11.5: E- Dynamics of Hamilton’s Equations
In this appendix we give a brief introduction to some of the characteristics and results associated with Hamiltonian differential equations (or, Hamilton’s equations or Hamiltonian vector fields). The Hamiltonian formulation of Newton’s equations reveals a great deal of structure about dynamics and it also gives rise to a large amount of deep mathematics that is the focus of much contemporary research. .
Our purpose here is not to derive Hamilton’s equations from Newton’s equations. Discussions of that can be found in many textbooks on mechanics (although it is often considered ‘’advanced mechanics”). For example, a classical exposition of this topic can be found in the classic book of Landau, and more modern expositions can be found in Abraham and Marsden and Arnold. Rather, our approach is to start with Hamilton’s equations and to understand some simple aspects and consequences of the special structure associated with Hamilton’s equations. Towards this end, our starting point will be Hamilton’s equations. Keeping with the simple approach throughout these lectures, our discussion of Hamilton’s equations will be for two dimensional systems.
We begin with a scalar valued function defined on \(\mathbb{R}^2\)
\[H = H(q, p), (q, p) \in \mathbb{R}^2. \label{E.1}\]
This function is referred to as the Hamiltonian. From the Hamiltonian, Hamilton’s equations take the following form:
\(\dot{q} = \frac{\partial H}{\partial p} (q, p)\),
\[\dot{p} = \frac{\partial H}{\partial q}(q, p), (q, p) \in \mathbb{R}^2. \label{E.2}\]
The form of Hamilton’s equations implies that the Hamiltonian is constant on trajectories. This can be seen from the following calculation:
\(\frac{dH}{dt} = \frac{\partial H}{\partial q} \dot{q} +\frac{\partial H}{\partial p} \dot{p}\)
\[= \frac{\partial H}{\partial q} \frac{\partial H}{\partial p} - \frac{\partial H}{\partial p} \frac{\partial H}{\partial q} = 0. \label{E.3}\]
Furthermore, this calculation implies that the level sets of the Hamiltonian are invariant manifolds. We denote the level set of the Hamiltonian as:
\[H_{E} = \{(q, p) \in \mathbb{R}^2 | H(q, p) = E\} \label{E.4}\]
In general, the level set is a curve (or possibly an equilibrium point). Hence, in the two dimensional case, the trajectories of Hamilton’s equations are given by the level sets of the Hamiltonian.
The Jacobian of the Hamiltonian vector field (E.2), denoted J, is given by:
\[J(p, q) = \begin{pmatrix} {\frac{\partial^{2}H}{\partial q \partial p}}&{\frac{\partial^{2}H}{\partial p^2}}\\ {-\frac{\partial^{2}H}{\partial q^2}}&{-\frac{\partial^{2}H}{\partial p \partial q}} \end{pmatrix}, \label{E.5}\]
at an arbitrary point \((q, p) \in \mathbb{R}^2\). Note that the trace of J(q, p), denoted trJ(q, p), is zero. This implies that the eigenvalues of J(q, p), denoted by \(\lambda_{1, 2}\), are given by:
\[\lambda_{1, 2} = \pm \sqrt{-det J(q, p)}, \label{E.6}\]
where detJ(q, p) denotes the determinant of J(q, p). Therefore, if \((q_{0}, p_{0})\) is an equilibrium point of (E.1) and \(detJ(q_{0}, p_{0}) = 0\), then the equilibrium point is a center for \(detJ(q_{0}, p_{0}) > 0\) and a saddle for \(detJ(q_{0}, p_{0}) < 0\).
Next we describe some examples of two dimensional, linear autonomous Hamiltonian vector fields.
Example \(\PageIndex{41}\) (The Hamiltonian Saddle )
We consider the Hamiltonian:
\[H(q, p) = \frac{\lambda}{2} (p^{2}-q^{2}) = \frac{\lambda}{2} (p-q)(p+q), (q, p) \in \mathbb{R}^2, \label{E.7}\]
with \(\lambda > 0\). From this Hamiltonian, we derive Hamilton’s equations:
\(\dot{q} = \frac{\partial H}{\partial p} (q, p) = \lambda p\),
\[\dot{p} = \frac{\partial H}{\partial p} (q, p) = \lambda q, \label{E.8}\]
or in matrix form:
\[\begin{pmatrix} {\dot{q}}\\ {\dot{p}} \end{pmatrix} = \begin{pmatrix} {0}&{\lambda}\\ {\lambda}&{0} \end{pmatrix} \begin{pmatrix} {q}\\ {p} \end{pmatrix}. \label{E.9}\]
The origin is a fixed point, and the eigenvalues associated with the linearization are given by \(\pm \lambda\). Hence, the origin is a saddle point.The value of the Hamiltonian at the origin is zero. We also see from (E.7) that the Hamiltonian is zero on the lines \(p - q = 0\) and p + q = 0. These are the unstable and stable manifolds of the origin, respectively. The phase portrait is illustrated in Fig. E.1.
The flow generated by this vector field is given in Chapter 2, Problem Set 2, problem 6.
Example \(\PageIndex{42}\) (The Hamiltonian Center)
We consider the Hamiltonian:
\[H(q, p) = \frac{\omega}{2} (p^{2}+q^{2}), (q, p) \in \mathbb{R}^2, \label{E.10}\]
with \(\omega > 0\). From this Hamiltonian, we derive Hamilton’s equations:
\(\dot{q} = \frac{\partial H}{\partial p} (q, p) = \omega p\),
\[\dot{p} = \frac{\partial H}{\partial p} (q, p) = -\omega q, \label{E.11}\]
or in matrix form:
\[\begin{pmatrix} {\dot{q}}\\ {\dot{p}} \end{pmatrix} = \begin{pmatrix} {0}&{\omega}\\ {-\omega}&{0} \end{pmatrix} \begin{pmatrix} {q}\\ {p} \end{pmatrix}. \label{E.12}\]
The level sets of the Hamiltonian are circles, and are illustrated in Fig. E.2.
The flow generated by this vector field is given in Chapter 2, Problem Set 2, problem 5.
We will now consider two examples of bifurcation of equilibria in two dimensional Hamiltonian systems. Bifurcation associated with one zero eigenvalue (as we studied in Chapter 8) is not possible since, following (E.6), if there is one zero eigenvalue the other eigenvalue must also be zero. We will consider examples of the Hamiltonian saddle-node and Hamiltonian pitchfork bifurcations. Discussions of the Hamiltonian versions of these bifurcations can also be found in Golubitsky et al.
Example \(\PageIndex{43}\) (Hamiltonian saddle-node bifurcation)
We consider the Hamiltonian:
\[H(q, p) = \frac{p^2}{2} - \lambda q+\frac{q^{3}}{3}), (q, p) \in \mathbb{R}^2, \label{E.13}\]
where \(\lambda\) is considered to be a parameter that can be varied. From this Hamiltonian, we derive Hamilton’s equations:
\(\dot{q} = \frac{\partial H}{\partial p} (q, p) = p\),
\[\dot{p} = -\frac{\partial H}{\partial p} (q, p) = \lambda - q^2, \label{E.14}\]
The fixed points for (E.14) are:
\[(q, p) = (\pm \sqrt{\lambda}, 0), \label{E.15}\]
from which it follows that there are no fixed points for \(\lambda < 0\), one fixed point for \(\lambda = 0\), and two fixed points for \(\lambda > 0\). This is the scenario for a saddle-node bifurcation.
Next we examine stability of the fixed points. The Jacobian of (E.14) is given by:
\[\begin{pmatrix} {0}&{1}\\ {-2q}&{0} \end{pmatrix}. \label{E.16}\]
The eigenvalues of this matrix are:
\(\lambda_{1, 2} = \pm \sqrt{-2q}\).
Hence \((q, p) = (-\sqrt{\lambda}, 0)\) is a saddle, \((q, p) = (\sqrt{\lambda}, 0)\) is a center, and (q, p) = (0, 0) has two zero eigenvalues. The phase portraits are shown in Fig. E.3.
Example \(\PageIndex{44}\) (Hamiltonian pitchfork bifurcation)
We consider the Hamiltonian:
\[H(q, p) = \frac{p^2}{2} - \lambda \frac{q^2}{2}+\frac{q^{4}}{4}), (q, p) \in \mathbb{R}^2, \label{E.17}\]
where \(\lambda\) is considered to be a parameter that can be varied. From this Hamiltonian, we derive Hamilton’s equations:
\(\dot{q} = \frac{\partial H}{\partial p} (q, p) = p\),
\[\dot{p} = -\frac{\partial H}{\partial p} (q, p) = \lambda q - q^3, \label{E.18}\]
The fixed points for (E.18) are:
\[(q, p) = (0, 0), (\pm p\lambda, 0), \label{E.19}\]
from which it follows that there is one fixed point for \(\lambda < 0\), one fixed point for \(\lambda = 0\), and three fixed points for \(\lambda > 0\). This is the scenario for a pitchfork bifurcation.
Next we examine stability of the fixed points. The Jacobian of (E.18) is given by:
\[\begin{pmatrix} {0}&{1}\\ {\lambda-3q^2}&{0} \end{pmatrix}. \label{E.20}\]
The eigenvalues of this matrix are:
\(\lambda_{1,2} = \pm \sqrt{\lambda-3q^2}\).
Hence (q, p) = (0, 0) is a center for \(\lambda < 0\), a saddle for \(\lambda > 0\) and has two zero eigenvalues for \(\lambda = 0\). The fixed points \((q, p) = (p\lambda, 0)\) are centers for \(\lambda > 0\). The phase portraits are shown in Fig. E.4.
We remark that, with a bit of thought, it should be clear that in two dimensions there is no analog of the Hopf bifurcation for Hamiltonian vector fields similar to to the situation we analyzed earlier in the non-Hamiltonian context. There is a situation that is referred to as the Hamiltonian Hopf bifurcation, but this notion requires at least four dimensions, see Van Der Meer.
In Hamiltonian systems a natural bifurcation parameter is the value of the level set of the Hamiltonian, or the "energy". From this point of view perhaps a more natural candidate for a Hopf bifurcation in a Hamiltonian system is described by the Lyapunov subcenter theorem, see Kelley. The setting for this theorem also requires at least four dimensions, but the associated phenomena occur quite often in applications.