Hori's Method Research Articles

Parametric Resonance in the Oscillations of a Charged Pendulum Inside a Uniformly Charged Circular Ring

We study the mechanical system consisting of the following variant of the planar pendulum. The suspension point oscillates harmonically in the vertical direction, with small amplitude $\varepsilon$, about the center of a circumference which is located in the plane of oscillations of the pendulum. The circumference has a uniform distribution of electric charges with total charge $Q$ and the bob of the pendulum, with mass $m$, carries an electric charge $q$. We study the motion of the pendulum as a function of three parameters: $\varepsilon$, the ratio of charges $\mu=\frac{q}{Q}$ and a parameter $\alpha$ related to the frequency of oscillations of the suspension point and the length of the pendulum. As the speed of oscillations of the mass $m$ are small magnetic effects are disregarded and the motion is subjected only to the gravity force and the electrostatic force. The electrostatic potential is determined in terms of the Jacobi elliptic functions. We study the parametric resonance of the linearized equations about the stable equilibrium finding the boundary surfaces of stability domains using the Deprit – Hori method.

Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point

We consider a planar pendulum with an oscillating suspension point and with the bob carrying an electric charge $$q$$ . The pendulum oscillates above a fixed point with a charge $$Q.$$ The dynamics is studied as a system in the small parameter $$\epsilon$$ given by the amplitude of the suspension point. The system depends on two other parameters, $$\alpha$$ and $$\beta,$$ the first related to the frequency of the oscillation of the suspension point and the second being the ratio of charges. We study the parametric stability of the linearly stable equilibria and use the Deprit – Hori method to construct the boundary surfaces of the stability/instability regions.

Optimal low-thrust transfers between coplanar orbits with small eccentricities

In this paper, a first-order analytical solution, which includes the short periodic terms, for the problem of optimal time-fixed low-thrust limited-power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is obtained through canonical transformation theory. Short periodic terms are eliminated from the maximum Hamiltonian, expressed in non-singular orbital elements, through an infinitesimal canonical transformation built through the Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton–Jacobi equation through the separation of variables technique. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition.

Application of the Hori Method in the Theory of Nonlinear Oscillations

Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near‐identity transformations. These simplified algorithms are applied in determining second‐order asymptotic solutions of two well‐known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation.

Optimization of Low‐Thrust Limited‐Power Trajectories in a Noncentral Gravity Field—Transfers between Orbits with Small Eccentricities

Numerical and first‐order analytical results are presented for optimal low‐thrust limited‐power trajectories in a gravity field that includes the second zonal harmonic J2 in the gravitational potential. Only transfers between orbits with small eccentricities are considered. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin Maximum Principle, successive canonical transformations are performed and a suitable set of orbital elements is introduced. Hori method—a perturbation technique based on Lie series—is applied in solving the canonical system of differential equations that governs the optimal trajectories. First‐order analytical solutions are presented for transfers between close orbits, and a numerical solution is obtained for transfers between arbitrary orbits by solving the two‐point boundary value problem described by averaged maximum Hamiltonian, expressed in nonsingular elements, through a shooting method. A comparison between analytical and numerical results is presented for some maneuvers.

Hori method for generalized canonical systems

In this paper, some special features on the canonical version of Hori method, when it is applied to generalized canonical systems (systems of differential equations described by a Hamiltonian function linear in the momenta), are presented. Two different procedures, based on a new approach for the integration theory recently presented for the canonical version, are proposed for determining the new Hamiltonian and the generating function for systems whose differential equations for the coordinates describe a periodic system with one fast phase. These procedures are equivalent and they are directly related to the canonical transformations defined by the general solution of the integrable kernel of the Hamiltonian. They provide the same near-identity transformation for the coordinates obtained through the non-canonical version of Hori method. It is also shown that these procedures are connected to the classic averaging principle through a canonical transformation. As examples, asymptotic solutions of a non-linear oscillations problem and of the elliptic perturbed problem are discussed.

A First‐Order Analytical Theory for Optimal Low‐Thrust Limited‐Power Transfers between Arbitrary Elliptical Coplanar Orbits

A complete first‐order analytical solution, which includes the short periodic terms, for the problem of optimal low‐thrust limited‐power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory. The optimization problem is formulated as a Mayer problem of optimal control theory with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin maximum principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a Mathieu transformation. The short periodic terms are then eliminated from the maximum Hamiltonian through an infinitesimal canonical transformation built through Hori method. Closed‐form analytical solutions are obtained for the average canonical system by solving the Hamilton‐Jacobi equation through separation of variables technique. For transfers between close orbits a simplified solution is straightforwardly derived by linearizing the new Hamiltonian and the generating function obtained through Hori method.

A High-Order Analytical Model for the Secular Dynamics of Irregular Satellites

We develop an analytical model for the long-term (secular) dynamics of irregular satellites of the giant planets. The disturbing potential in this model is represented by a high-order (in semimajor axis, eccentricity, and inclination) Legendre expansion. We use a third-order Hori's averaging method to eliminate terms in the original equations that are irrelevant for the long-term dynamics and to construct new second- and third-order secular terms. The resulting secular equations are valid for both direct and retrograde orbits (of any inclinations) and for eccentricities up to ≈0.7. In the present paper we describe the mathematical background of our method and test it in several applications. The method uses a Hamiltonian formulation of dynamics. The original Hamiltonian and its high-order secular forms are represented by series that have self-similar functional forms. The coefficients of these series are calculated by using an algebraic manipulator. This approach allows us to iterate Hori's perturbation method to high orders. To test our method, we (1) calculate the precession frequencies of the orbits of the irregular satellites at Jupiter and (2) determine the dynamical structure of the Kozai resonance. We show that this resonance occurs at progressively larger (proper) inclinations with increasing separation of the satellite from the parent planet. These results are compared to those obtained by numerically integrating the exact equations of motion. Our theory will be particularly useful for determining the locations and strengths of secular resonances in the space occupied by distant satellite orbits. Several irregular satellites have been trapped in secular resonances by some, likely primordial, mechanism.

Notes on Hori Method for Non-Canonical Systems

In this paper the new approach for the integration theory of the canonical version of Hori method recently proposed is extended to the non-canonical one. It will be shown that the non-homogeneous ordinary differential equation with an auxiliary parameter t* associated with the mth order equation of the algorithm can also be replaced by a non-homogeneous partial differential equation in the time t. Using a generalized canonical approach, the general algorithm proposed by Sessin is then revised; as well as the Lagrange variational equations for the non-canonical version of Hori method. A simplified algorithm derived from Sessin's algorithm is presented for non-linear oscillations problem.

The contribution of José Luis Sérsic to celestial mechanics

In this paper, we shall review some aspects of José Luis Sersic's Ph.D. thesis: "Application of a certain type of canonical transformations to Celestial Mechanics", presented in 1956 to obtain the degree of Doctor in Philosophy in Astronomy of the National University of La Plata (La Plata, Argentina). We have found that Sérsic's work shares deep similarity to the method now known as Hori's Method although this was published ten years later. We shall discuss possible connections between both works, and the circumstances whereby Sérsic's work remained hidden to the international astronomical community.

A Semianalytical Model for the Motion of the Trojan Asteroids: Proper Elements and Families

In this paper we develop a semianalytical model to describe the long-term motion of Trojan asteroids located in tadpole orbits around the L 4 and L 5 jovian Lagrangian points. The dynamical model is based on the spatial elliptic three-body problem, including the main secular variations of Jupiter's orbit and the direct perturbations of the remaining outer planets. Based on ideas introduced by A. H. Jupp (1969, Astron. J. 74, 35–43), we develop a canonical transformation which allows the transformation of the tadpole librating orbits into circulating orbits. The disturbing function is then explicitly expanded around each libration point by means of a Taylor–Fourier asymmetric expansion. Making use of the property in which the different degrees of freedom in the Trojan problem are well separated with regard to their periods of oscillation, we are able to find approximate action-angle variables combining Hori's method with the theory of adiabatic invariants. This procedure is applied to estimate proper elements for the sample of 533 Trojans with well determined orbits at December 2000. The errors of our semianalytical estimates are about 2–3 times larger than those previously obtained with numerical approaches by other authors. Finally, we use these results to search for asteroidal families among the Trojan swarms. We are able to identify and confirm the existence of most of the families previously detected by Milani (1993, Celest. Mech. Dynam. Astron. 57, 59–94). The families of Menelaus and Epeios, both around L 4, are the most robust candidates to be the by-product of catastrophic disruption of larger asteroids. On the other hand, no significant family is detected around L 5.

Effects of dissipation and a liquid core on forced nutations in Hamiltonian theory

In a previous paper, the authors considered the free rotation of an earth model composed of a rigid mantle and a liquid core in the presence of dissipation and under the Hamiltonian formalism, obtaining analytical expressions for the free nutation modes. In this paper we treat the forced motion. Approximate analytical solutions are worked out by means of Hori's perturbation method, the free solutions obtained in the former paper playing the role of the unperturbed solutions required in the application of the method. These solutions are consistent in the sense that, with the usual terminology, the rigid body solutions and the complex transfer functions are calculated with the same parameters. Besides in-phase terms, the dissipation at the core-mantle boundary studied in this paper gives rise to out-of-phase terms. From a qualitative perspective, we discuss the issue of the resonance in this context. The presence of dissipation changes dramatically the character of the FCN wobble; that is, it is no longer a regular oscillation but a damped one. A strict resonance phenomenon cannot take place thereby, since the forcing perturbations are oscillations with a real (non-complex) frequency.

Optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field

A complete first-order analytical solution is developed for the problem of optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements as state variables. After applying the Pontryagin maximum principle and determining the optimal thrust acceleration, an intrinsic canonical transformation is performed: the Cartesian elements are changed by suitable orbital elements. Hori's method is applied in determining a first-order analytical solution. Simple analytical solutions are obtained explicitly for long-time transfers.

An elimination of the critical terms of a first-order Uranus-Neptune theory by Hori's method

We eliminate the 1:2 critical terms — after a previous elimination of the short period terms — in the Hamiltonian of a first order U-N theory. We take into account terms of degree 0, 1, 2, 3, 4 in the eccentricity-inclination. We apply for this elimination the Hori-Lie technique through the Poincare canonical variables and the Jacobi coordinates. The purely principal first order secular U-N Hamiltonian admits a complete solution. We obtained the U-N equations of motion generated by the principal first order long period U-N Hamiltonian which will be solved later. This part III is closely related to the two previous papers (Kamel, 1982, 1983).

Generalized Canonical Systems. Formal Solutions and the Main Problem of Satellite Theory

The generalized canonical version of Hori's method is discussed using some properties of generalized canonical system and, then, applied in solving the main problem of satellite theory.

Optimal low-thrust transfer between neighbouring quasi-circular orbits around an oblate planet

The problem of optimal low-thrust, limited power transfer between quasi-circular orbits ( e ⋍ 0) around an oblate planet is analysed. It is assumed that the orbital changes due to thrust acceleration and Earth oblateness are of the same order. A first order solution to the problem is obtained by application of Pontryagin's Maximum Principle. Subsequently, by application of Hori's method for generalized canonical systems, a first order solution in a small parameter ϵ is derived. Finally, three particular cases of long-time transfer and the orbit maintenance manoeuvre are considered. The results obtained are in agreement and represent an extension of the work done by Marec.

Optimal low-thrust limited power transfer between neighbouring quasi-circular orbits of small inclinations around an oblate planet

A complete analytic study about the influence of Earth's oblateness on the optimal low-thrust limited power transfer of small amplitude (orbit correction) between quasi-circular orbits of small inclinations is carried out up to the first order in a small parameter defined by the nondimensional thrust acceleration. The coefficient for the second zonal harmonic J 20 and the nondimensional thrust acceleration are supposed to be the same order of magnitude. Hori's method for generalized canonical systems is applied in order to obtain the analytical solution for adjoint and state differential equations. Simple analytic solutions are obtained explicitly for long-time transfer.

Third-order Uranus-Neptune theory by Hori's method

We construct a U-N secular canonical planetary theory of the third order with respect to planetary masses. The Hori-Lie procedure is adopted to solve the problem. Expansions have been carried out by hand, neglecting powers higher than the second with respect to the eccentricity-inclination. We take into account the principal as well as the indirect part of the planetary disturbing function. The theory is expressed in terms of the Poincare canonical variables, referring to the Jacobi-Radau set of origins. We assume that the 1:2 U-N critical terms and its multiples are the only periodic terms.

The generalized canonical form of Hori's method for non-canonical systems

Any dynamical system can be put in generalized canonical form through the introduction of a set of auxiliary ‘conjugate’ variables or momenta and solved by perturbation theory based on Lie series. The application of Hori's method for generalized canonical system leads to a new canonical transformation — the Mathieu transformation — defined by the solution of the Hori auxiliary system. This new transformation simplifies the algorithm since the inversion of the solution of the Hori auxiliary system is no longer necessary. In this paper, we wish to show some peculiarities of this technique.

Analytical Stability Criteria for Gas-Lubricated Sliders for Magnetic Disk Drive

Stability criteria for gas-lubricated sliders for magnetic disk drives are analyzed using Hori's method. Some relations between flotational stability and float factors are discussed. Equations of slider motion for a rigid body slider are expressed using linearized coefficients for gas-lubricated films accounting for second-order slip flow. Stability of the slider is analyzed using the Routh-Hurwitz stability criteria. The calculated results make it clear that some types of sliders become unstable when ambient pressure is reduced, or gas compressibility number is increased. Furthermore, analytical results were confirmed not to conflict with experimental work. Presented as a Society of Tribologists and Lubrication Engineers Paper at the ASME/STLE Tribology Conference in Baltimore, Maryland, October 16–19, 1988