Hamiltonian Mechanics

Example 4.1. One of the simplest example of lagrangian submanifolds are

\begin{align} & \Lambda _{q_0} := \big \{(q,p)\in T^*M \mid q=q_0\big \} \\ & \Lambda _{p_0} := \big \{(q,p)\in T^*M \mid p=p_0\big \}. \end{align} Observe that the canonical transformation \((q,p) \mapsto (p,-q)\) exchanges one with the other.

Remark 4.1. For a hamiltonian system with hamiltonian \(H\), we can compute the generating function of a lagrangian submanifold from \(n\) first integrals \(f_1, \ldots , f_n\) in involution such that

\begin{equation} \label {eq:condlsgf} \det \left (\frac {\partial f_i}{\partial p_k}\right ) \neq 0. \end{equation}

For any fixed \(n\)-tuple \(a = (a_1, \ldots , a_n) \in \mathbb {R}^n\) and \(f=(f_1,\ldots ,f_n)\), let

\begin{equation} \Lambda _a = f^{-1}(a) := \{(q,p)\in T^*M \;\mid \; f(q,p) = a\} \end{equation}

be the lagrangian submanifold described by the equations

\begin{equation} f_k(q,p) = a_k, \quad k=1,\ldots ,n. \end{equation}

Thanks to (4.17), we can apply the implicit function theorem and invert the equations \(f_k(q,p) = a_k\) with respect to \(p_i\). In this way, we obtain the functions

\begin{equation} p_i = W_i(q,a). \end{equation}

Since the equation above describes by construction the lagrangian submanifold \(\Lambda _a\), Theorem 4.3 implies that the one form \(W_i\dd q^i\) is closed. Therefore, locally there is a function \(S(q, a)\) such that \(p_i = \partial _{q_i} S\).

Furthermore, since \(\big \{f_i, H\big \} = 0\), the hamiltonian vector field satisfies the additional condition \(\omega (X_{f_i}, X_H) = 0\), which means that \(X_H\in T\Lambda _a\). Said in other words, the trajectories of the hamiltonian flow which intersect \(\Lambda _a\) are completely contained in \(\Lambda _a\).

Remark 4.2. If \(q^i = Q^i\), \(i=1,\ldots ,n\), then the function \(S=S(q)\). In fact, in this case, (4.24) reads

\begin{align} p_i \dd q^i - P_i \frac {\partial Q^i}{\partial q^j}\dd q^j - P_i \frac {\partial Q^i}{\partial p_j}\dd p_j & = (p_i -P_i)\dd q^i \\ & = \frac {\partial S}{\partial q^i}\dd q^i + \frac {\partial S}{\partial p_i}\dd p_i, \end{align} which imply \(\frac {\partial S}{\partial p_i} = 0\) and, thus, \(p_i = P_i + \frac {\partial S(q)}{\partial q^i}\).

Remark 4.3. Not all canonical transformations are free: the identical transformation is not free, as \(q=Q\) and thus they are not independent.

Example 4.2. Let \(\omega > 0\) be some fixed constant. Consider the transformation \(\Phi \) given by

\begin{equation} (q,p) \mapsto (Q,P) = \left ( \sqrt {\frac {2p}{\omega }} \sin (q), \sqrt {2p\omega }\cos (q) \right ). \end{equation}

In this case, the conservation of the symplectic form can be checked immediately:

\begin{align} \dd P\wedge \dd Q & = \left (\sqrt {\frac {\omega }{2 p}}\cos (q)\dd p - \sqrt {2 p\omega } \sin (q)\dd q\right ) \\ & \quad \wedge \left (\frac {1}{\sqrt {2\omega p}}\sin (q)\dd p + \sqrt {\frac {2 p}{\omega }} \cos (q)\dd q\right ) \\ & = \dd p \wedge \dd q. \end{align}

For \(S_1(q,Q) = -\frac {\omega }{2}Q^2\cot (q)\), one gets

\begin{equation} \frac {\partial S_1}{\partial q} = \frac {\omega Q^2}{2\sin ^2(2q)}, \quad \frac {\partial S_1}{\partial Q} = - Q\omega \cot (2q). \end{equation}

That is, \(S_1\) generates the canonical transformation \(\Phi \) since

\begin{equation} \frac {P}{Q} = \omega \cot (q), \quad \frac {Q^2}{\sin ^2(q)} = \frac {p}{\pi \omega }, \end{equation}

which implies

\begin{equation} P = -\frac {\partial S_1}{\partial Q}, \quad p = \frac {\partial S_1}{\partial q}. \end{equation}

The hamiltonian \(H(Q,P) = \frac 12 (P^2 + \omega ^2 Q^2)\) is transformed into

\begin{equation} K(q,p) = \omega p, \end{equation}

and the linear equations of motion of \(H\) give rise to the equation for motion for \(K\):

\begin{equation} \dot q = \omega , \quad \dot p = 0. \end{equation}

Example 4.3. Let \(Q=Q(q)\) be a change of coordinates on the base \(M\) of the cotangent bundle \(T^*M\). This induces a transformation of the conjugate momenta \(p_i\mapsto P_i\) given by \(P_i = p_l \frac {\partial q^l}{\partial Q^i}\), and we already saw that \((q,p)\mapsto (Q,P)\) defines a canonical transformation. The generating function of such transformation is given by

\begin{equation} S_2(q,P) = \sum _{i=1}^n P_i Q^i(q). \end{equation}

Remark 4.4. Sometimes, in the literature you can find a different classification in the following terms:

• 1. \(S_1(q,Q) = S(q,p(q,Q))\), which depends only on the old and new generalized coordinates;

• 2. \(S_2(q,P) = - Q^i(q,P) P_i + S(q,p(q,P))\), which depends only on the old generalized coordinates and the new generalized momenta;

• 3. \(S_3(p,Q) = q^i(p,Q) p_i + S(q(p,Q),p)\), which depends only on the old generalized momenta and the new generalized coordinates;

• 4. \(S_4(p,P) = q^i(p,P) p_i - Q^i(p,P) P_i + S(q(p,P),p)\), which depends only on the old and new generalized momenta.

Of course, as we also have shown, mixtures of these four types can exist.

Example 4.4. Let \(H(q,p) = \frac {p^2}{2} + \alpha q\). Consider a time-dependent canonical transformation \(\Phi _t\) generated by \(H\): let \(q(0) = Q\) and \(p(0) = P\) and

\begin{equation} \label {eq:unifaccm} p(t) = P - \alpha t, \quad q(t) = Q + Pt - \frac {1}{2}\alpha t^2. \end{equation}

How are these equations related to each other? Do they ring any bell?

We can compute the generating function for \(\Phi _t\) with a trick. Writing the change of coordinates in the form

\begin{equation} p = \frac {q-Q}{t}-\frac 12\alpha t, \quad P = \frac {q-Q}{t} + \frac 12\alpha t, \end{equation}

we see that \(S(q,Q,t)\) has to satisfy

\begin{align} & p = \frac {\partial S(q,Q,t)}{\partial q} = \frac {q-Q}{t} - \frac 12\alpha t, \\ & P = - \frac {\partial S(q,Q,t)}{\partial Q} = \frac {q-Q}{t} + \frac 12\alpha t. \end{align} These can be integrated to find

\begin{equation} S(q,Q,t) = \frac {(q-Q)^2}{2t} - \frac 12 \alpha (q+Q) t + f(t), \end{equation}

where \(f\) is arbitrary and does only depend on time.

The new hamiltonian \(K(q,Q,t)\) is then computed as follows

\begin{align} K(q,Q,t) & = K(Q, P(q,Q,t), T(t)) \\ & = H(q, p(q,Q,t)) + \frac {\partial S (q,Q,t)}{\partial t} \\ & = \frac {(q-Q)^2}{2t^2} + \frac 12 \alpha ^2 t^2 - \frac 12 \alpha (q-Q) + \alpha q \\ & \quad - \frac {(q-Q)^2}{2t^2} - \frac 12 \alpha (q+Q) + f'(t) \\ & = \frac {1}{8}\alpha ^2t^2 + f'(t). \end{align} Since \(f\) does not affect Hamilton’s equation for the new hamiltonian \(K\), we can use it to our advantage: choosing \(f'(t) = - \frac {1}{8}\alpha ^2t^2\) the hamiltonian vanishes identically, \(K\equiv 0\).

As a side remark, observe that the same result can be obtained also by directly checking that \(\dot P = \dot Q = 0\).

Remark 4.5. Choosing \((q,Q,t)\) as independent variables in the proof above one gets

\begin{equation} H(q, p(q,Q,t), t) = \frac {\partial S(q,Q,t)}{\partial t}. \end{equation}

Example 4.5. Cyclic coordinates are an immediate example of separable coordinates. Assume for example that \(q^1\) is cyclic, then the conjugate momentum \(p_1 = b_1\) is constant. The generating function has the form

\begin{equation} S(q,b) = S_0(q^2, \ldots , q^n, b) + q^1 b_1, \end{equation}

and the hamiltonian has the form

\begin{equation} H(q^2, \ldots , q^n, \frac {\partial S_0}{\partial q^2}, \ldots , \frac {\partial S_0}{\partial q^n}, b_1). \end{equation}

Example 4.6. A simple example of separable system is given by a sum of hamiltonians depending each on just one pair of conjugate coordinates:

\begin{equation} H(q,p) = \sum _{i=1}^n h_i(q^i, p_i). \end{equation}

Example 4.7 (Planar harmonic oscillator). Let’s reconsider an old friend of ours, the planar harmonic oscillator defined by a point particle of mass \(m\) attracted by an elastic force centered at \(0\in \mathbb {R}^2\) with stiffness \(k\).

The hamiltonian in cartesian coordinates \(x=(x_1, x_2)\) has the form

\begin{equation} H(x,p) = \frac {1}{2m}(p_{x_1}^2 + p_{x_2}^2) + k(x_1^2 + x_2^2). \end{equation}

Choosing the generating function

\begin{equation} S(x_1, x_2, b, E, t) = W_1(x_1, b, E) + W_2(x_2, b, E) - E t, \end{equation}

we obtain the Hamilton-Jacobi equation

\begin{equation} \left (\frac {\partial W_1}{\partial x_1}\right )^2 + \left (\frac {\partial W_2}{\partial x_2}\right )^2 + 2m k (x_1^2 + x_2^2) = 2m E, \end{equation}

which can be separated into the two equations

\begin{equation} \left (\frac {\partial W_i}{\partial x_i}\right )^2 + 2m k x_i^2 = 2m k b_i, \quad i=1,2, \end{equation}

such that \(K(b_1,b_2) := k(b_1 + b_2) = E\).

The rest now follows from the Jacobi method. Since we are in the case described by (4.105), it holds

\begin{equation} \frac {\partial W_i}{\partial b_i} = \frac {\partial K}{\partial b_i}t + c_i, \quad i=1,2, \end{equation}

where \(c_i\) are integration constants, we get

\begin{equation} \frac {\partial W_i}{\partial b_i} = \sqrt {\frac {mk}{2}}\arcsin \left (\frac {x_i}{\sqrt {b_i}}\right ) = kt + c_i, \quad i=1,2. \end{equation}

Some algebra and some care in checking domains of definition lead to

\begin{equation} x_i = \sqrt {b_i} \sin \left (\sqrt {\frac {2k}{m}} t + c_i\right ), \quad i=1,2. \end{equation}

The complete integral is thus given by the generating function

\begin{equation} S(x_1,x_2,b_1,b_2,t) = W_1(x_1, b_1) + W_2(x_2,b_2) - K(b_1, b_2) t, \end{equation}

which depends on time, the two (generalized) coordinates \((x_1, x_2)\) and the two integration constants \((b_1, b_2)\). Note that you may as well stick with \(S\) in terms of \((E, b)\), as we will do in the next example: what you end up choosing is just a matter of convenience.

Example 4.8. Let’s consider a more involved example of a point of mass \(m\) moving in \(\mathbb {R}^3\) under the influence of an arbitrary potential \(U\).

In spherical coordinates, the hamiltonian has the form

\begin{equation} H(r,\phi ,\theta , p) = \frac {1}{2m}\left (p_r^2 + \frac {p_\theta ^2}{r^2} + \frac {p_\phi ^2}{r^2\sin ^2\theta }\right ) + U(r,\phi ,\theta ). \end{equation}

Separation of variables is possible if

\begin{equation} U(r,\phi ,\theta ) = \alpha (r) + \frac {\gamma (\phi )}{r^2 \sin ^2(\theta )} + \frac {\beta (\theta )}{r^2}, \end{equation}

for some functions \(\alpha (r)\), \(\beta (\theta )\), \(\gamma (\phi )\).

For simplicity, let’s assume that \(\gamma (\phi ) = 0\), then \(\phi \) becomes a cyclic coordinate and \(p_\phi = \mathrm {const} =: b_\phi \). The corresponding Hamilton-Jacobi equation for \(S\) becomes

\begin{equation} \left (\frac {\partial S}{\partial r}\right )^2 + 2m\alpha (r) + \frac {1}{r^2}\left ( \left (\frac {\partial S}{\partial \theta }\right )^2 + 2m\beta (\theta ) \right ) + \frac {1}{r^2\sin ^2(\theta )}\left (\frac {\partial S}{\partial \phi }\right )^2 = 2mE, \end{equation}

where we are looking for a solution of the form

\begin{equation} S_0(r,\phi ,\theta ) = \phi b_\phi + W_1(r) + W_2(\theta ). \end{equation}

Replacing the ansatz for \(S\) in the equation above, we get

\begin{align} & \left (\frac {\partial W_2}{\partial \theta }\right )^2 + 2m \beta (\theta ) + \frac {b_\phi ^2}{\sin ^2(\theta )} = b_\theta , \\ & \left (\frac {\partial W_1}{\partial r}\right )^2 + 2m \alpha (r) + \frac {b_\theta }{r^2} = 2mE. \end{align}

Integrating, we end up with

\begin{align} S_0 & = \phi b_\phi \\ & + \int \sqrt {2m(E-\alpha (r))-{b_\theta }/{r^2}}\dd r \\ & + \int \sqrt {b_\theta -2m\beta (\theta )-{b_\phi ^2}/{\sin ^2(\theta )}}\dd \theta . \\ \end{align}

The complete integral is then the generating function

\begin{equation} S(r,\phi ,\theta ,E,b_\phi ,b_\theta ,t) = \phi b_\phi + W_1(r, E, b_\theta ) + W_2(\theta , b_\phi , b_\theta ) - E t, \end{equation}

depending on time, the three (generalized) coordinates \((r,\phi ,\theta )\) and the three integration constants \((E,b_\phi ,b_\theta )\).

To obtain \(r(t)\) and \(\theta (t)\) we then need to invert the integrals

\begin{equation} \frac {\partial S}{\partial E} = Q_E = t-t_0, \quad \frac {\partial S}{\partial b_\theta } = Q_\theta = c_1, \quad \frac {\partial S}{\partial b_\phi } = Q_\phi = c_2, \end{equation}

where \(c_1\) and \(c_2\) are arbitrary constants. The momenta \(p_\theta \) and \(p_r\) are then given by

\begin{equation} p_r = \frac {\partial S}{\partial r}, \quad p_\theta = \frac {\partial S}{\partial \theta }. \end{equation}

Doing these last computations analytically is generally hard or impossible because of complicated integrals and people refer to numerical investigations. On the other hand, by setting the constants to \(0\), they often allow to compute explicit equations for the trajectories – without time dependence – and to obtain useful asymptotic approximations.

Exercise 4.1. Show that the infinitesimal transformation

\begin{equation} q = Q + \epsilon Q^2 \sin (P), \quad p = P + 2\epsilon Q(1+\cos (P)), \end{equation}

is a canonical transformation and determine its domain. Compute the hamiltonian \(K(Q,P)\) associated to the transformation.

Consider then the transformation

\begin{align} q & = Q + \epsilon Q^2 \sin (P) + \sum _{j=2}^\infty \frac {(2j-1)!!}{(j+1)!}2^j \epsilon ^j \sin ^j(P) Q^{j+1}, \\ p & = P + 2\epsilon Q^2\left (1+\cos (P)\right )\left ( 1 + \epsilon Q \sin (P) + \sum _{j=2}^\infty \frac {(2j-1)!!}{(j+1)!}2^j \epsilon ^j \sin ^j(P) Q^{j} \right ). \end{align} Show that it is a canonical transformation and compute its generating function.
Hint: use the following series expansion \(\sqrt {1-x} = 1 - \frac {x}{2} + \sum _{j=2}^\infty \frac {(2j-3)!!}{j! 2^j} x^j\).

Exercise 4.2. Let \(Q = q^2 + \frac 12 \cos (q)\). Find \(P(q,p)\) such that the corresponding coordinate transformation is canonical. Compute the corresponding generating function.

Exercise 4.3 (Parabolic coordinates). Let \((r,\phi ,z)\) denote the cylindrical coordinates in \(\mathbb {R}^3\). Find the complete integral of the Hamilton-Jacobi equations for a point particle of mass \(m\) under the influence of the potential

\begin{equation} U(r,\phi ,z) = \frac {\alpha }{\sqrt {r^2+z^2}} - F z. \end{equation}

To this end, introduce the parabolic coordinates \((\xi , \eta , \phi )\) obtained from the equations

\begin{equation} z = \frac 12(\xi - \eta ), \quad r = \sqrt {\xi \eta }. \end{equation}

Exercise 4.4 (Prolate spheroidal coordinates). Consider a point particle of mass \(m\) under the gravitational attraction of two bodies which are fixed at the points \((d,0,0)\) and \((-d,0,0)\), \(d>0\). Show that the Hamilton-Jacobi equation is separable introducing the prolate spheroidal coordinates obtained by

\begin{equation} x = d \cosh \xi \cos \eta , \quad y = d \sinh \xi \sin \eta \cos \phi , \quad z = d \sinh \xi \sin \eta \sin \phi , \end{equation}

where \(\xi \in \mathbb {R}_+\), \(0\leq \eta \leq \phi \) and \(0\leq \phi \leq 2\pi \).

Exercise 4.5. Consider the motion of a free particle of mass \(m\) on a surface \(S\subset \mathbb {R}^3\) parametrized by

\begin{equation} x = u \cos v, \quad y = u \sin v, \quad z = \psi (u), \end{equation}

with \(u\in \mathbb {R}\) and \(0\leq v\leq 2\pi \). Determine the conjugate momenta associated to the variables \(v\) and \(u\) and show that the corresponding hamiltonian system is separable.

Remark 4.6. In the integral (4.148) you can choose the initial point arbitrarily as long as it varies smoothly with respect to \(E\).

Remark 4.7. A separable hamiltonian system is completely integrable. Indeed, for a separable system there exists a canonical transformation \((q,p) \mapsto (Q,P)\) such that the hamiltonian \(H\) in the new coordinates does not depend on \(Q\), \(H=H(P)\). The functions \(P_1, \ldots , P_n\), which are in involution by construction, commute with \(H\).

Exercise 4.8. Extend the proof to completely integrable system on an arbitrary symplectic manifold. (Solution: find the differences between our proof and [Kna18, Theorem 13.3]).

Remark 4.8. Consider the (inverse) canonical transformation

\begin{equation} q_k = q_k(\phi , I), \quad p_k = p_k(\phi , I), \quad k=1,\ldots ,n. \end{equation}

The smooth functions \(q\) and \(p\) are \(2\pi \)-periodic with respect to any of the angular variables \(\phi = (\phi _1, \ldots , \phi _n)\), and therefore they can be expanded in Fourier series:

\begin{align} & x_j = \sum _{\vb * k = (k_1, \ldots , k_n)\in \mathbb {Z}^n} A^j_{\vb * k}(I) e^{i(k_1 \phi _1 + \cdots + k_n \phi _n)} = \sum _{\vb * k\in \mathbb {Z}^n} A^j_{\vb * k}(I) e^{i\langle \vb * k, \phi \rangle } \\ & x = (q,p), \quad j=1,\ldots ,2n. \end{align} Then, the dynamics of the hamiltonian system in the original coordinates is of the form

\begin{align} & x_j(t) = \sum _{\vb * k\in \mathbb {Z}^n} A^j_{\vb * k}(I) e^{i t\langle \vb * k, \omega (I)\rangle + i\langle \vb * k,\phi ^0\rangle } \\ & x = (q,p), \quad j=1,\ldots ,2n, \end{align} where \(\omega (I) = (\omega _1(I), \ldots , \omega _n(I))\) and the parameters \(I = (I_1, \ldots , I_n)\) and \(\phi ^0 = (\phi _1^0, \ldots , \phi _n^0)\) can be considered as constants of integration.

The solutions \(x(t) = (q(t), p(t))\) are periodic in \(t\) (and the period does not depend on the parameter \(\phi ^0\)) if the frequencies \(\omega (I)\) are rationally dependent, i.e., if there is \((k_1, \ldots , k_n)\in \mathbb {Z}^d\) such that \(k_1\omega _1(I) + \ldots + k_n \omega _n(I) = 0\). If, instead, the frequencies are rationally independent, or non–resonant, then the trajectories are dense on the invariant torus \(M_{E(I)}\).

Example 4.9 (Harmonic oscillator). Consider, once again, the hamiltonian of a harmonic oscillator with mass \(m\) and stiffness \(k>0\):

\begin{equation} H(q,p) = \frac {p^2}{2m} + \frac {k q^2}{2}. \end{equation}

We have seen a long time ago that for \(E>0\) the curve \(H(q,p) = E\) is an ellipse and the motion on the ellipse is periodic with period

\begin{equation} T = \frac {2\pi }{\omega } = 2\pi \sqrt {\frac {m}{k}}. \end{equation}

The canonical action variable

\begin{equation} I = \frac 1{2\pi }\oint _{H(q,p) = E} p \dd q \end{equation}

is proportional to the area of the ellipse:

\begin{equation} I = \frac 1{2\pi } \mathrm {area}\left (\frac {p^2}{2m} + \frac {k q^2}{2} = E\right ) = E \sqrt {\frac {m}{k}} = \frac E \omega . \end{equation}

The variable conjugate to \(I\), is generally non-trivial to obtain. In this case, it is obtained from the canonical transformation

\begin{equation} q(I, \phi ) = \rho ^{-1} \sqrt {2I} \cos \phi , \quad p(I, \phi ) = \rho \sqrt {2I} \sin \phi , \quad \rho = \sqrt {m \omega }. \end{equation}

The change of coordinates \((q,p) \mapsto (I, \phi )\) maps the phase space flow onto a cylinder parametrized by \((I, \phi )\) so that the flows are now all straight lines in the new coordinates.

If you recall Example 4.2 you can now see what we were doing. The construction in Example 4.7 is also not dissimilar: if you observe that \(\dot W_i = \frac {\partial W_i}{\partial q^i} \dot q^i = p_i \dot q^i\), then you get \(W_i = \oint p_i\dd q^i\) which hints at the role of the \(W_i\) of the separated generating function in the context of the action–angle variables.

Example 4.10 (Kepler problem – elliptic motion). Let’s consider once again the Kepler problem for negative energies and, therefore, elliptic motions. For conciseness, we assume that the system is already reduced to the planar system. In this case, the invariant tori are two dimensional:

\begin{align} & p_\phi = m r^2 \dot \phi = M \\ & \frac {p_r^2}{2 m} + \frac {p_\phi ^2}{2 m r^2} - \frac {k}{r} = E <0, \end{align} where \((r,\phi )\) are polar coordinates on the orbit plane and the constants \((M,E)\), the angular momentum and the energy, parametrize the torus. The canonical actions, then, are

\begin{align} I_\phi & = \frac {1}{2\pi } \int _0^{2\pi } p_\phi \dd \phi = M \\ I_r & = \frac {2}{2\pi } \int _{r_{\mathrm {min}}}^{r_{\mathrm {max}}} p_r \dd r \\ & = \frac 1\pi \int _{r_{\mathrm {min}}}^{r_{\mathrm {max}}} \sqrt {2m(E + k/r) - M^2/r^2}\dd r \\ & = - M + k \sqrt {\frac {m}{2|E|}}, \end{align} where the latter follows from the identity

\begin{equation} \int _{r_{m}}^{r_{M}} \sqrt {\left (1-\frac {r_m}r\right )\left (\frac {r_M}r-1\right )}\dd r = \frac {\pi }2(r_m + r_M) - \pi \sqrt {r_m r_M}. \end{equation}

Then, the hamiltonian in the action angle variables takes the form

\begin{equation} H = - \frac {mk^2}{2(I_r + I_\phi )^2}. \end{equation}

Observe that \(H\) is symmetric in the actions. This means that the associated frequencies,

\begin{equation} \omega _\phi = \frac {\partial H}{\partial I_\phi }, \quad \omega _r = \frac {\partial H}{\partial I_r}, \end{equation}

coincide for all values of \((M,E)\): they are degenerate. Which, in particular, means that the trajectories are always periodic.

This has major consequences in the quantum mechanics of the hydrogen atom. The actions are all quantized as integer multiples of Planck’s constant: from the formula above, the energy depends only on the sum of the quantum numbers. Thus, above the ground state, energy levels are degenerate, which is why the energy spectrum of the hydrogen atom has the simple form so well explained by the Bohr model.

Recall the parametrization of the ellipse from (2.129), where the eccentricity, the focal parameter and the semiaxes of the ellipse where given respectively by

\begin{equation} e = \sqrt {1+ \frac {2EM^2}{mk^2}}, \quad p = \frac {M^2}{mk}, \quad a = \frac {p}{1-e^2}, \quad b = \frac {p}{\sqrt {1-e^2}}, \quad \end{equation}

We can derive the following identities,

\begin{equation} |E| = \frac {k}{2a}, \quad e^2 = 1 - \left (\frac {I_\phi }{ I_\phi + I_r}\right ), \end{equation}

which imply

\begin{equation} \frac {b}{a} = \sqrt {1-e^2} = \frac {I_\phi }{I_\phi + I_r} = \frac {|m|}{n} \end{equation}

where the last equality is just a rewriting using the notation for the hydrogen quantum numbers: \(|m| \leq n\) is the orbital quantum number, \(n\) is the principal quantum number and the energy is \(E_n \sim n^{-2}\).

Compare this example with Example 4.8. It is instructive to derive the action–angle coordinates for the general system in \(\mathbb {R}^3\) using this procedure or the separation of variables. The energy will have a similar form with the denominator including \(I_\theta \) in the sum and thus three degenerate frequencies. Reasoning on the frequencies and the actions, you can identify 5 constants of motion, which can be associated to the angular momentum vector, the energy and the Laplace-Runge-Lenz vector. With some algebra, these constants can be associated to the five parameters of the planetary orbit: inclination angle, longitude of the ascending node, semi–major axis, eccentricity and the angle of orientation in the orbital plane. You can read much more about this and how it is useful in perturbation theory, like the spin–orbit problem or the various restricted three-body problems, in [AKN06] or [Arn10].

Exercise 4.9 (Anharmonic oscillator). Consider the following hamiltonian system with one degree of freedom for a point particle of mass \(m\):

\begin{equation} H(q,p) = \frac {p^2}{2m} + V_0 \tan ^2\left (\frac {q}{\alpha }\right ), \end{equation}

where \(V_0>0\) and \(\alpha >0\) are given constants.

Compute the action variable \(I\) and show that the energy \(E = E(I)\) is given by

\begin{equation} E(I) = \frac {I (I + 2\alpha \sqrt {2m V_0})}{2m\alpha ^2}. \end{equation}

Verify that the period of the motion is

\begin{equation} T(I) = \frac {4 \pi m \alpha ^2}{I - \alpha \sqrt {2mV_0}} \end{equation}

and compute the angular variable.