Actionangle variables and novel superintegrable systems. In many cases there is a natural choice of variables, which simplify the problem they are known as \ action angle variables. Operator angleaction variables are studied in the frame of the su2 algebra, and their eigenstates and coherent states are discussed. As an example, we construct the spherical and pseudospherical generalizations of the twodimensional superintegrable models introduced by tremblay, turbiner and winternitz and by post and winternitz. The paper ddlls03 contains a detailed study of the singularity of the actionangle variables and shows that tori near a separatrix play an important.
They are especially useful for building perturbation theory for more complicated systems that in the lowest approximation reduce to a linear oscillator. A suitable choice of new conjugate variables can simplify the situation drastically. They degenerate to action angle variables on the union of circular level sets. We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for. We shall see that apart from their role as the natural variables for a separable system, the action angle variables act in an interesting way under adiabatic changes to the parameters of the system. We have presented the action angle variables for three twodimensional singular oscillator. In classical mechanics, actionangle coordinates are a set of canonical coordinates useful in solving many integrable systems. Here well study dynamics with the hamiltonian formalism. The actionangle variables constitute a system of coordinates and momenta in which the hamiltonian is a function only of the momentum. Actionangle variables from eric weissteins world of.
Action angle variables in classical and quantum mechanics. I wouldnt say this is a very standard definition of actionangle variables. The frequency of the nonisochronous system is obtained, which agrees well. Pdf actionangle variables for the purely nonlinear oscillator. During a cycle of adiabatic change one revolution of the earth, the polar angle change can be divided into a dynamic phase and a geometric phase. They degenerate to actionangle variables on the union of circular level sets.
Find the conditions necessary for this to be a canonical transformation, and nd a generating function fx. This is exactly the case of the simple pendulum, where it is shown that in order to make explicit the transformation to action angle variables, one needs to resort to nontrivial expansions of special functions and series reversion. Actionangle variables for the continuous spectrum philip j. Hence, the hamiltonian does not depend explicitly on time, i. In this paper we demonstrate the effectiveness of the actionangle variables in the study of superintegrable systems. If you just have a symplectic manifold, all you have is darbouxs theorem. Let us recall briefly some basic properties of the actionangle variables in classical mechanics. Actionangle variables on dirac manifolds 2012 zetamu. We shall see that apart from their role as the natural variables for a separable system, the actionangle variables act in an interesting way under adiabatic changes to the parameters of the system. This is the case classically and is the case quantummechanically if the action angle variables are properly defined. Coordinates from actionangle variables stack exchange. A new treatment of this problem is found in leacock 17 and at various points in this thesis. Pdf quantum ring models and actionangle variables a.
Sep 16, 2012 in this paper we demonstrate the effectiveness of the action angle variables in the study of superintegrable systems. Pdf actionangle variables for generic 1d mechanical. The full set of lecture notes, weighing in at around pages, can be downloaded here. An individual user may print out a pdf of a single chapter of a monograph in oso for personal use. Actionangle coordinates are chiefly used when the hamiltonjacobi equations are completely separable. Systems that can be so transformed are said to be integrable, which is to say that they can be solved, at least in principle. Pdf actionangle variables for the purely nonlinear. A meas for at taching krein energy signature to a continuum eigenmode is given. Morrison departmentofphysicsandinstituteforfusionstudies. In this paper we develop a general conceptual approach to the problem of existence of actionangle variables for dynamical systems, which establishes and uses the fundamental conservation property of associated torus actions.
Then we know that the characteristic function wq i, p i is the generator of a canonical transformation to new constant momenta p i, all q i, are ignorable, and the new hamiltonian depends only on the p i. In this paper we develop a general conceptual approach to the problem of existence of action angle variables for dynamical systems, which establishes and uses the fundamental conservation property of associated torus actions. Actionangle variables, ladder operators and coherent states. Action angle variables 199 b consider the transformation to new phasespace variables p p12, q xp1 2. The hamiltonian is derived for the foucault pendulum in polar coordinates in terms of the action variables of the system. Actionangle variables are widely accepted to be a very useful set of canonical coordinates in the solution of integrable problems. As far as the author has been able to determine, while the above authors have treated various specific aspects of the problem of defining. Pdf actionangle variables and novel superintegrable. Find materials for this course in the pages linked along the left. A phase space poisson bracket structure induced by the schroedinger equation is addressed in order to show the existence of a phase space representation of quantum dynamics where action angle variables can be defined. This is a second course in classical mechanics, given to final year undergraduates. Each such torus is labelled by action variables i ii. The method of action angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion.
Individual chapters and problem sheets are available below. The essence of newtons insight, encoded in his second law f ma, is that the motion of a particle described by its trajectory, rt, is completely determined once its initial position and velocity are known. Expanding the simple pendulums rotation solution in. The particular behaviour of the foucault pendulum can also be interpreted in terms of the parallel transport of the velocity vector. Invariant tori, actionangle variables, and phase space. I have read the wikipedia page on action angle variables and canonical transfor. Initially we have a onedimensional classical system whose evolution is determined by a hamiltonian h x, p with canonical variables x, p. Actionangle variables the systems under consideration are multiply periodic. D s 1 1 m ar 2 02 0 we give a complete and quantitative description of the analyticity properties of such integrating transformations, estimating, in particular, how such. Kam theory without actionangle variables 857 see ne84,her83. In this letter, we study the purely nonlinear oscillator by the method of action angle variables of hamiltonian systems.
On every connected component we perform a arnold liouville symplectic actionangle transformation which integrates the system. For such systems we can introduce an especially convenient set of canonical variables called actionangle variables. Chapter 4 canonical transformations, hamiltonjacobi equations, and actionangle variables weve made good use of the lagrangian formalism. Morrison department of physics and institute for fusion studies university of texas at austin austin, texas 78712, u. In the following we will assume that the hamiltonian does not depend explicitly on time. Actionangle variables in quantum mechanics sciencedirect. Actionangle variables lagrangian and hamiltonian mechanics. The main purpose of this paper is to show the existence of actionangle variables for integrable hamiltonian systems on dirac manifolds under some natural regularity and compactness conditions, using the torus action approach. We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and actionangle variables. A phase space poisson bracket structure induced by the schroedinger equation is addressed in order to show the existence of a phase space representation of quantum dynamics where actionangle variables can be defined. Chapter 4 canonical transformations, hamiltonjacobi equations, and action angle variables weve made good use of the lagrangian formalism. Initially we have a onedimensional classical system whose evolution is determined by a.
We use actionangle variables to describe the geodesic motions in the 5dimensional sasakieinstein spaces y p, q this formulation allows us to study thoroughly the complete integrability of the system. The action angle variables constitute a system of coordinates and momenta in which the hamiltonian is a function only of the momentum. Calculating the bound spectrum by path summation in action. The quantum mechanical addition of actionangle variables is shown to lead to a noncommutative hopf algebra. Actionangle variables suppose a mechanical system has a periodic motion libration return in phase space rotation oscillating momentum in some situations knowledge of the full motion is not so interesting as knowing the frequencies of motion in the system frequencies determined by the following procedure 1. Besides, the following canonical equations are valid. In many cases there is a natural choice of variables, which simplify the problem they are known as \actionangle variables. Semiclassical mechanics with molecular applications. In summary, the classical actionangle transformation, together with the correspondence principle, has been used to transform to quantum actionangle operators and thereby to effect an exact diagonalization of the hamiltonian in the standard 1 d solvable problems of quantum mechanics, and in the separable coulomb problem. Actionangle variables from eric weissteins world of physics.
We have suggested using the actionangle variables for the study of a quasiparticle in quantum ring. Action angle variables and novel superintegrable systems. Hannay angle study of the foucault pendulum in action. The analysis of the decision table involved rough setsbased detection of dependencies between state variables and the control action variable, identification of redundant state variables, and derivation of the optimized decision table table 1 with all redundant variables removed. We find that the hamiltonian involves a reduced number of action variables. Aug 23, 2010 we have suggested using the action angle variables for the study of a quasiparticle in quantum ring. May 12, 2015 i wouldnt say this is a very standard definition of actionangle variables. Problems can be greatly simpli ed by a good choice of generalized coordinates. Actionangle variables define an invariant torus, so called because holding the action constant defines the surface of a torus, while the angle. The paper ddlls03 contains a detailed study of the singularity of the actionangle variables and shows that tori near a. This means that they possess n independent constants of motion i. Actionangle variables, ladder operators and coherent. Actionangle variables, ladder operators and coherent states 2.
This is the case classically and is the case quantummechanically if the actionangle variables are properly defined. In the case of the simple pendulum, the solution of the equations of motion in actionangle variables can be achieved in closed form by standard application of the hamiltonjacobi method. For such systems we can introduce an especially convenient set of canonical variables called action angle variables. Aeronauticos, plaza cardenal cisneros, 3, 28040 madrid, spain. Global actionangle variables for the periodic toda lattice. Angleaction variables angleaction variables chapter. We consider the one degreeoffreedom hamiltonian system defined by the morse potential energy function the morse oscillator. Imagine a particle in one dimension oscillating back and forth in some potential. Or at least, with that definition you wont necessarily have them. Action variable article about action variable by the. Padgett, michael john, action angle variables in classical and quantum mechanics 1980. Action angle variables the systems under consideration are multiply periodic. A conceptual approach to the problem of actionangle variables.
The i pair is called the actionangle coordinates for this particular case. In this letter, we study the purely nonlinear oscillator by the method of actionangle variables of hamiltonian systems. The potential doesnt have to be harmonic, but it must be such as to trap the particle, which is executing periodic motion with period. Calculating the bound spectrum by path summation 373 3. The complete solution to the foucault pendulum is found in the small oscillation limit from the action. Action variable article about action variable by the free.
Classical mechanics has not really changed, in substance, since the days of isaac newton. Actionangle variables 199 b consider the transformation to new phasespace variables p p12, q xp1 2. Although there is no rigorous unitary transformation from cartesian p,q to angleaction i. Expanding the simple pendulums rotation solution in action. Equations 1, 1416 are functional equations for the unknown function k and the unknown. Expanding the simple pendulums rotation solution in actionangle variables martin lara a,1, sebastian ferrer. We construct actionangle variables for the dynamics on the union of 2tori, which occupy all but a measure zero subset of the phase space. We have presented the actionangle variables for three twodimensional singular oscillator. We construct action angle variables for the dynamics on the union of 2tori, which occupy all but a measure zero subset of the phase space. In summary, the classical action angle transformation, together with the correspondence principle, has been used to transform to quantum action angle operators and thereby to effect an exact diagonalization of the hamiltonian in the standard 1 d solvable problems of quantum mechanics, and in the separable coulomb problem.
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