Canonical Transformation Research Articles

Hybrid Unscented Kalman Filter: Application to the Simplest Walker

State estimation of hybrid dynamic systems, such as legged robots, is challenging because of the presence of non-smooth dynamics. This paper applies the Unscented Kalman Filter (UKF) state estimator and two novel hybrid extensions (HUKF) to a hybrid system, the simplest walking model. These estimators are identical far from the switching boundary, which partitions dynamic domains, but apply different time update algorithms at the switching boundary. (1) UKF permits sigma points to propagate through the system's hybrid dynamics. (2) HUKF-SPG (Sigma Point Generation) generates new sigma points when the weighted mean of the initial sigma points is on the switching boundary. (3) HUKF-SPT (Sigma Point Transformation) transforms the sigma points forward and backward in time through the system's hybrid dynamics only when the weighted mean of the initial sigma points is on the switching boundary. Results here shows that HUKF-SPG and HUKF-SPT have a lower absolute error but modestly more computations compared to UKF. A caveat of HUKF-SPT is it can only apply to conservative systems. HUKF-SPG is more general and could be applied to any hybrid system.

Kepler problem and chiral effective dynamics.

It is shown that by an appropriate canonical transformation, Kepler dynamics can be put in the form which allows one to exhibit the structure of the symmetry transformations related to the superintegrability. They appear to fit nicely into a general scheme of nonlinear realizations. In new coordinates, the Kepler dynamics results from dimensional reduction of that describing low-energy mesons with spontaneously broken chiral symmetry.

An Effective Field Theory for large oscillons

We consider oscillons — localized, quasiperiodic, and extremely long-living classical solutions in models with real scalar fields. We develop their effective description in the limit of large size at finite field strength. Namely, we note that nonlinear long-range field configurations can be described by an effective complex field ψ(t,x) which is related to the original fields by a canonical transformation. The action for ψ has the form of a systematic gradient expansion. At every order of the expansion, such an effective theory has a global U(1) symmetry and hence a family of stationary nontopological solitons — oscillons. The decay of the latter objects is a nonperturbative process from the viewpoint of the effective theory. Our approach gives an intuitive understanding of oscillons in full nonlinearity and explains their longevity. Importantly, it also provides reliable selection criteria for models with long-lived oscillons. This technique is more precise in the nonrelativistic limit, in the notable cases of nonlinear, extremely long-lived, and large objects, and also in lower spatial dimensions. We test the effective theory by performing explicit numerical simulations of a (d + 1)-dimensional scalar field with a plateau potential.

On the cosmological constant in the deformed Einstein–Cartan gauge gravity in De Donder–Weyl Hamiltonian formulation

AbstractA modification of the Einstein–Hilbert theory, the Covariant Canonical Gauge Gravity (CCGG), leads to a cosmological constant that represents the energy of the space–time continuum when deformed from its (A)dS ground state to a flat geometry. CCGG is based on the canonical transformation theory in the De Donder–Weyl (DW) Hamiltonian formulation. That framework modifies the Einstein–Hilbert Lagrangian of the free gravitational field by a quadratic Riemann–Cartan concomitant. The theory predicts a total energy‐momentum of the system of space–time and matter to vanish, in line with the conjecture of a “Zero‐Energy‐Universe” going back to Lorentz (1916) and Levi‐Civita (1917). Consequently, a flat geometry can only exist in presence of matter where the bulk vacuum energy of matter, regardless of its value, is eliminated by the vacuum energy of space–time. The observed cosmological constant Λobs is found to be merely a small correction attributable to deviations from a flat geometry and effects of complex dynamical geometry of space–time, namely torsion and possibly also vacuum fluctuations. That quadratic extension of General Relativity, anticipated already in 1918 by Einstein, thus provides a significant and natural contribution to resolving the “cosmological constant problem”.

Analytical ray tracing based on Hamilton principal function and conjugate variable pairs.

A transformation method based on optical Hamilton equations is proposed for 3D ray tracing in axially inhomogeneous gradient-index (GRIN) media with cylindrical symmetry. For a given GRIN field, the optical conjugate variable pairs of physical space can be transformed into a virtual space by applying canonical transformation. The virtual trace can be simply solved as a uniform expression regardless of what the GRIN field is, and one can inversely transform it into the physical space. The transformation is intimately related to a Hamilton principal function, called the S function, which simultaneously gives the real ray trace and its conjugate. The "conjugate trace" displays the direction cosines of the ray trace and thus shows the information about propagation direction at every point, and it can be independently derived from the S function without knowing the real trace. In addition, as two special dimension-reduction cases, the S function is also applicable for 2D structures with only axial inhomogeneity or cylindrical symmetry.

Study of Noether symmetry analysis for a cosmological model with variable G and Lambda gravity theory

In this present model we have discussed about the solution of the matter-dominated cosmological equations using the expression of Lambda =Lambda (G). Using Noether symmetry analysis we have analyzed both the classical and quantum cosmology. We have made a suitable point transformation in order to make one variable cyclic. As a result the evolution equations become simpler to solve. Wheeler–DeWitt (WD) equation has been constructed for this cosmological model and using conserved charge we have found the solution of WD equation. Finally the solutions have been analyzed in the cosmological point of view.

Equivalent Integrable Metrics on the Sphere with Quartic Invariants

We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we can construct new integrable systems on the sphere with quartic invariants.

Lewis–Riesenfeld invariants for PT-symmetrically coupled oscillators from two-dimensional point transformations and Lie algebraic expansions

We construct Lewis–Riesenfeld invariants from two-dimensional point transformations for two oscillators that are coupled to each other in space in a PT-symmetrical and time-dependent fashion. The non-Hermitian Hamiltonian of the model is conveniently expressed in terms of generators of the symplectic sp(4) Lie algebra. This allows for an alternative systematic approach to find Lewis–Riesenfeld invariants leading to a set of coupled differential equations that we solve by using time-ordered exponentials. We also demonstrate that point transformations may be utilized to directly construct time-dependent Dyson maps from their respective time-independent counterparts in the reference system.

Geometry of Inhomogeneous Poisson Brackets, Multicomponent Harry Dym Hierarchies, and Multicomponent Hunter–Saxton Equations

We introduce a natural class of multicomponent local Poisson structures $\mathcal P_k + \mathcal P_1$, where $\mathcal P_1$ is a local Poisson bracket of order one and $\mathcal P_k$ is a homogeneous Poisson bracket of odd order $k$ under assumption that is has Darboux coordinates (Darboux-Poisson bracket) and non-degenerate. For such brackets we obtain the general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples) and provide the description of the Casimirs, using purely algebraic procedure. In two-component case we completely classify such brackets up to the point transformation. From the description of Casimirs we derive new Harry Dym (HD) hierarchies and new Hunter-Saxton (HS) equations for arbitrary number of components. In two component case our HS equation differs from the well-known HS2 equation.

An Exact Hypergraph Matching algorithm for posture identification in embryonic C. elegans.

The nematode Caenorhabditis elegans (C. elegans) is a model organism used frequently in developmental biology and neurobiology [White, (1986), Sulston, (1983), Chisholm, (2016) and Rapti, (2020)]. The C. elegans embryo can be used for cell tracking studies to understand how cell movement drives the development of specific embryonic tissues. Analyses in late-stage development are complicated by bouts of rapid twitching motions which invalidate traditional cell tracking approaches. However, the embryo possesses a small set of cells which may be identified, thereby defining the coiled embryo's posture [Christensen, 2015]. The posture serves as a frame of reference, facilitating cell tracking even in the presence of twitching. Posture identification is nevertheless challenging due to the complete repositioning of the embryo between sampled images. Current approaches to posture identification rely on time-consuming manual efforts by trained users which limits the efficiency of subsequent cell tracking. Here, we cast posture identification as a point-set matching task in which coordinates of seam cell nuclei are identified to jointly recover the posture. Most point-set matching methods comprise coherent point transformations that use low order objective functions [Zhou, (2016) and Zhang, (2019)]. Hypergraphs, an extension of traditional graphs, allow more intricate modeling of relationships between objects, yet existing hypergraphical point-set matching methods are limited to heuristic algorithms which do not easily scale to handle higher degree hypergraphs [Duchenne, (2010), Chertok, (2010) and Lee, (2011)]. Our algorithm, Exact Hypergraph Matching (EHGM), adapts the classical branch-and-bound paradigm to dynamically identify a globally optimal correspondence between point-sets under an arbitrarily intricate hypergraphical model. EHGM with hypergraphical models inspired by C. elegans embryo shape identified posture more accurately (56%) than established point-set matching methods (27%), correctly identifying twice as many sampled postures as a leading graphical approach. Posterior region seeding empowered EHGM to correctly identify 78% of postures while reducing runtime, demonstrating the efficacy of the method on a cutting-edge problem in developmental biology.

One-Loop effective action approach to quantum MHV theory

It is well known that the MHV action, i.e. the action containing all the maximally helicity violating vertices, is alone not sufficient for loop computations. In order to develop loop contributions systematically and to ensure that there are no missing terms, we propose to formulate the quantum MHV action via one-loop effective action approach. The quadratic field fluctuations in the light cone Yang-Mills theory are explicitly integrated, followed by the classical canonical field transformation. We test the approach by calculating one loop (++++) and (+++) amplitudes, i.e. amplitudes that cannot be calculated from ordinary MHV action. Such an approach can be further used to unambiguously define loop corrections in other theories related to Yang-Mills theory by field transformations.

Temperature dependent nonequilibrium magneto-transport in a correlated polar single molecular transistor with quantum dissipation

Quantum magneto-transport in a dissipative single molecular transistor is investigated at finite temperature in the presence of electron correlation and electron–phonon interaction within the framework of the Anderson–Holstein–Caldeira–Leggett Hamiltonian. The electron–phonon interaction and dissipation are dealt with by canonical transformations and the Coulomb correlation is treated at the mean-field level. The transport properties such as spectral function, tunneling current, differential conductance, and spin polarization are determined using the Keldysh method.

Alternative symplectic structures in second–class constrained systems

The classical description of second-class constrained systems using alternative pairs, comprising a symplectic structure and a primary Hamiltonian other than the canonical ones, is investigated. It is exhibited that the dynamical equations do not fix a unique pair and instead there may exist infinitely many of them whose difference is not exhausted by canonical transformations. The consequences on the symplectic structure of the reduced phase space are addressed, namely the effect of different pairs on the construction of Dirac bracket is explored. Since constrained systems can also be studied via the Faddeev–Jackiw symplectic analysis, the implementation of alternative pairs on this formalism is considered as well. It is shown that Dirac’s algorithm outcomes are independent of the type of alternative pairs considered here irrespectively to whether there are also first-class constraints in the system. Finally, some comments on the implications of our results in the path integral quantisation scheme are pointed out.

Aspects of the Equivalence between the fμ and cνμ Terms in Lorentz-Violating Quantum Field Theory

It is known that in Lorentz-violating effective field theory, there is a classical equivalence between certain coefficients (c and f), in spite of the fact that the operators of the two types of coefficients described appear to have opposite behaviors under CPT. This paper is a continuation of previous work extending this equivalence to the quantum level: generalizing the explicit spinorial point transformations that interconvert the c and f terms; demonstrating that the transformations do not give rise to any additional anomaly terms as the quantum level; and giving explicit prescriptions for modifying the C, P, and T operators in the f theory, so that they correspond to the correct interchanges of physical particle states.

Escape dynamics of a particle from a purely nonlinear truncated quartic potential well under harmonic excitation

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.

Davydov–Kyslukha model as the starting point in the development of integrable multi-component nonlinear dynamical systems on quasi-one-dimensional lattices

The Davydov–Kyslukha nonlinear exciton-phonon model on a regular one-dimensional lattice is asserted to be the driving force for the development of integrable multi-component nonlinear dynamical systems encompassing excitonic, vibrational and orientational degrees of freedom. The two most representative quasi-one-dimensional integrable multi-component nonlinear systems inspired by the Davydov–Kyslukha model are presented explicitly in their concise Hamiltonian forms. The new six-subsystem integrable nonlinear model on a regular quasi-one-dimensional lattice is proposed and its derivation based upon the appropriate zero-curvature representation is presented. The constructive aspect of the famous Davydov motto is illustrated by the examples of semi-discrete integrable nonlinear dynamical systems canonicalizeable via the proper point transformations to the physically motivated field variables.

Dynamics of a hybrid cubic vibro-impact oscillator and nonlinear energy sink

In the current work, we model the HCVI oscillator by a forced particle in a hybrid quartic-square potential well with infinite depth under periodic external forcing. The slow-flow dynamics of the system is analyzed in the framework of isolated resonance approximation by canonical transformation to action–angle (AA) variables and the corresponding reduced resonance manifold (RM). Two types of bifurcation are examined and described analytically. The former is associated with the SNO-regime and the HCVI-regime and vice versa, and the latter with reaching a chosen maximal transient energy level. Unlike previous studies, three underlying dynamical mechanisms that govern the occurrence of bifurcations are identified: two maximum mechanisms and one 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 when takes place in engineering systems and more effective for energy pumping when takes place in the HCVI-NES. Both mechanism types are universal for systems that undergo escape from a potential well. The maximal transient energy is predicted analytically over the space of excitation parameters and described using iso-energy contours. The response curves of the system are obtained analytically, allowing a full perspective on the HCVI-oscillator dynamical response, and the HCVI-NES performances and design optimization. All theoretical results are supported by numerical simulations.

Two-mode photon added Schrödinger cat states: nonclassicality and entanglement

The concept of a photon added two-mode Schrödinger cat state in which both modes are independent is introduced, and their nonclassical properties and entanglement are studied. The introduced states emerge as eigenstates of f 1 f 2 a 1 a 2 , where f 1 , f 2 are nonlinear functions of the number operator, and a 1 , a 2 are annihilation operators. We study the evolution of these states under canonical transformation using the parity operator for the case of standard coherent states of the harmonic oscillator. The nonclassical properties of these states are evaluated especially by considering sub-Poissonian photon statistics and photon number distribution. Interestingly, the addition of photons leads to shifting the region in which photon number distribution shows oscillatory behavior. In addition, the entanglement of introduced states is quantitatively analyzed using concurrence. We observe that the state approaches maximum entanglement more rapidly after the addition of photons.

Non-Gaussian entanglement of magnetically coupled modes: A dynamical analysis

The quantum entanglement of two connected harmonic oscillators via angular momentum yielding a magnetic coupling [Formula: see text] is discussed in this study. The corresponding Hamiltonian is diagonalized by using three canonical transformations, and then the stationary wave function is obtained. The Schmidt decomposition is used to explicitly determine the modes [Formula: see text], with [Formula: see text], [Formula: see text] and [Formula: see text] being two quantum numbers associated with the two oscillators. We summarize our findings by looking at the effects of anisotropy [Formula: see text], [Formula: see text], asymmetry [Formula: see text], and dynamics on entanglement. (i) With increasing [Formula: see text], the entanglement grows exceedingly large. (ii) The sensitivity to [Formula: see text] is determined by [Formula: see text] and [Formula: see text]. The physical parameters and quantum numbers play a great role in the periodic resuscitation of entanglement.

The perturbation theory of matrix mechanics based on its canonical transformations

We elucidate that the canonical transformations of matrix mechanics are just the transformations of different representations of the standard quantum mechanics, reproduce the non-degenerate, degenerate and time-dependent perturbation theory in three men’s paper, we solve the mystery of time-dependent perturbation theory of matrix mechanics. The Kramers’ dispersion formula from time-dependent perturbation theory of matrix mechanics is also derived from that of wave mechanics.