Extended Camus Theory and Higher Order Conjugated Curves

2019 ◽  
Author(s):  
Cody Leeheng Chan ◽  
Kwun-Lon Ting
Keyword(s):  
Stationary Point ◽  
Higher Order ◽  
The Body ◽  
The Other ◽  
Body System ◽  
Straight Line ◽  
Pair Generation ◽  
Curvature Theory ◽  
Three Body ◽  
Two Bodies

Abstract According to Camus’ theorem, for a single DOF 3-body system with the three instant centers staying coincident, a point embedded on a body traces a pair of conjugated curves on the other two bodies. This paper discusses a fundamental issue not addressed in Camus’ theorem in the context of higher order curvature theory. Following the Aronhold-Kennedy theorem, in a single degree-of-freedom three-body system, the three instant centers must lie on a straight line. This paper proposes that if the line of the three instant centers is stationary (i.e. slide along itself), on the line of the instant centers a point embedded on a body traces a pair of conjugated curves on the other two bodies. Another case is that if the line of the three instant centers rotate about a stationary point, the stationary point embedded on the body also traces a pair of conjugated curves on the other two bodies. The paper demonstrates the use of instantaneous invariants to synthesize such a three-body system leading to a conjugate curve-pair generation. It is a supplement or extension of the Camus’ theorem. The Camus’ theorem may be regarded as a special singular case, in which all three instant centers are coincident.

Extended Camus Theory and Higher Order Conjugated Curves

2019 ◽  
Vol 11 (5) ◽  
Author(s):  
Cody Leeheng Chan ◽  
Kwun-Lon Ting
Keyword(s):  
Stationary Point ◽  
Higher Order ◽  
Degree Of Freedom ◽  
The Other ◽  
Body System ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Pair Generation ◽  
Three Body ◽  
Two Bodies

According to Camus’ theorem, for a single degree-of-freedom (DOF) three-body system with the three instant centers staying coincident, a point embedded on a body traces a pair of conjugated curves on the other two bodies. This paper discusses a fundamental issue not addressed in Camus’ theorem in the context of higher order curvature theory. Following the Aronhold–Kennedy theorem, in a single degree-of-freedom three-body system, the three instant centers must lie on a straight line. This paper proposes that if the line of the three instant centers is stationary (i.e., slide along itself) on the line of the instant centers, a point embedded on a body traces a pair of conjugated curves on the other two bodies. Another case is that if the line of the three instant centers rotates about a stationary point, the stationary point embedded on a body also traces a pair of conjugated curves on the other two bodies. The paper demonstrates the use of instantaneous invariants to synthesize such a three-body system leading to a conjugate curve-pair generation. It is a supplement or extension of Camus’ theorem. Camus’ theorem may be regarded as a special singular case, in which all three instant centers are coincident.

scholarly journals New Phenomena in the Spatial Isosceles Three-Body Problem

2015 ◽  
Vol 25 (09) ◽  
pp. 1550116 ◽  
Author(s):  
Duokui Yan ◽  
Tiancheng Ouyang
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Initial Condition ◽  
Collision Orbit ◽  
Three Body Problem ◽  
Straight Line ◽  
Three Body ◽  
New Phenomena ◽  
Two Bodies

In the three-body problem, it is known that there exists a special set of periodic orbits: spatial isosceles periodic orbits. In each period, one body moves up and down along a straight line, and the other two bodies rotate around this line. In this work, we revisit this set of orbits by applying variational method. Two unexpected phenomena are discovered. First, this set is not always spatial. It actually bifurcates from the circular Euler (central configuration) orbit to the Broucke (collision) orbit. Second, one of the orbits in this set encounters an oscillating behavior. By running its initial condition, the orbit stays periodic for only a few periods before it becomes irregular. However, it moves close to another periodic shape in a while. Shortly it falls apart again and starts running close to a third periodic shape after a moment. This oscillation continues as t increases. Actually, up to t = 1.2 × 105, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions.

On the generalized restricted problem of three bodies

1974 ◽  
Vol 22 ◽  
pp. 85
Author(s):  
G. N. Duboshin
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Active Material ◽  
The Body ◽  
The Other ◽  
Three Body Problem ◽  
Straight Line ◽  
Second Derivatives ◽  
Material Points ◽  
Three Body

AbstractThe particular case of the complete generalized three-body problem where one of the body-points does not exert influence on the other two is analysed. These active material points act on the passive point and also on each other with forces (attraction or repulsion), proportional to the product of masses of both points and a certain function of the time, their mutual distances and their first and second derivatives. Furthermore it is not supposed that generally the th ird axiom of mechanics (action = reaction) takes place.One determines the conditions for some particular solutions which exist, when the three points form the equilateral triangle or remain always on a straight line.Finally, some conclusions on the Liapunov stability in the simplest cases are drawn.

CFD Solution of a Two-Canopy Parachute in a Top-to-Top Formation With a Vent of Air From One of Its Canopy

2008 ◽  
Author(s):  
Mohammad J. Izadi
Keyword(s):  
Cross Sectional Area ◽  
The Body ◽  
The Other ◽  
Bluff Bodies ◽  
Variable Cross ◽  
Sectional Area ◽  
Cross Sectional ◽  
Drag Forces ◽  
Varied Form ◽  
Two Bodies

A CFD study of a 3 Dimensional flow field around two bodies (Two Canopies of a Parachutes) as two bluff bodies in an incompressible fluid (Air) is modeled here. Formations of these two bodies are top-to-top (One on the top of the other) with respect to the center of each other. One canopy with a constant cross sectional area with a vent of air at its apex, and the other with a variable cross sectional area with no vent is studied here. Vertical distances of these two bodies are varied form zero to half, equal, double and triple radius of the body with a vent on it. The flow condition is considered to be 3-D, unsteady, turbulent, and incompressible. The vertical distances between the bluff bodies, cross sectional area, and also vent ratio of bluff bodies are varied here. The drag forces with static pressures around the two bodies are calculated. From the numerical results, it can be seen that, the drag coefficient is constant on the range of zero to twenty percent of the vent ratio and it decreases for higher vent ratios for when the upper parachute is smaller than the lower one, and it increases for when the upper parachute is larger than the lower one. Both Steady and Unsteady cases gave similar results especially when the distance between the canopies is increased.

scholarly journals V. An account of an experiment explaining a mechanical paradox, viz. that two bodies of equal weight suspended on a certain sort of balance [as in Tab. Fig. 2.] do not lose their Æquilibrium, by being removed one farther from, the other nearer to the center.

1731 ◽  
Vol 37 (419) ◽  
pp. 125-129
Keyword(s):  
The Body ◽  
The Other ◽  
Equal Weight ◽  
Two Bodies

If the two Weights, P, W, in Fig. 3. hang at the Ends of the Balance A B, whose Center of Motion is C; those Weights will act against each other (because their Directions are contrary) with Forces made up of the Quantity of Matter in each multiplied by its Velocity; that is, by the Velocity which the Motion of the Balance turning about C will give to the Body suspended.

Memoirs: Materials for a Monograph of the Ascons.--I. On the Origin and Growth of the Triradiate and Quadriradiate Spicules in the Family Clathrinidæ

1898 ◽  
Vol s2-40 (160) ◽  
pp. 469-587
Author(s):  
E. A. MINCHIN
Keyword(s):  
Distal Portion ◽  
The Body ◽  
The Other ◽  
Ground Substance ◽  
Connective Tissue Layer ◽  
Dermal Layer ◽  
Straight Line ◽  
The Family ◽  
A Cell ◽  
Pore Cell

1. The first appearance of a calcareous spicule or spicular element, both ancestrally and in the actual development, was probably a minute vacuole in a cell of the dermal layer, filled with an organic substance perhaps identical with the intercellular ground substance, within which the minute sclerite appeared as a crystal or concretion. 2. The ancestral sclerite, though crystalline in structure, soon assumed a non-crystalline form as a whole, as an adaptation to its secondarily acquired function of support, and as it grew in size the contents of the vacuole formed the spicule sheath. 3. The ancestral form of spicule in the Calcarea was a simple monaxon, placed tangentially and completely embedded in the body-wall, lying between two adjacent pores. 4. From this ancestral spicule the forms of spicule now occurring in the Calcarea arose as follows: (a) the primitive monaxon acquired a distal portion projecting from the surface, as in the existing primary monaxons; (b) groups consisting each of three primitive monaxons became united by their contiguous ends to form a single triradiate system; (c) to some of the triradiate systems thus formed a fourth ray was added, secreted by the pore-cell, giving rise to the quadriradiate system ; (d) some of the triradiate systems, by loss of one ray and placing of the other two in a straight line, or by loss of two rays, perhaps became modified into secondary monaxon spicules. 5. The power of secreting a monaxon sclerite was primitively possessed by every cell of the dermal layer, and this condition appears to be retained in Leucosolenia. In Clathrina, on the other hand, all the skeletogenous cells migrate inwards from the dermal epithelium, and form a connective-tissue layer distinct in function from the contractile, undifferentiated dermal epithelium. In Leucosolenia also the actinoblasts of the triradiate systems form a deeper layer, but the dermal epithelium secretes primary monaxons--at least in the young form--and is non-contractile. 6. The forms of the spicules are the result of adaptation to the requirements of the sponge as a whole, produced by the action of natural selection upon variation in every direction.

A new measurement of the decay rate of the negative positronium ion:status and preliminary results

2005 ◽  
Vol 83 (4) ◽  
pp. 413-423 ◽  
Author(s):  
F Fleischer ◽  
K Degreif ◽  
G Gwinner ◽  
M Lestinsky ◽  
V Liechtenstein ◽  
...  
Keyword(s):  
Experimental Data ◽  
Decay Rate ◽  
Lifetime Measurement ◽  
The Other ◽  
Body System ◽  
Positron Source ◽  
Preliminary Results ◽  
The Status ◽  
Set Up ◽  
Three Body

A great number of theoretical papers have been published dealing with the negative positronium ion Ps–. On the other hand, experimental data on this purely leptonic three-body system (e+e–e–) is very limited. Apart from a first observation, a lifetime measurement with an accuracy of 4% has been published. We have built a set-up to produce Ps– making use of moderated positrons from a 22Na source, and we are presently running an experiment to improve on its decay rate. This paper discusses the status of the project as well as the possibilities of extending these investigations to other properties of Ps–, the latter becoming possible using the NEPOMUC positron source at the FRM II reactor in Munich.PACS No.: 36.10.Dr

scholarly journals Simulation of Marine Towing Cable Dynamics Using a Finite Elements Method

2020 ◽  
Vol 8 (2) ◽  
pp. 140 ◽  
Author(s):  
Álvaro Rodríguez Luis ◽  
José Antonio Armesto ◽  
Raúl Guanche ◽  
Carlos Barrera ◽  
César Vidal
Keyword(s):  
Experimental Data ◽  
Finite Element Method ◽  
The Body ◽  
The Other ◽  
Damping Coefficient ◽  
Internal Damping ◽  
Cable Dynamics ◽  
Proposed Model ◽  
Real Scale ◽  
Two Bodies

A numerical model to study the towing maneuver for floating and submerged bodies has been developed. The proposed model is based on the dynamic study of a catenary line moving between two bodies, one body with imposed motion, and the other free to move. The model improves previous models used to study the behavior of mooring systems based on a finite element method by reducing the noise of the numerical results considering the Rayleigh springs model for the tension of the line. The code was successfully validated using experimental results for experimental data from different authors and experiments found in the literature. Sensitivity analysis on the internal damping coefficient and the number of elements has been included in the present work, showing the importance of the internal damping coefficient. As an example of the application of the developed tool, simulations of towing systems on a real scale were analyzed for different setups. The variation of the loads at the towed body and the position of the body were analyzed for the studied configurations. The reasonable results allow us to say that the proposed model is a useful tool with several applications to towing system design, study or optimization.

II. Early history of the Moon - Dynamical capture of the Moon by the Earth

1967 ◽  
Vol 296 (1446) ◽  
pp. 285-292 ◽  
Keyword(s):  
Orbital Motion ◽  
The Body ◽  
Rotational Instability ◽  
The Moon ◽  
Body System ◽  
The Earth ◽  
Geometrical Features ◽  
Planetary Orbits ◽  
History Of ◽  
Three Body

The process of extracting the Moon from the Earth through some mechanism of rotational instability, and one that can also set it into orbital motion round the Earth, has nowadays come to be widely recognized as almost certainly dynamically impossible. Accordingly ideas have turned towards the notion that the Moon originated as a separate planet and was later captured by the Earth. It is reasonable to conjecture beforehand that this could happen in a three-body system consisting of the Sun, Earth, and Moon, but nevertheless it is of interest and importance to establish that such a capture is possible within the laws of dynamics, and moreover we should like to have some numerical indications of the initial dimensions that the lunar orbit would have on the basis of such an origin. Dissipative action may well be operative upon the planetary orbits to a minute extent, and there may have been eras in the history of the solar system when such dissipation was greater than average, but it seems certain that the main geometrical features of the process of capture of a satellite must in the final stages be governed purely by dynamical forces arising only from the mutual attractions of the bodies. Thus the first stage towards demonstrating the possibility of capture will be to study motions under purely conservative dynamical forces. Considerations of a phase-space or ergodic nature suggest that if the moon were captured in such a way it would eventually escape again, but we cannot on such a basis form any notion of the period of time for which the body might remain a satellite before escaping again. Actual numerical instances are needed to determine this.

Analomical description of the foot of a chinese female

1833 ◽  
Vol 2 ◽  
pp. 373-373
Keyword(s):  
Club Foot ◽  
The Body ◽  
The Other ◽  
Natural Growth ◽  
Dead Body ◽  
Muscular Power ◽  
Great Toe ◽  
External Edge ◽  
Straight Line ◽  
Sole Of The Foot

The foot, of which an account is here given, was obtained from the dead body of a female found floating in the river at Canton, and had all the characters of deformity, consequent upon the prevailing practice of early bandaging, for the purpose of checking its natural growth. To an unpractised eye, it has more the appearance of a congenital malformation, than of being the effect of art, however long continued; and appears at first sight like a club-foot, or an unre­duced dislocation. From the heel to the great toe, the length of the foot measures only five inches; the great toe is bent abruptly back­ wards, and its extremity points directly upwards, while the phalanges of the other toes are doubled in beneath the sole of the foot, leaving scarcely any breadth across the foot, where it is naturally broadest. The heel, instead of projecting backwards, descends in a straight line from the bones of the leg, and imparts a singular appearance to the foot, as if it were kept in a state of permanent extension. From the doubling in of the toes into the sole of the foot, the external edge of the foot is formed in a great measure by the extremities of the meta­tarsal bones, and a deep cleft or hollow appears in the sole of the foot, across its whole breadth. The author gives a minute anatomical description of all these parts, pointing out the deviations from the natural conformation. He remarks that from the diminutive size of the foot, the height of the instep, the deficiency of breadth, and the density of the cellular texture of the foot, all attempts to walk with so deformed a foot, must be extremely awkward; and that in order to preserve an equilibrium in an erect position, the body must neces­sarily be bent forwards with a painful effort, and with a very consi­derable exertion of muscular power.

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