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"source": [
"# The one Degree-of-Freedom Morse Oscillator"
]
},
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"metadata": {},
"source": [
"## Introduction and Development of the Problem"
]
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"The potential energy function derived by P. M. Morse is truly a ''workhorse'' potential energy function in theoretical chemistry {% cite morse1929diatomic --file action_angle %}. Originally it was devised to describe the intermolecular force between the two atoms in a diatomic molecule. It has the functional form:\n",
"\n",
"\\begin{equation}\n",
"V(q) = D \\left( 1-e^{-\\alpha q} \\right)^2,\n",
"\\label{eq:mpot}\n",
"\\end{equation}\n",
"\n",
"where $q$ represents the distance between the two atoms, $D >0$ represents the depth of the potential well (defined relative to the dissociated atoms), and $\\alpha >0$ controls the width of the potential well ($\\alpha$ small corresponds to a ''wide'' well, $\\alpha$ large corresponds to a narrow well). \n",
"\n",
"The Morse potential defines a one degree-of-freedom Hamiltonian system, i.e. the phase space is two dimensional described by coordinates $(q, p)$, where $p$ is the momentum conjugate to the position variable $q$. The Hamiltonian has the form of the sum of the kinetic energy and the potential energy (the Morse potential). The Hamiltonian system is integrable, and all trajectories lie on the level sets of the Hamiltonian function. The level sets can be used to derive integrals that give (time) parametrizations of trajectories. Regions of closed (bounded) trajectories can be used to construct special coordinates--*action-angle coordinates*, where the angle denotes a particular location on the closed level set and the action is the area enclosed by a closed level set (divided by $2 \\pi$). The transformation to action-angle coordinates for integrable Hamiltonan systems is a standard topic in good classical mechanics textbooks, see, e.g. {% cite landau1960mechanics arnold2013mathematical --file action_angle %}. The transformation preserves the Hamiltonian nature of the system, i.e. it is a canonical transformation, and therefore the standard approach to constructing such transformations is through the use of generating functions. However, for one degree-of-freedom time-independent Hamiltonian systems (such as the one described by the Morse potential) there is a simpler approach to generating the action-angle transformation that uses the geometry of the closed level set of the Hamiltonian function and the explicit (time) parametrization of the trajectories that can be obtained (in principle, if the necessary integrals can be explicitly computed) for one degree-of-freedom Hamiltonian systems. The approach is inspired by the seminal paper of Melnikov {% cite melnikov1963vk --file action_angle %}. This approach was developed in detail in {% cite wiggins1990introduction --file action_angle %} (the 1990 edition, *not* the 2003 edition) and is also described in {% cite mezic1994integrability --file action_angle %}. This is the approach that we will follow here.\n",
" \n",
"Action-angle variables are important in Hamiltonian mechanics for a number of reasons. From the point of view of classical mechanics they are the coordinate system used for the development of the Kolmogorov-Arnold-Moser and Nekhoroshev theorems {% cite dumas2014kam --file action_angle %}. They also play a central role in the quantization of classical Hamiltonian systems (in fact, the action and the constant $\\hbar$ have the same units) {% cite stone2005einstein keller1958corrected keller1960asymptotic keller1985semiclassical --file action_angle %}. \n"
]
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"The dynamical system defined by the Morse potential is Hamiltonian, with Hamiltonian function given by:\n",
"\n",
"\\begin{equation}\n",
"H (q, p) = \\frac{p^2}{2m} + D \\left( 1-e^{-\\alpha q} \\right)^2, \\qquad (q, p) \\in \\mathbb{R}^2,\n",
"\\label{eq:ham}\n",
"\\end{equation}\n",
"\n",
"and Hamilton's equations defined by:\n",
"\n",
"\\begin{eqnarray}\n",
"\\dot{q} = \\frac{\\partial H}{\\partial p} & = & \\frac{p}{m}, \\nonumber \\\\\n",
"\\dot{p} = - \\frac{\\partial H}{\\partial q} %& = & -2D \\left( 1-e^{-\\alpha q} \\right) \\alpha e^{-\\alpha q}, \\nonumber \\\\\n",
"& = & -2D \\alpha \\left(e^{-\\alpha q} - e^{-2\\alpha q} \\right). \n",
"\\label{eq:hameq}\n",
"\\end{eqnarray}\n",
"\n",
"Each level set of the Hamiltonian function (energy surface) has the form:\n",
"\n",
"\\begin{equation}\n",
"\\left\\{(q, p) \\in \\mathbb{R}^2 \\, | \\, H(q, p) = h = \\mbox{constant} \\right\\}.\n",
"\\label{eq:levelset}\n",
"\\end{equation}\n",
"\n",
"For one DoF systems, they are one dimensional curves that are invariant under the Hamiltonian dynamics. Since the trajectories lie on these curves, we can use the form of the level curves to obtain parametrizations of certain trajectories, as we will demonstrate. \n"
]
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"metadata": {},
"source": [
"## Revealing the Phase Space Structures\n",
"\n",
"### Equilibria and their Stability"
]
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"metadata": {},
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"It is straightforward to verify the following two points are equilibria for Hamilton's equations:\n",
"\n",
"\\begin{equation}\n",
"(q, p) = (\\infty, 0), \\, (0, 0).\n",
"\\label{eq:equil}\n",
"\\end{equation}\n",
"\n",
"Next we check their linearized stability properties. The Jacobian matrix, denoted $J$, of Hamilton's equation is given by:\n",
"\n",
"\\begin{equation}\n",
"J = \\left(\n",
"\\begin{array}{cc}\n",
"0 & \\frac{1}{m} \\\\\n",
"- 2D \\alpha^2 \\left(- e^{-\\alpha q} + 2 e^{-2\\alpha q} \\right) & 0\n",
"\\end{array}\n",
"\\right).\n",
"\\label{eq:jac}\n",
"\\end{equation}\n",
"\n",
"The eigenvalues of $J$ are given by:\n",
"\n",
"\\begin{equation}\n",
"\\pm \\sqrt{\\rm{det}\\, J}.\n",
"\\label{eq:eivs}\n",
"\\end{equation}\n",
"\n",
"Hence for the two equilibria we have:\n",
"\n",
"\\begin{equation}\n",
"(q, p) = (0, 0) \\Rightarrow \\rm{det} \\, J = -\\frac{2D \\alpha ^2}{m},\n",
"\\label{eq:eqstab}\n",
"\\end{equation}\n",
"\n",
"\n",
"with corresponding eigenvalues:\n",
"\n",
"\\begin{equation}\n",
"\\pm i\\sqrt{\\frac{2D}{m}} \\alpha,\n",
"\\label{eq:imageivs}\n",
"\\end{equation}\n",
"\n",
"\n",
"and\n",
"\n",
"\\begin{equation}\n",
"(q, p) = (\\infty, 0) \\Rightarrow \\rm{det} \\, J =0,\n",
"\\label{eq:eqsad}\n",
"\\end{equation}\n",
"\n",
"\n",
"where both eigenvalues are zero.\n",
"\n",
"The equilibrium $(q, p) = (0, 0)$ is stable (''elliptic'' in the Hamiltonian dynamics terminology) and the $(q, p) = (\\infty, 0)$ is unstable (a ''parabolic'' saddle point in the Hamiltonian dynamics terminology). "
]
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"source": [
"### Periodic Orbits"
]
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"Using the Hamiltonian \\eqref{eq:ham} it is straightforward to verify that the equilibria have the following energies:\n",
"\n",
"\n",
"\\begin{equation}\n",
"(q, p) = (\\infty, 0) \\Rightarrow H=D >0.\n",
"\\end{equation}\n",
"\n",
"\n",
"and\n",
"\n",
"\\begin{equation}\n",
"(q, p) = (0, 0) \\Rightarrow H=0.\n",
"\\end{equation}\n",
"\n",
"\n",
"Trajectories with energies larger than $D$ have unbounded motion in $q$. Trajectories having energies $h$ satisfying $0< h < D$ correspond to periodic motions. The level sets of these periodic orbits are given by:\n",
"\n",
"\\begin{equation}\n",
"h= \\frac{p^2}{2m} + D \\left( 1-e^{-\\alpha q} \\right)^2, \\quad 0< h < D,\n",
"\\label{eq:tp1}\n",
"\\end{equation}\n",
"\n",
"\n",
"and surround the stable equilibrium point $(q, p) = (0, 0)$ as shown in figure [fig:1](#fig:phase). The periodic orbits intersect the $q$ axis at two distinct points, $q_+ > 0$ and $q_- <0$, which are referred to as *turning points*. These turning points are computed as follows.\n",
"\n",
"Rewriting \\eqref{eq:tp1} gives:\n",
"\n",
"\n",
"\n",
"\\begin{equation}\n",
"\\frac{p^2}{2m} = h- D \\left( 1-e^{-\\alpha q} \\right)^2.\n",
"\\label{eq:tp2}\n",
"\\end{equation}\n",
"\n",
"\n",
"The turning points are obtained from \\eqref{eq:tp2} by setting $p=0$:\n",
"\n",
"\\begin{equation}\n",
"h=D \\left( 1-e^{-\\alpha q} \\right)^2\n",
"\\label{eq:tp3}\n",
"\\end{equation}\n",
"\n",
"\n",
"Note that we have:\n",
"\n",
"\\begin{equation}\n",
"0 \\le \\sqrt{\\frac{h}{D}} \\le 1,\n",
"\\label{eq:tp4}\n",
"\\end{equation}\n",
"\n",
"\n",
"from which we obtain the following relations: \n",
"\n",
"\n",
"\\begin{eqnarray}\n",
"&& 1 \\le 1 + \\sqrt{\\frac{h}{D}} \\le 2, \\label{eq:tp5} \\\\\n",
"&& 0 \\le 1 - \\sqrt{\\frac{h}{D}} \\le 1, \\label{eq:tp6}\n",
"\\end{eqnarray}\n",
"\n",
"\n",
"Taking the positive root of \\eqref{eq:tp3} gives:\n",
"\n",
"\\begin{equation}\n",
"1-e^{-\\alpha q} = \\sqrt{\\frac{h}{D}}.\n",
"\\label{eq:tp7}\n",
"\\end{equation}\n",
"\n",
"\n",
"from which we obtain the positive turning point:\n",
"\n",
"\\begin{equation}\n",
"q_+ = - \\frac{1}{\\alpha} \\log \\left(1- \\sqrt{\\frac{h}{D}} \\right) > 0.\n",
"\\label{eq:tppos}\n",
"\\end{equation}\n",
"\n",
"\n",
"Taking the negative root of \\eqref{eq:tp3} gives:\n",
"\n",
"\\begin{equation}\n",
"1-e^{-\\alpha q} = -\\sqrt{\\frac{h}{D}},\n",
"\\label{eq:tp8}\n",
"\\end{equation}\n",
"\n",
"\n",
"from which we obtain the negative turning point:\n",
"\n",
"\\begin{equation}\n",
"q_- = - \\frac{1}{\\alpha} \\log \\left(1 + \\sqrt{\\frac{h}{D}} \\right) < 0.\n",
"\\label{eq:tpneg}\n",
"\\end{equation}\n",
"\n",
"The level curve with energy equal to the dissociation energy $h=D$ has the form:\n",
"\n",
"\\begin{equation}\n",
"D= \\frac{p^2}{2m} + D \\left( 1-e^{-\\alpha q} \\right)^2,\n",
"\\label{eq:homo1}\n",
"\\end{equation}\n",
"\n",
"and is a separatrix connecting the (parabolic) saddle point. In the terminology of Hamiltonian dynamics it is a homoclinic orbit. It separates bounded from unbounded motion as illustrated in figure [fig:1](#fig:phase)."
]
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"\n",
"\n",
"\n",
"