Data processing: vehicles – navigation – and relative location – Navigation – Space orbits or paths
Reexamination Certificate
1999-03-29
2001-08-21
Zanelli, Michael J. (Department: 3661)
Data processing: vehicles, navigation, and relative location
Navigation
Space orbits or paths
C701S013000
Reexamination Certificate
active
06278946
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates in general to methods for space travel, and in particular, to methods for an object, such as a satellite, space craft, and the like, to be placed in lunar orbit around the moon, and/or orbit around other planets including earth.
2. Background of the Related Art
The study of motion of objects, including celestial objects, originated, in part, with Newtonian mechanics. During the eighteenth and nineteenth centuries, Newtonian mechanics, using a law of motion described by acceleration provided an orderly and useful framework to solve most of the celestial mechanical problems of interest for that time. In order to specify the initial state of a Newtonian system, the velocities and positions of each particle must be specified.
However, in the mid-nineteenth century, Hamilton recast the formulation of dynamical systems by introducing the so-called Hamiltonian function, H, which represents the total energy of the system expressed in terms of the position and momentum, which is a first-order differential equation description. This first order aspect of the Hamiltonian, which represents a universal formalism for modeling dynamical systems in physics, implies a determinism for classical systems, as well as a link to quantum mechanics.
By the early 1900s, Poincare understood that the classical Newtonian three-body problem gave rise to a complicated set of dynamics that was very sensitive to dependence on initial conditions, which today is referred to as “chaos theory.” The origin of chaotic motion can be traced back to classical (Hamiltonian) mechanics which is the foundation of (modern) classical physics. In particular, it was nonintegrable Hamiltonian mechanics and the associated nonlinear problems which posed both the dilemma and ultimately the insight into the occurrence of randomness and unpredictability in apparently completely deterministic systems.
The advent of the computer provided the tools which were hitherto lacking to earlier researchers, such as Poincare, and which relegated the nonintegrable Hamiltonian mechanics from the mainstream of physics research. With the development of computational methodology combined with deep intuitive insights, the early 1960s gave rise to the formulation of the KAM theorem, named after A. N. Kolmogorov, V. l. Arnold, and J. Moser, that provided the conditions for randomness and unpredictability for nearly nonintegrable Hamiltonian systems.
Within the framework of current thinking, almost synonymous with certain classes of nonlinear problems is the so-called chaotic behavior. Chaos is not just simply disorder, but rather an order without periodicity. An interesting and revealing aspect of chaotic behavior is that it can appear random when the generating algorithms are finite, as described by the so-called logistic equations.
Chaotic motion is important for astrophysical (orbital) problems in particular, simply because very often within generally chaotic domains, patterns of ordered motion can be interspersed with chaotic activity at smaller scales. Because of the scale characteristics, the key element is to achieve sufficiently high resolving power in the numerical computation in order to describe precisely the quantitative behavior that can reveal certain types of chaotic activity. Such precision is required because instead of the much more familiar spatial or temporal periodicity, a type of scale invariance manifests itself. This scale invariance, discovered by Feigenbaum for one-dimensional mappings, provided for the possibility of analyzing renormalization group considerations within chaotic transitions.
Insights into stochastic mechanics have also been derived from related developments in nonlinear analysis, such as the relationship between nonlinear dynamics and modern ergodic theory. For example, if time averages along a trajectory on an energy surface are equal to the ensemble averages over the entire energy surface, a system is said to be ergodic on its energy surface. In the case of classical systems, randomness is closely related to ergodicity. When characterizing attractors in dissipative systems, similarities to ergodic behavior are encountered.
An example of a system's inherent randomness is the work of E. N. Lorenz on thermal convection, which demonstrated that completely deterministic systems of three ordinary differential equations underwent irregular fluctuations. Such bounded, nonperiodic solutions which are unstable can introduce turbulence, and hence the appellation “chaos,” which connotes the apparent random motion of some mappings. One test that can be used to distinguish chaos from true randomness is through invocation of algorithmic complexity; a random sequence of zeros and ones can only be reproduced by copying the entire sequence, i.e., periodicity is of no assistance.
The Hamiltonian formulation seeks to describe motion in terms of first-order equations of motion. The usefulness of the Hamiltonian viewpoint lies in providing a framework for the theoretical extensions into many physical models, foremost among which is celestial mechanics. Hamiltonian equations hold for both special and general relativity. Furthermore, within classical mechanics it forms the basis for further development, such as the familiar Hamilton-Jacobi method and, of even greater extension, the basis for perturbation methods. This latter aspect of Hamiltonian theory will provide a starting point for the analytical discussions to follow in this brief outline.
As already mentioned, the Hamiltonian formulation basically seeks to describe motion in terms of first-order equations of motion. Generally, the motion of an integrable Hamilton system with N degrees of freedom is periodic and confined to the N-torus as shown in FIG.
1
.
FIG. 1
depicts an integrable system with two degrees of freedom on a torus, and a closed orbit of a trajectory. The KAM tori are concentric versions of the single torus. Hamiltonian systems for which N=1 are all integrable, while the vast majority of systems with N greater than or equal to 2 become nonintegrable.
An integral of motion which makes it possible to reduce the order of a set of equations, is called the first integral. To integrate a set of differential equations of the order 2N, that same number of integrals are generally required, except in the case of the Hamiltonian equations of motion, where N integrals are sufficient. This also can be expressed in terms of the Liouville theorem, which states that any region of phase space must remain constant under any (integrable) Hamiltonian formalism. The phase space region can change its shape, but not its phase space volume. Therefore, for any conservative dynamical system, such as planetary motion or pendula that does not have an attracting point, the phase space must remain constant.
Another outcome of the Hamiltonian formulation, which started out as a formulation for regular motion, is the implication of the existence of irregular and chaotic trajectories. Poincare realized that nonintegrable, classical, three-body systems could lead to chaotic trajectories. Chaotic behavior is due neither to a large number of degrees of freedom nor to any initial numerical imprecision. Chaotic behavior arises from a nonlinearity in the Hamiltonian equations with initially close trajectories that separate exponentially fast into a bounded region of phase space. Since initial conditions can only be measured with a finite accuracy and the errors propagate at an exponential rate, the long range behavior of these systems cannot be predicted.
The effects of perturbations in establishing regions of nonintegrability can be described for a weak perturbation using the KAM theorem. The KAM theorem, originally stated by Kolmogorov, and rigorously proven by Arnold and Moser, analyzed perturbative solutions to the classical many-body problem. The KAM theorem states that provided the perturbation is small, the perturbation is confined to an N-torus except for a negligible set of initial conditions which
Donner Irah H.
Galaxy Development LLC
Gibson Eric M.
Hale and Dorr LLP
Zanelli Michael J.
LandOfFree
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