Canonical Equations Of Motion Research Articles

Investigating Suitable Combinations of Dynamic Models and Control Techniques for Offline Reinforcement Learning Based Navigation: Application of Universal Omni-Wheeled Robots

Abstract Omnidirectional locomotion provides wheeled mobile robots (WMR) with better maneuverability and flexibility, which enhances their energy efficiency and dexterity. Universal omni-wheels are one of the best categories of wheels that can be used to develop a WMR (Amarasiri et al., 2022, “Robust Dynamic Modeling and Trajectory Tracking Controller of a Universal Omni-Wheeled Mobile Robot,” ASME Letters Dyn. Sys. Control., 2(4), p. 040902. 10.1115/1.4055690). We study dynamic modeling and controllers for mobile robots to train in a reinforcement learning (RL)-based navigation algorithm. RL tasks require copious amounts of learning iteration episodes, which makes training very time consuming. The choice of dynamic model and controller has a significant impact on training time. In this paper, we compare a traditional Kane’s equations model to a non-holonomic canonical momenta model (Barhorst, 2019, “Generalized Momenta in Constrained Non-Holonomic Systems—Another Perspective on the Canonical Equations of Motion,” Int. J. Non-Linear Mech., 113, pp. 128–145.). We implemented four controllers: proportional integral derivative, linear quadratic regulator with integral action, pole placement, and a full nonlinear sliding mode controller. We summarize the pros and cons of each of the modeling techniques, and control laws implemented. The outcomes of our analysis will improve RL training time for path generation in unstructured environments.

Hasimoto frames and the Gibbs measure of the periodic nonlinear Schrödinger equation

The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto’s theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric.

Evolution equations of multi-planet systems with variable masses

In celestial mechanics and astrodynamics, the study  of the  dynamical evolution of exoplanetary systems is the   relevant topics.    For today more than 3,000 exoplanetary systems are known. In this paper, we study the dynamic evolution of extrasolar systems, when the leading factor of evolution  is the variability of the masses of gravitating bodies. The problem of spherically symmetric bodies with  variable masses is considered in a relative coordinate system, this bodies inter-gravitating according to Newton's law. The quasi-elliptical motions of planets whose orbits do not intersect during evolution are investigated. It is believed that the mass of bodies under consideration  varies isotropically by various known laws with different velocities. The mass of the parent star is considered to be the most massive than its planets and the origin of the relative coordinate system is in the center of the parent star.. Due to the variability of the masses, the differential equations of motion become non-autonomous and the task is difficult. The problem is investigated by methods of perturbation theory. The canonical perturbation theory based on a periodic motion over a quasi-canonical section is used.  Canonical equations of motion are obtained in analogues of the second Poincare system, which are effective in the case when the analogues of eccentricities and the analogues of the inclination of the orbital plane of planets are sufficiently small.  The secular perturbations of the planets, which determine the behavior of the orbital parameters over long time intervals, are studied. The evolutionary equations of many planetary systems with isotropically varying masses in analogues of the second system of Poincare variables are derived in an analytical form which are obtained using the Wolfram Mathematica computer algebra system. This takes into account the effects of the decreasing mass of the parent star and the growth of the masses of the planets due to the accretion of matter from the remnants of the protoplanetary disk. For the three-planetary problem of four bodies with variable masses, the evolutionary equations in dimensionless variables are obtained explicitly. In the future, these results will be used to study the dynamics of the three-planet system K2-3 in the non-stationary stage of its evolution.

Brownian motion theory of the two-dimensional quantum vortex gas.

A theory of Brownian motion is presented for an assembly of vortices. The attempt is motivated by a realization of Dyson's Coulomb gas in the context of quantum condensates. By starting with the time-dependent Landau-Ginzburg (LG) theory, the dynamics of the vortex gas is constructed, which is governed by the canonical equationof motion. The dynamics of point vortices is converted to the Langevin equation, which results in the generalized Fokker-Planck (GFP) (or Smolkovski) equationusing the functional integral on the ansatz of the Gaussian white noise. The GFP, which possesses a non-Hermitian property, is characterized by two regimes called the overdamping and the underdamping regimes. In the overdamping regime, where the dissipation is much larger that the vortex strength, the GFP becomes the standard Fokker-Planck equation, which is transformed into the two-dimensional many-particle system. Several specific applications are given of the Fokker-Planck equation. An asymptotic limit of small diffusion is also discussed for the two-vortices system. The underdamping limit, for which the vortex charge is much larger than the dissipation, is briefly discussed.

Hamiltonian direct differentiation and adjoint approaches for multibody system sensitivity analysis

SummaryThe design of multibody systems involves high fidelity and reliable techniques and formulations that should help the analyst to make reasonable decisions. Given that constrained equations of motion for the simplest of multibody systems are highly nonlinear, determining the sensitivity terms is a computationally intensive and complex process that requires the application of special procedures. In this article, two novel Hamiltonian‐based approaches are presented for efficient sensitivity analysis of general multibody systems. The developed direct differentiation and the adjoint methods are based on constrained Hamilton's canonical equations of motion. This formulation provides solutions, which are more stable as compared to the results of direct integration of equations of motion expressed in terms of accelerations due to a reduced differential index of the underlying system of differential‐algebraic equations and explicit constraint imposition at the velocity level.The proposed Hamiltonian based methods are both capable of calculating the sensitivity derivatives and keeping the growth of constraint violation errors at a reasonable rate. The Hamiltonian‐based procedures derived herein appear to be good alternatives to existing methods for sensitivity analysis of general multibody systems.

Canonical Kane’s Equations of Motion for Discrete Dynamical Systems

The generalized inversion-based canonical equations of motion (CEOMs) for ideally constrained discrete dynamical systems are introduced in the framework of Kane’s method. Upon selection of the inertia-scaled canonical generalized acceleration variables, the proposed formulation employs the acceleration form of constraints and the generalized inversion Greville formula for parameterized solutions of underdetermined algebraic equations together with the nonminimal constrained momentum balance equations. The resulting CEOMs are explicit and full order in the acceleration variables. Moreover, the geometry of constrained motion is revealed by the CEOMs intuitively by partitioning the canonical accelerations column matrix into two portions at every time instant: a portion that drives the dynamical system to abide by the constraints, and a portion that generates the momentum balance dynamics such that the system abides by the Newton–Euler laws of motion. Some insightful geometrical perspectives of the CEOMs are illustrated via vectorial visualizations, which lead to verifying Gauss’s principle of least constraints and its Udwadia–Kalaba interpretation. The procedure is illustrated by formulating a third-order CEOMs dynamic model of two particles experiencing a single degree of freedom, as well as a sixth-order CEOMs dynamic model of a three-degrees-of-freedom disk that is rolling without slipping on an inertial plane under the effect of friction.

Analytical theory in terms of J2, J3, J4 with eccentric anomaly for short-term orbit predictions using uniformly regular KS canonical elements

A new non-singular analytical theory with respect to the Earth’s zonal harmonic terms J2, J3, J4 has been developed for short-periodic motion, by analytically integrating the uniformly regular KS canonical equations of motion using generalized eccentric anomaly ‘E’ as the independent variable. Only one of the eight equations need to be integrated analytically to generate the state vector, due to the symmetry in the equations of motion, and the computation for the other equations is done by changing the initial conditions. King-Hele’s expression for radial distance ‘r’ with J2 is also considered in generating the solution. The results obtained from the analytical expressions in a single step during half a revolution match quite well with numerically integrated values. Numerical results also indicate that the solution is reasonably accurate for a wide range of orbital elements during half a revolution and is an improvement over Sharon et al. [17] theory, which is generated in terms of KS regular elements. It can be used for studying the short-term relative motion of two or more space objects and in collision avoidance studies of space objects. It can be also useful for onboard computation in the navigation and guidance packages. Â

Generalized momenta in constrained non-holonomic systems—Another perspective on the canonical equations of motion

In this paper generalized momenta are defined for non-holonomic rigid body systems in a possibly novel formalism. The momenta are defined for Lagrangian/energy systems as well as projective d’Alembert systems, thus allowing canonical equations to be developed. The momenta can be defined in terms of quasi-variables or with ultimately an excess of the time derivatives of the generalized coordinates. This work may lay a framework to build non-holonomic generalizations to the Hamilton–Jacobi canonical transformation equations. In the sense that these equations are fundamental and minimalistic, they too can be considered canonical in their nature.For immediate practical consideration, demonstrated is significant reductions of the time required to numerically integrate the expository examples modeled with these techniques, indicating that this modeling formalism may have practical applicability

Generalized Langevin equation and the linear regression model with memory.

Because the existence of cheap emulators can give rise to potentially drastic computational savings as compared to direct numerical simulations of complex turbulence models, the study of reduced models has good practical relevance, and the guidelines regarding their construction from the true dynamical system model are much in demand. In accordance with this contemporary trend in science and engineering, we provide a new approach for the rigorous derivation of the linear reduced model with memory from the one-dimensional Majda-McLaughlin-Tabak (MMT) prototypical wave turbulence in thermal equilibrium. The basic idea in obtaining the physics-constrained autoregressive model is to perform the discretization in time of the generalized Langevin equation (GLE) governing a single wave profile of the true dynamical system model; the GLE formalized in the Mori-Zwanzig (MZ) projection theory is the exact reduced-order equation and the closed-form rearrangement of the canonical equation of motion for the Hamiltonian system. We study the performance of the resulting linear non-Markovian model in addressing two different manners of uncertainty quantification (UQ) problems: prediction and filtering. In doing so, we perform a comparison analysis against the linear and nonlinear Markovian models describing the MMT system, which are also built upon the GLE under the MZ framework. Finally we discuss an optimal selection of the statistical model in applying the reduced-model approach to the UQ of the turbulent signal.

On the adiabatic representation of Meyer-Miller electronic-nuclear dynamics.

The Meyer-Miller (MM) classical vibronic (electronic + nuclear) Hamiltonian for electronically non-adiabatic dynamics-as used, for example, with the recently developed symmetrical quasiclassical (SQC) windowing model-can be written in either a diabatic or an adiabatic representation of the electronic degrees of freedom, the two being a canonical transformation of each other, thus giving the same dynamics. Although most recent applications of this SQC/MM approach have been carried out in the diabatic representation-because most of the benchmark model problems that have exact quantum results available for comparison are typically defined in a diabatic representation-it will typically be much more convenient to work in the adiabatic representation, e.g., when using Born-Oppenheimer potential energy surfaces (PESs) and derivative couplings that come from electronic structure calculations. The canonical equations of motion (EOMs) (i.e., Hamilton's equations) that come from the adiabatic MM Hamiltonian, however, in addition to the common first-derivative couplings, also involve second-derivative non-adiabatic coupling terms (as does the quantum Schrödinger equation), and the latter are considerably more difficult to calculate. This paper thus revisits the adiabatic version of the MM Hamiltonian and describes a modification of the classical adiabatic EOMs that are entirely equivalent to Hamilton's equations but that do not involve the second-derivative couplings. The second-derivative coupling terms have not been neglected; they simply do not appear in these modified adiabatic EOMs. This means that SQC/MM calculations can be carried out in the adiabatic representation, without approximation, needing only the PESs and the first-derivative coupling elements. The results of example SQC/MM calculations are presented, which illustrate this point, and also the fact that simply neglecting the second-derivative couplings in Hamilton's equations (and presumably also in the Schrödinger equation) can cause very significant errors.

Extension of the Standard CD Algebra in the Axiomatic Approach for Spinor Field and Fermi–Bose Duality

The exclusive representations of the extended (29-dimensional) real-number Clifford–Dirac algebra are constructed for the spinor field. In the canonical Foldy–Wouthuysen representation for a \({e^-e^+}\)-dublet these representations contain physically justified on equal footing and conserved in time fermion and boson spins, and the canonical equation of motion for a dublet coincides with the quantum mechanical equation in the Hilbert space \({L_2({\mathcal R}^3)\times {\mathcal C}^4 \equiv H^{3,4}}\) with definite metrics. In \({H^{3,4}}\) the experimentally observed dublet energy is always positive. The Fermi and Bose spins define sets of both equal status Fermi and Bose states, which univocally elucidate the physical content of the Fermi–Bose (FB)-dualism of the \({e^-e^+}\) microobject. We briefly review the ad hoc boson object in the same space as partner of the \({e^-e^+}\)-dublet and treat issues, related to its BF-dualism. The mathematical correctness of the technique is acquired by the application of the simplified variant of the axiomatic approach (A-approach) to the spinor field.

Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined by a (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton–Jacobi equation applicable in the field theory then follows readily. The general method is illustrated with three examples: non-relativistic Hamiltonian mechanics, De Donder–Weyl scalar field theory, and string theory.

Importance of Accurate Static Equilibrium Orbit Calculation in Cyclotron Design

Knowledge of the static equilibrium orbit properties is crucial for cyclotron design and development. With the increase of computational speed, numerical orbit calculations are more and more often employed as a part of other methods. Direct numerical integration of the canonical equations of motion is used most often for orbit calculations. We propose an original method for the calculation of static equilibrium orbit properties by direct trajectory tracking in the real, configuration space, and compare it with the method most commonly used.

SPACE TIME COVARIANCE OF CANONICAL QUANTIZATION OF GRAVITY: A (FORMAL) GENERAL RESULT AND THE (RIGOROUS) EXPLICIT CASE OF 2+1 QUANTUM COSMOLOGY

A general classical theorem is presented according to which all invariant relations among the space time metric scalars, when turned into functions on the Phase Space of full Pure Gravity (using the Canonical Equations of motion), be-come weakly vanishing functions of the Quadratic and Linear Constraints. The implication of this result is that (formal) Dirac consistency of the Quantum Operator Constraints (annihilating the wave Function) suffices to guarantee space time covariance of the ensuing quantum theory: An ordering for each invariant relation will always exist such that the emanating operator has an eigenvalue identical to the classical value. The example of 2+1 Quantum Cosmology is explicitly considered: The four possible Cosmological Solutions -two for pure Einstein’s equations plus two more when a Λ term is present- are exhibited and the corresponding models are quantized. The invariant relations describing the geometries are explicitly calculated and promoted to operators whose eigenvalues are their corresponding classical values.

Hamiltonian Formulation for Water Wave Equation

This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressible, inviscid fluid have been constructed in cartesian coordinates and also in cylindrical coordinates. Then Lagrangian function within a certain flow region is expanded under the assumption that the dispersion μ and the nonlinearity ε satisfied . Using Hamilton’s principle for water wave evolution Hamiltonian formulation is derived. It is obvious that the motion of the system is conservative. Then Hamilton’s canonical equation of motion is also derived.

The Distances in the Stable Systems Due to the Virial Theorem

The virial theorem is written by using the canonical equations of motion in classical mechanics. A moving particle with an initial speed in an n-particle system is considered. The distance of the moving particle from the origin of the system to the final position is derived as a function of the kinetic energy of the particle. It is thought that the considered particle would not collide with other particles in the system. The relation between the final and initial distance of the particle from the origin of the system is given by a single equation.

Stochastic theory of quantum vortex on a sphere

A stochastic theory is presented for a quantum vortex in superfluid films coated on a two-dimensional sphere S^{2}. The starting point is the canonical equation of motion (Kirchhoff equation) for a point vortex, which is derived using the time-dependent Landau-Ginzburg theory. The vortex equation, which is equivalent to the spin equation, turns out to be the Langevin equation in presence of random forces. This is converted to the Fokker-Planck (FP) equation for the distribution function of a point vortex by using a functional integral technique. The FP equation is analyzed with special emphasis on the role of the pinning potential. By considering a typical form of the pinning potential, we address two problems: (i) The one is concerning an interplay between strength of the pinning potential and effective temperature, which discriminates the weak and strong coupling scheme to determine the solutions of the FP equation. (ii) The other is concerning a small diffusion limit, for which an asymptotic analysis is given using the functional integral to lead a compact expression of the distribution function. An extension to the vortex in nonspherical geometry is briefly discussed for the case of vortex on a plane and a pseudosphere.

A review of the time-dependent variational principle

The time-dependent variational principle associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space ℳ with canonical equations of motion. When the quantum observables generate a Lie group G and the states are taken as functions on an appropriate homogeneous quotient space space ℳ := G/G0 under under G, they can be equipped with a classical Poisson bracket which reproduces the commutators of the Lie group. The quantum system maps into a classical system whose equations of motion are governed by the expectation value ℋ of the quantum Hamiltonian H. We consider examples of this construction and show that the analysis of generalized classical systems provides insight into quantum many-body dynamics like chemical reactions.

Exact invariants and adiabatic invariants of Raitzin's canonical equations of motion for a nonlinear nonholonomic mechanical system

The exact invariants and the adiabatic invariants of Raitzin's canonical equations of motion for a nonlinear nonholonomic mechanical system are studied. The relations between the invariants and the symmetries of the system are established. Based on the concept of higher-order adiabatic invariant of a mechanical system under the action of a small perturbation, the forms of the exact invariants and adiabatic invariants and the conditions for their existence are proved. Finally, the inverse problem of the perturbation to symmetries of the system is studied and an example is also given to illustrate the application of the results.

Fisher information, canonical ensemble, and Hamiltonian systems

We show that the mathematical form of the information measure of Fisher's I [ f ] , if expressed as a functional of Gibbs' canonical probability distribution f (the most important one in statistical mechanics), incorporates important features of the intrinsic structure of classical mechanics and has a universal form in terms of “forces” (derivatives of generalized momenta with respect to time) and “generalized velocities”, i.e., one that is valid whenever Hamilton's equations of motion hold. Additionally, if the system of differential equations associated with Hamilton's canonical equations of motion is linear (a very important special instance), we show that there is an equal amount of Fisher information per degree of freedom, proportional to the inverse temperature. This gives temperature an “information” meaning. This “equipartition” of I is also shown to hold in a simple but relevant (magnetic) example involving a non-linear system of differential equations, where the Fisher measure turns out to be proportional to Langevin's function.