a value in the \(U(1)\)-direction, then plugging these into the usual symplectic form. This is well defined because the symplectic form is preserved by \(U(1)\) so the basepoint doesn’t matter. The \(U(1)\) direction is in the null space of the symplectic form, so t)$$ direction, so those component of the vector don’t matter.

Notice that the symplectic quotient reduces the dimension by two: Not only does it remove a factor of \(U(1)\), but it restricts to a level set of the moment map. The symmetry acts twice as much as you’d naïvly expect, hence the double-slash notation. (These double-acting symmetries are a very general phenomena in symplectic geometry.) Perhaps one could think of this as “removing one harmonic oscillator”, first fixing its energy then dividing out by its phase degree of freedom.

Orbits of a 2D oscillator

Consider once again phase space \(\mathbb{C}^2\), with \(U(1)\) action generated by \(H = \Vert z_1\Vert ^2 + \Vert z_2\Vert ^2\). For a regular value, say a, \(H^{-1}(1)\) is the unit 3-sphere. Since \(\text{d} H\) does not vanish on \(H^{-1}(1)\), by general grounds \(U(1)\) must act freely on \(S^3\). In particular, there is a fibration

\[\require{AMScd} \begin{CD} U(1) \cong S^1 @>>> S^3;\\ @. @VVV \\ @. S^3/S^1; \end{CD}\]

The only fibration of \(S^3\) by \(S^1\) is the famous Hopf fibration, with base \(S^2\). So, the space of orbits is a 2-sphere. But seeing is believing: the Hamiltonian \(H = \Vert z_1\Vert ^2 + \Vert z_2\Vert ^2\) represents a 2D harmonic oscillator with equal periods, with orbits that are ellipses. Let’s look at the ellipses with constant energy:

(Simulation is here, yet to be incorperated)

First we fix the value of the \(U(1)^2\) moment map, and understand the space of orbits. he pre-image of a generic value of the moment map is a torus \(U(1)^2\), as described in [[#^ex–2d-oscillator-torus-fibration]]. \(H\) generates a \(U(1)\) subgroup, so the space of orbits of \(H\) is the complementary copy of \(U(1)\): It is a circle. We can visualize this with a trick:make the period of one oscillator (say, the \(x\)-oscillator) ever so slightly sorter than the other. Once \(y\) has made a full orbit, \(x\) won’t quite line up, so the orbit’s phase in \(x\) rotates slightly. This orbit tightly spirals around the torus, eventually wrapping it densely. (In general, any Hamiltonian whose orbit densely fills a torus is called almost-periodic. For a 2D oscillator this happens when the ratio of periods is irrational). Tracking the orbit for one y-period, this acts just like a slowly varying \(U(1)\) orbit in \(U(1)^2\)

==add picture of torus==

(Simulation is here, yet to be incorperated)

It looks like a circle in 3D, slowly rotating around the \(y\) axis! Every orbit, the \(x\)-phase of each point in the orbit increases by a small angle, so they slowly traces an \(x\)-oscillator orbit. This is especially clear when we mark a point. So, it slowly rotates. The “z-axis” it rotates through is the x-momentum.

(Simulation is here, yet to be incorperated)

Now we return to visualizing the orbits of a level set for \(H = \Vert z_1\Vert ^2 + \Vert z_2\Vert ^2\). The equation \(H=1\) defines a line in the values of the \(U(1)^2\) moment map \((H_1 = \Vert z_1\Vert ^2, H_2 = \Vert z_2\Vert ^2)\) as shown below. The space of orbits in \(H^{-1}(1)\) is composed of the space of orbits for all allowed values of the moment map. For most values of \((H_1,H_2)\) this is a circle, but this is not true at the boundaries. When \(H_1=1, H_2=0\), the orbit is purely in the x-axis, and is fixed by rotation – the circle of orbits degenerates to a point. Likewise when \(H_1=1, H_2=0\) the orbit lies in the \(y\) axis, and the circle again degenerates. Thus, we see a sphere. And indeed, showing all these orbits at once, they trace out a sphere in 3D on front of our eyes!

(Simulation is here, yet to be incorperated)

Recalling example [[#^ex–U1-action-S2]], the construction of the space of orbits is identical to \(S^2\) as a fibration, according to its standard \(U(1)\) moment map. The height function for the space of orbits is \(H_1 - H_2\), which is \(1\) at \((1,0)\) and \(-1\) at \((0,1)\). Its image is naturally identified with the intersection of the line \(H_1+H_2 = 1\) with the allowed values of the moment map. Said another way, the moment map of the sphere of orbits is just the cross-section of the higher dimensional moment map! Our trick perturbing the \(x\)-period essentially added an extra Hamiltonian \(\epsilon H_1\) to the flow of \(H\). Since \(H_1\) is conserved under \(H\) it descends to a function on the space of orbits, and so it induces a flow. (Incidentally, this relation between dynamics of the full space and dynamics on the space of orbits is the source of the reduced symplectic form.) Projecting onto the cross-section, \(H_1\) is proportional to the height function, so its flow is rotation. This is why we see rotation of the sphere.

Our choice of \(x\)-perturbation instead of \(y\) was arbitrary. A \(y\) perturbation rotates points about the \(y\) axis, instead of the \(x\). But, the function \(-H_2\) projects to an identical function as \(H_1\) on the cross-section. So, they give the same function on the sphere, and have the same dynamics: Rotation through the space of orbits. This remarkably means that the evolution of ellipses could either be a circle rotating in 3D around the Y-axis, or the X-axis!

(Simulation is here, yet to be incorperated)

This becomes more mind-bending with other \(U(1)\) generating Hamiltonians, like \(H'=3H_1 + 2H_2\) shown below. Once again, both \(x\) and \(y\) perturbations generate identical dynamics (up to scale) on the space of \(H'\) orbits. We can produce the resulting evolving curve as the shadow of a wire-sculpture in 3D rotating around the \(x\)-axis, or a Different wire sculpture rotating around the \(y\)-axis. Your brain can see either! This is called the Dual-Axis illusion, and it won the 2019 best illusion of the year award. It works because of toric geometry, for the shape is the orbit of a slightly perturbed periodic 2D harmonic oscillator. I suspect harmonic oscillator orbits are the only family of shapes with this dual-axis property, but I’m not sure.

\(U(1)\) orbits on toric manifolds

When the Hamiltonian \(H\) is defined from a \(U(1)^n\) moment map, we can do the symplectic reduction in stages. First, the level set of \(H\) is a cross-section of the image of the \(U(1)^n\) moment map. Second, the \(U(1)\) acts only on the fibers of the moment map. When the manifold is toric, the fibers are \(U(1)^n\) and it is very easy to divide out by \(U(1)\): You get the complementary subgroup \(U(1)^{n-1}\) of \(U(1)^n\). The resulting manifold is still toric. Instead of the original \(n\)-dimensional space of Hamiltonians, we get the \(n-1\) dimensional orthogonal complement to the Hamiltonian we reduced with. The tori are generated by the flows of these remaining Hamiltonians.

Symplectic reduction gives us some more exciting examples of toric manifolds:

Example: \(\mathbb{C} P^n\) as a toric manifold

Consider \(n\) harmonic oscillators, with phase space \(\mathbb{C}^n\) and the usual \(U(1)^n\) action. The [[Moment map]] \(H = \Vert z_1\Vert ^2 + \dots + \Vert z_n\Vert ^2\) generates a \(U(1)\) subaction. What is the space of orbits of \(H\) with energy 1? The \(U(1)\) action generated by \(H\) is multiplication by an overall phase

\[(z_1, \dots , z_n) \mapsto e^{2 \pi i t} (z_1, \dots , z_n)\]

The space of orbits is complex vectors of norm 1 modulo phase. Equivalently, the Symplectic reduction is \(\mathbb{C}^n - \{0\}\) modulo the overall scaling action of \(\mathbb{C}^*\). The quotient is called Complex projective space \(\mathbb{C} P^{n-1}\).

\(\mathbb{C} P^{n-1}\) has an induced toric structure from \(\mathbb{C}^n\). The image of the \(U(1)^n\) moment map is \(\mathbb{RR}_{\geq 0}^n\). Reducing by \(H\) selects the subspace \(H_1 + \dots + H_n =1\), which is isomorphic to the image of the moment map of \(\mathbb{C} P ^{n-1}\). In particular, the moment map image is an \(n-1\)- dimensional simplex. For \(\mathbb{C} P^2\), this can be written as the triangle in \(\mathbb{R}^2\) with corners \((0,0), (1,0),\) and \((0,1)\).

Orbits of general groups

The machinery of symplectic reduction works for arbitrary group, though it is somewhat easier to state for \(U(1)^n\). Suppose a symplectic manifold \(\mathcal{P}\) has symmetry represented by a Lie group \(G\), meaning \(G\) acts on \(\mathcal{P}\) and preserves the symplectic form. We further assume \(G\) is a Hamiltonian action, meaning its infinitesimal flows are generated by Hamiltonian functions. The infinitesimal flow is represented by a homomorphism from the lie algebra \(\mathfrak{g}\) (The infinitesimal symmetries of the group) to the lie algebra of vector fields on \(\mathcal{P}\) (The infinitesimal symmetries of the manifold structure on \(\mathcal{P}\)). We (by assumption) have a hamiltonian function \(H_g\) for \(g \in \mathfrak{g}\). However, The transformations from \(\mathfrak{g}\) to vector fields to hamiltonian functions, so \(H_g\) must be linear in \(g\). Therefore, we define the Moment map of a general group

\[\mu:M \to \mathfrak{g}^* \qquad \mu(x)g = H_g(x)\]

The moment map is a machine for converting an element of a lie algebra into the Hamiltonian which generates the associated flow. In our case with \(U(1)^n\), the Lie algebra is \(\mathfrak{u}(1)^n \cong \mathbb{R}^n\). Since \(U(1)^n\) is abelian, its lie algebra has trivial lie bracket. So, the moment map is a map \(\mu:\mathcal{P} \to \mathbb{R}^n\) such that each factor commutes with one another, i.e it is described by \(n\) commuting Hamiltonians. We can define symplectic reduction: if \(0 \in \mathfrak{g}^*\) is a regular value of \(\mu\), then \(G\) acts freely on \(\mu^{-1}(0)\), so we define

\[\mathcal{P} // G \equiv \mu^{-1}(0) / G\]

All of the results in this paper extend in one way or another to general groups. For the most part, we will stick with $U(1)^n$.