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In this paper, we examine the role of the Jacobi last multiplier in the context of two-dimensional oscillators. We first consider two-dimensional unit-mass oscillators admitting a separable Hamiltonian description, i.e., H = H 1 + H 2 , where H 1 and H 2 are the Hamiltonians of two one-dimensional unit-mass oscillators. It is shown that there exists a third functionally-independent first integral Θ , ensuring superintegrability. Various examples are explicitly worked out. We then consider position-dependent-mass oscillators and the Bateman pair, where the latter consists of a pair of dissipative linear oscillators. Quite remarkably, the Bateman pair is found to be superintegrable, despite admitting a Hamiltonian which cannot be separated into two isolated (non-interacting) one-dimensional oscillators. [ABSTRACT FROM AUTHOR]

Based on the Lenard gradient sequence, a hierarchy of the nonlocal nonlinear Schrödinger (NNLS) equations is obtained. Using the Lax representation, the nonlocal finite-dimensional integrable system with Lie–Poisson structure is presented. Then, under coordinate transformation, the nonlocal finite-dimensional integrable system with Lie–Poisson structure can be expressed as the canonical Hamiltonian system of the standard symplectic structures. Moreover, the parametric representation of the NNLS equation and the nonlocal complex modified Korteweg–de Vries (NcmKdV) equation are constructed. Finally, according to the Hamilton–Jacobi theory, the action–angle variables are built and the inversion problem related to the Lie–Poisson Hamiltonian systems is discussed. [ABSTRACT FROM AUTHOR]

In this work, we employ the inverse scattering transform (IST) to study the action–angle variables for the Hirota equation. Based on variational principle, we demonstrate that an Euler–Lagrange equation is equivalent to the Hirota equation. By using the IST, several properties of scattering data for the equation are discussed, and we calculate their Poisson brackets successfully with the help of tensor product. Interestingly, we reveal that the action–angle variables can be constructed by the scattering data. Furthermore, the spectral parameter expressions of conservation laws for the equation are derived, related to the Hamiltonian formulation for the equation. [ABSTRACT FROM AUTHOR]

In this work, we present two finite-dimensional Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq equation by using the nonlinearization method. Moreover, the separation of variables on the common level set of Casimir functions is introduced to study these systems which are associated with a non-hyperelliptic algebraic curve. Finally, in light of the Hamilton–Jacobi theory, the action-angle variables for these systems are constructed, and the Jacobi inversion problem associated with the Hirota–Satsuma modified Boussinesq equation is obtained.

We discuss the holomorphic properties of the complex continuation of the classical Arnol'd – Liouville action-angle variables for real analytic 1 degree-of-freedom Hamiltonian systems depending on external parameters in suitable Generic Standard Form, with particular regard to the behaviour near separatrices. In particular, we show that near separatrices the actions, regarded as functions of the energy, have a special universal representation in terms of affine functions of the logarithm with coefficients analytic functions. Then, we study the analyticity radii of the action-angle variables in arbitrary neighborhoods of separatrices and describe their behaviour in terms of a (suitably rescaled) distance from separatrices. Finally, we investigate the convexity of the energy functions (defined as the inverse of the action functions) near separatrices, and prove that, in particular cases (in the outer regions outside the main separatrix, and in the case the potential is close to a cosine), the convexity is strictly defined, while in general it can be shown that inside separatrices there are inflection points. [ABSTRACT FROM AUTHOR]

This paper focuses on the escape problem of a harmonically forced classical particle from a purely quartic truncated potential well. The latter corresponds to various engineering systems that involve purely cubic restoring force and absence of linear stiffness even under the assumption of small oscillations, such as pre-tensioned metal wires and springs, and compliant structural components made of polymer materials. This is in contrast to previous studies where the equivalent potential well could be treated as linear at first approximation under the assumption of small perturbations. Due to the strong nonlinearity of the current potential well, traditional analytical methods are inapplicable for describing the transient bounded and escape dynamics of the particle. The latter is analyzed in the framework of isolated resonance approximation by canonical transformation to action–angle variables and the corresponding reduced resonance manifold. The escape envelope is formulated analytically. Surprisingly, despite the essential nonlinearity of the well investigated, it exhibits a universal property of a sharp minimum due to the existence of multiple intersecting escape mechanisms. Unlike previous studies, three underlying mechanisms that govern the transient dynamics of the particle were identified: two maximum mechanisms and a saddle mechanism. The first two correspond to a gradual increase in the system's response amplitude for a proportional increase in the excitation intensity, and the latter corresponds to an abrupt increase in the system's response and therefore more potentially hazardous. The response of the particle is described in terms of energy-based response curves. The maximal transient energy is predicted analytically over the space of excitation parameters and described using iso-energy contours. All theoretical predictions are in complete agreement with numerical results. [ABSTRACT FROM AUTHOR]

The gl3(C) rational Gaudin model governed by 3×3 Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalized Kepler potential. The angular components, on the contrary, are given implicitly by a generally transcendental equation. In the present note, devoted to the previously less studied models with the radial potential of the generalized Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator. [ABSTRACT FROM AUTHOR]

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical system used to exemplify and illustrate modern concepts that are crucial in order to understand more complicated dynamical systems. The Foucault’s pendulum first installed in 2005 in the collegiate church of Sainte-Waudru (Mons, Belgium) will allow us to numerically estimate the different quantities introduced.

It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if theWigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory--e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states--and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements. [ABSTRACT FROM AUTHOR]

Hybrid vibro-impact (HVI) oscillations is a strongly nonlinear dynamical regime that involves both linear oscillations and collisions under periodic, impulsive, or stochastic excitation. This regime arises in various engineering systems, such as mechanical components under tight rigid constraints, seismic-induced sloshing in partially-filled liquid storage tanks, and more. The adaptive nonlinearity of the HVI oscillator is used by the HVI-nonlinear energy sink as an effective vibration mitigation solution for broad energy and frequency range. Due to the extreme nonlinearity of this regime, traditional analytical methods are inapplicable for the description of its transient dynamics. In the current work, we model the HVI oscillator by a forced particle in a truncated quadratic potential well with infinite depth. The slow flow dynamics of the system in the vicinity of primary resonance is described by canonical transformation to action-angle (AA) variables and the corresponding reduced resonance manifold (RM). Two types of bifurcation are examined. The former is associated with transition between linear oscillations and the HVI-regime and vice versa, and the latter with reaching a chosen maximal transient energy level. The transition boundaries on the forcing parameters plane associated with both bifurcation types are obtained analytically. The maximal transient energy level obtained for any given set of forcing parameters is described analytically as well. The energy jumps associated with the bifurcation of type I and crossing the corresponding transition boundary are obtained. Two underlying dynamical mechanisms that govern the occurrence of bifurcations are identified. They correspond to two distinct scenarios: in the first scenario, the energy of the slow flow gradually reaches the threshold energy level and is thus referred to as the "maximum" mechanism. The second, potentially more dangerous scenario, involves abrupt transitions of the system's energy response from a relatively small value to the threshold energy level. This pattern is related to the passage of the slow-flow phase trajectory through the saddle point of the RM, and thus is referred to as the "saddle" mechanism. Both mechanisms are universal for systems that undergo escape from a potential well. All theoretical results are in complete agreement with full-scale numerical simulations. [ABSTRACT FROM AUTHOR]

Classical Floquet theory describes motion near a periodic orbit. But comparing Floquet theory to action angle methods shows which Jordan form is desirable. A new eigenvector algorithm is developed ensuring a canonical transform and handling the typical for the case of repeated eigenvalues, a chronic problem in conservative Hamiltonian systems. This solution also extends the Floquet decomposition to adjacent trajectories, and is fully canonical. This method yields the matrix of frequency partial derivatives, extending the solution's validity. Some numerical examples are offered. [ABSTRACT FROM AUTHOR]

In this work, the Poisson brackets for the scattering data of an integrable hierarchy for the Zakharov–Ito system are computed by employ the inverse scattering approach. Then, the action–angle variables and infinite conservation laws are expressed in terms of the scattering data. It is worth noting that, compared to previous works on Camassa–Holm equation and KdV-type equations, we have made some further progress. We have, for the first time, obtained action–angle variables that holds for the entire integrable hierarchy associated with the Zakharov–Ito system. Furthermore, we prove that the variational derivative of action–angle variables are eigenfunctions of the recursive operator, thus obtaining an infinite set of conservation laws in terms of scattering data. • Time evolution of scattering data consistent with Hamiltonian formulations. • Poisson brackets for scattering data within entire hierarchy are computed. • We present Hamiltonian and action–angle variables for entire hierarchy. • Derivative of action–angle variables are eigenfunctions of recursive operator. • Infinite conservation laws expressed by scattering data are derived. [ABSTRACT FROM AUTHOR]

In this paper, we employ the inverse scattering transform to investigate the action-angle variables of the Fokas-Lenells equation. Firstly, the Fokas-Lenells equation is derived by the variational principle, and the definition of the Poisson structure is presented. Then, the Poisson brackets between the scattering data are successfully determined by introducing the matrix tensor product. Thus the action-angle variables can be constructed by scattering data. Furthermore, the Hamiltonian of the Fokas-Lenells equation in terms of the scattering data is presented, and the Hamiltonian formalism of the equation is derived. [ABSTRACT FROM AUTHOR]

The current work is primarily devoted to the asymptotic analysis of the instability zones existing in the bi-linear Mathieu equation. In this study, we invoke the common asymptotical techniques such as the method of averaging and the method of multiple time scales to derive relatively simple analytical expressions for the transition curves corresponding to the 1:n resonances. In contrast to the classical Mathieu equation, its bi-linear counterpart possesses additional instability zones (e.g. for n > 2). In this study, we demonstrate analytically the formation of these zones when passing from linear to bi-linear models as well as show the effect of the stiffness asymmetry parameter on their width in the limit of low amplitude parametric excitation. We show that using the analytical prediction devised in this study one can fully control the width of the resonance regions through the choice of asymmetry parameter resulting in either maximum possible width or it's complete annihilation. Results of the analysis show an extremely good correspondence with the numerical simulations of the model. [ABSTRACT FROM AUTHOR]

Various models and systems involving the escape of periodically forced particle from the potential well demonstrate a common pattern. The minimal forcing amplitude required for the escape exhibits sharp minimum for the excitation frequency below the natural frequency of small oscillations in the well. The paper explains this regularity by detailed exploration of the transient dynamics of the escape from a number of benchmark potential wells. In the truncated parabolic well, in the absence of the damping, the minimal forcing amplitude required for the escape obviously tends to zero, as the excitation frequency approaches the natural frequency of oscillations in the well. Perturbation of the parabolic truncated well by weak symmetric softening nonlinearity leads to a shift of the minimum forcing to a nonzero value at the frequency below the natural. We explicitly compute this shift in the principal approximation by considering the slow-flow dynamics in conditions of the principal 1:1 resonance. The addition of nonlinearity also leads to appearance of two qualitatively different scenarios of the escape transition. In the first scenario, the slow flow just approaches the maximum value required for the escape. In the second scenario, the slow flow approaches dynamical saddle point that appears due to the external forcing. The presence of two competing and qualitatively different escape scenarios makes it impossible to formulate empirical escape criteria based on the steady-state solutions. Essentially, nonlinear φ 4 model requires slightly more complex mathematical tools (transformation to action-angle variables), but demonstrates very similar qualitative features of the transient dynamics, including two competing scenarios of the escape transition. [ABSTRACT FROM AUTHOR]

In this letter, we study the purely nonlinear oscillator by the method of action-angle variables of Hamiltonian systems. The frequency of the non-isochronous system is obtained, which agrees well with the previously known result. Exact analytic solutions of the system involving generalized trigonometric functions are presented. We also present arguments to show the adiabatic invariance of the action variable for a time-dependent purely nonlinear oscillator. • Action-angle variables defined for purely nonlinear oscillators. • Exact solutions written in terms of generalized trigonometric function. • Discussion on period function and frequency. • Proposed connection between ateb and generalized trigonometric functions. • Adiabatic invariance of action variable for purely nonlinear oscillator. [ABSTRACT FROM AUTHOR]

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the "Morse oscillator"). 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 action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach. [ABSTRACT FROM AUTHOR]

In this paper, we employ the inverse scattering approach to study the Poisson structure and action-angle variables of the complex modified Korteweg-de Vries (cmKdV) equation. We first derive the cmKdV equation via the principle of variation. Then, we successfully obtain the Poisson brackets for the scattering data of the equation. Furthermore, the action-angle variables are expressed in terms of the scattering data. Interestingly, our results show that the coordinate expression and the spectral parameter expression of the Hamiltonian can be related by the conservation laws. [ABSTRACT FROM AUTHOR]

The paper considers a transient process of escape of a classical particle from a one-dimensional potential well. We address a particular model of the infinite-range potential well that allows independent adjustment of the well depth and of the frequency of small oscillations. The problem can be conveniently reformulated in terms of action-angle variables. Further averaging provides a nontrivial conservation law for the slow flow. Then, one can consider the problem in terms of averaged transient dynamics on primary 1:1 resonance manifold. This simplification allows efficient analytic exploration of the escape process. As a result, one obtains a theoretical prediction for minimal forcing amplitude required for the escape, as a function of the excitation frequency. This function exhibits a single sharp minimum for a certain intermediate frequency value, below the frequency of small free oscillations. This result conforms to earlier numeric and semi-analytic estimations for similar escape models, considered, in particular, in connection with problems of ship capsize and dynamic pull-in in microelectromechanical systems. The results presented in the paper allow conjecturing the generic dynamical mechanism, responsible for these regularities. In particular, the aforementioned sharp minimum in the frequency-amplitude domain is related to formation of heteroclinic connection between the saddle points on the resonance manifold. Numeric simulations are in complete qualitative and reasonable quantitative agreement with the theoretical predictions. [ABSTRACT FROM AUTHOR]

Methods of Hamiltonian dynamics are applied to study the geodesic flow on the resolved conifolds (rcs) over Sasaki–Einstein space T1,1. We construct explicitly the constants of motion and prove complete integrability of geodesics in the five-dimensional Sasaki–Einstein space T1,1 and its Calabi–Yau metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. Using the small resolution, the geodesic motion on the rc remains completely integrable. Instead, in the case of the deformation of the conifold, the complete integrability is lost. [ABSTRACT FROM AUTHOR]

In this paper we express the linearized dynamics of interacting interfacial waves in stratified shear flows in the compact form of action-angle Hamilton’s equations. The pseudo-energy serves as the Hamiltonian of the system, the action coordinates are the contribution of the interfacial waves to the wave action and the angles are the phases of the interfacial waves. The term ‘generalized action angle’ aims to emphasize that the action of each wave is generally time dependent and this allows for instability. An attempt is made to relate this formalism to the action at a distance resonance instability mechanism between counter-propagating vorticity waves via the global conservations of pseudo-energy and pseudo-momentum. [ABSTRACT FROM AUTHOR]

In this work, we employ the inverse scattering approach to investigate the Poisson structure and action–angle variables for the Wadati–Konno–Ichikawa equation. The Poisson brackets for the scattering data are deduced. Consequently, the action–angle variables are expressed in terms of the scattering data. In addition, we also discuss on the conservation laws and the spectral parameter expression of Hamiltonian for this equation. [ABSTRACT FROM AUTHOR]

Due to the usually complicated and anisotropic nature of the fast-ion distribution function, diagnostic velocity-space weight functions, which indicate the sensitivity of a diagnostic to different fastion velocities, are used to facilitate the analysis of experimental data. Additionally, when velocityspace weight functions are discretized, a linear equation relating the fast-ion density and the expected diagnostic signal is formed. In a technique known as velocity-space tomography, many measurements can be combined to create an ill-conditioned system of linear equations that can be solved using various computational methods. However, when velocity-space weight functions (which by definition ignore spatial dependencies) are used, velocity-space tomography is restricted, both by the accuracy of its forward model and also by the availability of spatially overlapping diagnostic measurements. In this work, we extend velocity-space weight functions to a full 6D generalized coordinate system and then show how to reduce them to a 3D orbit-space without loss of generality using an action-angle formulation. Furthermore, we show how diagnostic orbit-weight functions can be used to infer the full fast-ion distribution function, i.e., orbit tomography. In depth derivations of orbit weight functions for the neutron, neutral particle analyzer, and fast-ion D-α diagnostics are also shown. [ABSTRACT FROM AUTHOR]

In this work, we consider the periodic Kostant-Toda flow on matrix loops in $${\mathfrak{s}\mathfrak{l}(n,\mathbb{C})}$$ of level k, which correspond to periodic infinite band matrices with period n with lower bandwidth equal to k and fixed upper bandwidth equal to 1 with 1's on the first superdiagonal. We show that the coadjoint orbits through the submanifold of such matrix loops can be identified with those of a finite-dimensional Lie group, which appears in the form of a semi-direct product. We then characterize the generic coadjoint orbits and obtain an explicit global cross-section for such orbits. We also establish the Liouville integrability of the periodic Kostant-Toda flow on such orbits via the construction of action-angle variables. [ABSTRACT FROM AUTHOR]

The generalization of the Mishchenko-Fomenko theorem for symplectic superintegrable systems to the case of an arbitrary, not necessarily compact, invariant submanifold allows giving a global description of a superintegrable Hamiltonian system, which can be split into several nonequivalent globally superintegrable systems on nonoverlapping open submanifolds of the symplectic phase manifold having both compact and noncompact invariant submanifolds. A typical example of such a composition of globally superintegrable systems is motion in a centrally symmetric field, in particular, the two-dimensional Kepler problem. [ABSTRACT FROM AUTHOR]

We give an elementary description of the relationship between the classical and quantum monodromy of a completely integrable system, from the point of view of geometric quantization, as a consequence of the construction of action–angle variables. We also describe the relation to Symington’s notion of “affine monodromy”. [ABSTRACT FROM AUTHOR]

Both the three-dimensional density of red clump giants and the gas kinematics in the inner Galaxy indicate that the pattern speed of the Galactic bar could be much lower than previously estimated. Here, we show that such slow bar models are unable to reproduce the bimodality observed in local stellar velocity space. We do so by computing the response of stars in the solar neighbourhood to the gravitational potential of slow and fast bars, in terms of their perturbed distribution function in action-angle space up to second order, as well as by identifying resonantly trapped orbits. We also check that the bimodality is unlikely to be produced through perturbations from spiral arms, and conclude that, contrary to gas kinematics, local stellar kinematics still favour a fast bar in the Milky Way, with a pattern speed of the order of almost twice (and no less than 1.8 times) the circular frequency at the Sun's position. This leaves open the question of the nature of the long flat extension of the bar in the Milky Way. [ABSTRACT FROM AUTHOR]

The near integrability property is studied for a class of perturbed Hamiltonian systems with two degrees of freedom on a phase space whose symplectic form depends non-uniformly on a small parameter.

A fully geometric approach is used to study the mutual spectral properties of different classes of MHD equilibria with symmetries defined by action-angle coordinates. We refer mostly to the helical equilibria of tokamak and reversed field pinch plasmas compared to the axisymmetric counterpart. Based on the existence of different coordinate systems, we show how the magnetic field and the vector potential are correspondingly evaluated, and how their spectral decompositions may largely vary. Most notably, a monochromatic perturbation in a reference frame can appear with a rich spectrum in another frame. The consequences on the interpretation of the measurements and the implications on the effectiveness of the externally applied magnetic fields on the plasma are discussed with practical examples. [ABSTRACT FROM AUTHOR]

In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of "microdiffusion": under generic assumptions on h and f, there exists an orbit of the system for which the drift of its action variables is at least of order √ ε, after a time of order √ ε -1. The assumptions, which are essentially minimal, are that there exists a resonant point for h and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting. [ABSTRACT FROM AUTHOR]

In this article we prove an action-angle theorem for b -integrable systems on b -Poisson manifolds improving the action-angle theorem contained in [14] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b -Poisson manifolds. [ABSTRACT FROM AUTHOR]

In this talk, we consider the vector Schwinger model with a photon mass term with Faddeevian Regularization, describing two-dimensional (2D) electrodynamics with mass-less fermions and study its Hamiltonian and path integral quantization. This theory is seen to be gauge-non-invariant (GNI). We then construct a gauge-invariant (GI) theory corresponding to this GNI theory using the Stueckelberg mechanism and then recover the physical content of the original GNI theory from the newly constructed GI theory under some special gauge-fixing conditions. [ABSTRACT FROM AUTHOR]

This paper presents closed-form solutions for the problem of long-term satellite relative motion in the presence of $$J_2$$ perturbations, and introduces a design methodology for long-term passive distance-bounded relative motion. There are two key ingredients of closed-form solutions.One is the model of relative motion; the other is the Hamiltonian model and its canonical solution of the $$J_2$$ -perturbed absolute motion. The model of relative motion makes no assumptions on the eccentricity of the reference orbit or on the magnitude of the relative distances. Besides, the relative motion model is concise with straightforward physical insight, and consistent with the Hamiltonian model. The Hamiltonian model takes into account the secular, long-periodic and short-periodic effects of the $$J_2$$ perturbation. It also remains separable in terms of spherical coordinates to ensure the application of the Hamilton-Jacobi theory to derive the canonical solution. When deriving the canonical solution, pseudo-circular and pseudo-elliptical orbits are treated separately and Carlson's method is employed to calculate elliptic integrals, which takes advantage of the symmetry of the integrand. These symmetry properties hold physical insights of the $$J_2$$ -perturbed absolute motion. To design the long-term distance-bounded relative motion, the nodal period and the drift of right ascension of the ascending node (RAAN) per nodal period are, respectively, matched non-instantaneously. Even though the nodal period and the drift of RAAN per nodal period can be obtained via the canonical solution, action-angle variables are used to obtain the frequency of the system without finding the complete solution to the perturbed orbital motion. [ABSTRACT FROM AUTHOR]

I describe dynamical modelling of the Milky Way using action-angle coordinates. I explain what action-angle coordinates are, and what progress has been made in the past few years to ensuring they can be used in reasonably realistic Galactic potentials. I then describe recent modelling efforts, and progress they have made in constraining the potential of the Milky Way and the local dark matter density. [ABSTRACT FROM AUTHOR]