Misplaced Pages

#607392

47-533: Kozai may refer to: Kozai-Lidov mechanism , a dynamical phenomenon in celestial mechanics 3040 Kozai , an asteroid Persons with the surname Kozai [ edit ] Yoshihide Kozai (1928–2018), Japanese astronomer Kazuteru Kozai (born 1986), Japanese badminton player Kaori Kozai (born 1963), Japanese singer Takeshi Kozai (1974–2006), Japanese judoka See also [ edit ] Kosai (disambiguation) Kōzai Station Topics referred to by

94-478: A mechanical system forms the cotangent bundle T ∗ Q {\displaystyle T^{*}Q} of the configuration manifold Q {\displaystyle Q} . This larger manifold is called the phase space of the system. In quantum mechanics , configuration space can be used (see for example the Mott problem ), but the classical mechanics extension to phase space cannot. Instead,

141-448: A rather different set of formalisms and notation are used in the analogous concept called quantum state space . The analog of a "point particle" becomes a single point in C P 1 {\displaystyle \mathbb {C} \mathbf {P} ^{1}} , the complex projective line , also known as the Bloch sphere . It is complex, because a quantum-mechanical wave function has

188-455: A robot arm to obtain a particular end-effector location, and it is even possible to have the robot arm move while keeping the end effector stationary. Thus, a complete description of the arm, suitable for use in kinematics, requires the specification of all of the joint positions and angles, and not just some of them. The joint parameters of the robot are used as generalized coordinates to define configurations. The set of joint parameter values

235-424: A specific manifold . For example, if the particle is attached to a rigid linkage, free to swing about the origin, it is effectively constrained to lie on a sphere. Its configuration space is the subset of coordinates in R 3 {\displaystyle \mathbb {R} ^{3}} that define points on the sphere S 2 {\displaystyle S^{2}} . In this case, one says that

282-402: A subspace of the n {\displaystyle n} -rigid-body configuration space. Note, however, that in robotics, the term configuration space can also refer to a further-reduced subset: the set of reachable positions by a robot's end-effector . This definition, however, leads to complexities described by the holonomy : that is, there may be several different ways of arranging

329-414: A system composed of three bodies system acting under their mutual gravitational attraction is complex. In general, the behaviour of a three-body system over long periods of time is enormously sensitive to any slight changes in the initial conditions , including even small uncertainties in determining the initial conditions, and rounding-errors in computer floating point arithmetic. The practical consequence

376-468: A system refers to the position of all constituent point particles of the system. The configuration space is insufficient to completely describe a mechanical system: it fails to take into account velocities. The set of velocities available to a system defines a plane tangent to the configuration manifold of the system. At a point q ∈ Q {\displaystyle q\in Q} , that tangent plane

423-418: Is a one-parameter family of orbital solutions having the same L z but different amounts of variation in e or i . Remarkably, the degree of possible variation in i is independent of the masses involved, which only set the timescale of the oscillations. The basic timescale associated with Kozai oscillations is where a indicates the semimajor axis, P is orbital period, e is eccentricity and m

470-403: Is called the joint space . A robot's forward and inverse kinematics equations define maps between configurations and end-effector positions, or between joint space and configuration space. Robot motion planning uses this mapping to find a path in joint space that provides an achievable route in the configuration space of the end-effector. In classical mechanics , the configuration of

517-618: Is conventional to use the symbol q {\displaystyle q} for a point in configuration space; this is the convention in both the Hamiltonian formulation of classical mechanics , and in Lagrangian mechanics . The symbol p {\displaystyle p} is used to denote momenta; the symbol q ˙ = d q / d t {\displaystyle {\dot {q}}=dq/dt} refers to velocities. A particle might be constrained to move on

SECTION 10

#1732848363608

564-418: Is denoted by T q Q {\displaystyle T_{q}Q} . Momentum vectors are linear functionals of the tangent plane, known as cotangent vectors; for a point q ∈ Q {\displaystyle q\in Q} , that cotangent plane is denoted by T q ∗ Q {\displaystyle T_{q}^{*}Q} . The set of positions and momenta of

611-486: Is described using generalized coordinates ; thus, three of the coordinates might describe the location of the center of mass of the rigid body, while three more might be the Euler angles describing its orientation. There is no canonical choice of coordinates; one could also choose some tip or endpoint of the rigid body, instead of its center of mass; one might choose to use quaternions instead of Euler angles, and so on. However,

658-435: Is different from Wikidata All article disambiguation pages All disambiguation pages Kozai-Lidov mechanism The effect has been found to be an important factor shaping the orbits of irregular satellites of the planets, trans-Neptunian objects , extrasolar planets , and multiple star systems . It hypothetically promotes black hole mergers . It was described in 1961 by Mikhail Lidov while analyzing

705-477: Is farthest from the equatorial plane. This effect is part of the reason that Pluto is dynamically protected from close encounters with Neptune . The Lidov–Kozai mechanism places restrictions on the orbits possible within a system. For example: The mechanism has been invoked in searches for Planet Nine , a hypothetical planet orbiting the Sun far beyond the orbit of Neptune. A number of moons have been found to be in

752-467: Is found already at the lowest, quadrupole order in the perturbative expansion. The octupole term becomes dominant in certain regimes and is responsible for a long-term variation in the amplitude of the Lidov–Kozai oscillations. The Lidov–Kozai mechanism is a secular effect, that is, it occurs on timescales much longer compared to the orbital periods of the inner and the outer binary. In order to simplify

799-589: Is mass; variables with subscript "2" refer to the outer (perturber) orbit and variables lacking subscripts refer to the inner orbit; M is the mass of the primary. For example, with Moon 's period of 27.3 days, eccentricity 0.055 and the Global Positioning System satellites period of half a (sidereal) day, the Kozai timescale is a little over 4 years; for geostationary orbits it is twice shorter. The period of oscillation of all three variables ( e , i , ω –

846-421: Is mostly used, in which the diagonals, representing "colliding" particles, are removed. The position of a single particle moving in ordinary Euclidean 3-space is defined by the vector q = ( x , y , z ) {\displaystyle q=(x,y,z)} , and therefore its configuration space is Q = R 3 {\displaystyle Q=\mathbb {R} ^{3}} . It

893-460: Is said to have six degrees of freedom . In this case, the configuration space Q = R 3 × S O ( 3 ) {\displaystyle Q=\mathbb {R} ^{3}\times \mathrm {SO} (3)} is six-dimensional, and a point q ∈ Q {\displaystyle q\in Q} is just a point in that space. The "location" of q {\displaystyle q} in that configuration space

940-426: Is that, the three-body problem cannot be solved analytically for an indefinite amount of time, except in special cases. Instead, numerical methods are used for forecast-times limited by the available precision. The Lidov–Kozai mechanism is a feature of hierarchical triple systems, that is systems in which one of the bodies, called the "perturber", is located far from the other two, which are said to comprise

987-408: Is then expanded in the orders of parameter α {\displaystyle \alpha } , defined as the ratio of the semi-major axes of the inner and the outer binary and hence small in a hierarchical system. Since the perturbative series converges rapidly, the qualitative behaviour of a hierarchical three-body system is determined by the initial terms in the expansion, referred to as

SECTION 20

#1732848363608

1034-401: The N bodies in the system ( N = 3 {\displaystyle N=3} for the von Zeipel-Kozai–Lidov effect). The number of ( x k , p k ) {\displaystyle (x_{k},p_{k})} pairs required to describe a given system is the number of its degrees of freedom . The coordinate pairs are usually chosen in such a way as to simplify

1081-628: The far side of the Moon . It burned in the Earth's atmosphere after completing eleven revolutions. However, according to Gkolias et al. . (2016) a different mechanism must have driven the decay of the probe's orbit since the Lidov–Kozai oscillations would have been thwarted by effects of the Earth's oblateness . The von Zeipel-Lidov–Kozai mechanism, in combination with tidal friction , is able to produce Hot Jupiters , which are gas giant exoplanets orbiting their stars on tight orbits. The high eccentricity of

1128-460: The inner binary . The perturber and the centre of mass of the inner binary comprise the outer binary . Such systems are often studied by using the methods of perturbation theory to write the Hamiltonian of a hierarchical three-body system as a sum of two terms responsible for the isolated evolution of the inner and the outer binary, and a third term coupling the two orbits, The coupling term

1175-409: The quadrupole ( ∝ α 2 {\displaystyle \propto \alpha ^{2}} ), octupole ( ∝ α 3 {\displaystyle \propto \alpha ^{3}} ) and hexadecapole ( ∝ α 4 {\displaystyle \propto \alpha ^{4}} ) order terms, For many systems, a satisfactory description

1222-481: The secondary , is a test particle – an idealized point-like object with negligible mass compared to the other two bodies, the primary and the distant perturber. These assumptions are valid, for instance, in the case of an artificial satellite in a low Earth orbit that is perturbed by the Moon , or a short-period comet that is perturbed by Jupiter . Under these approximations, the orbit-averaged equations of motion for

1269-419: The semimajor axis constant reduces the distance between the objects at periapsis , this mechanism can cause comets (perturbed by Jupiter ) to become sungrazing . Lidov–Kozai oscillations will be present if L z is lower than a certain value. At the critical value of L z , a "fixed-point" orbit appears, with constant inclination given by For values of L z less than this critical value, there

1316-411: The tangent space T Q {\displaystyle TQ} corresponds to the velocities of the points q ∈ Q {\displaystyle q\in Q} , while the cotangent space T ∗ Q {\displaystyle T^{*}Q} corresponds to momenta. (Velocities and momenta can be connected; for the most general, abstract case, this is done with

1363-455: The Kozai–Lidov or just the Kozai mechanism. Configuration space (physics) In classical mechanics , the parameters that define the configuration of a system are called generalized coordinates , and the space defined by these coordinates is called the configuration space of the physical system . It is often the case that these parameters satisfy mathematical constraints, such that

1410-476: The Lidov–Kozai resonance with their planet, including Jupiter's Carpo and Euporie , Saturn's Kiviuq and Ijiraq , Uranus's Margaret , and Neptune's Sao and Neso . Some sources identify the Soviet space probe Luna 3 as the first example of an artificial satellite undergoing Lidov–Kozai oscillations. Launched in 1959 into a highly inclined, eccentric, geocentric orbit, it was the first mission to photograph

1457-742: The Swedish astronomer Edvard Hugo von Zeipel had also studied this mechanism in 1909, and his name is sometimes now added. In Hamiltonian mechanics, a physical system is specified by a function, called Hamiltonian and denoted H {\displaystyle {\mathcal {H}}} , of canonical coordinates in phase space . The canonical coordinates consist of the generalized coordinates x k {\displaystyle x_{k}} in configuration space and their conjugate momenta p k {\displaystyle p_{k}} , for k = 1 , . . . N {\displaystyle k=1,...N} , for

Kozai - Misplaced Pages Continue

1504-415: The calculations involved in solving a particular problem. One set of canonical coordinates can be changed to another by a canonical transformation . The equations of motion for the system are obtained from the Hamiltonian through Hamilton's canonical equations , which relate time derivatives of the coordinates to partial derivatives of the Hamiltonian with respect to the conjugate momenta. The dynamics of

1551-649: The configuration space is not all of R 3 n {\displaystyle \mathbb {R} ^{3n}} , but the subspace (submanifold) of allowable positions that the points can take. The set of coordinates that define the position of a reference point and the orientation of a coordinate frame attached to a rigid body in three-dimensional space form its configuration space, often denoted R 3 × S O ( 3 ) {\displaystyle \mathbb {R} ^{3}\times \mathrm {SO} (3)} where R 3 {\displaystyle \mathbb {R} ^{3}} represents

1598-515: The coordinates of the origin of the frame attached to the body, and S O ( 3 ) {\displaystyle \mathrm {SO} (3)} represents the rotation matrices that define the orientation of this frame relative to a ground frame. A configuration of the rigid body is defined by six parameters, three from R 3 {\displaystyle \mathbb {R} ^{3}} and three from S O ( 3 ) {\displaystyle \mathrm {SO} (3)} , and

1645-402: The last being the argument of periapsis ) is the same, but depends on how "far" the orbit is from the fixed-point orbit, becoming very long for the separatrix orbit that separates librating orbits from oscillating orbits. The von Zeipel-Lidov–Kozai mechanism causes the argument of pericenter ( ω ) to librate about either 90° or 270°, which is to say that its periapse occurs when the body

1692-563: The manifold Q {\displaystyle Q} is the sphere, i.e. Q = S 2 {\displaystyle Q=S^{2}} . For n disconnected, non-interacting point particles, the configuration space is R 3 n {\displaystyle \mathbb {R} ^{3n}} . In general, however, one is interested in the case where the particles interact: for example, they are specific locations in some assembly of gears, pulleys, rolling balls, etc. often constrained to move without slipping. In this case,

1739-712: The motion of periodic comets in Astronomische Nachrichten . In 1961, the Soviet space scientist Mikhail Lidov discovered the effect while analyzing the orbits of artificial and natural satellites of planets. Originally published in Russian, the result was translated into English in 1962. Lidov first presented his work on artificial satellite orbits at the Conference on General and Applied Problems of Theoretical Astronomy held in Moscow on 20–25 November 1961. His paper

1786-419: The orbits of artificial and natural satellites of planets. In 1962, Yoshihide Kozai published this same result in application to the orbits of asteroids perturbed by Jupiter . The citations of the papers by Kozai and Lidov have risen sharply in the 21st century. As of 2017 , the mechanism is among the most studied astrophysical phenomena. It was pointed out in 2019 by Takashi Ito and Katsuhito Ohtsuka that

1833-426: The parameterization does not change the mechanical characteristics of the system; all of the different parameterizations ultimately describe the same (six-dimensional) manifold, the same set of possible positions and orientations. Some parameterizations are easier to work with than others, and many important statements can be made by working in a coordinate-free fashion. Examples of coordinate-free statements are that

1880-464: The planet HD 80606 b in the HD 80606/80607 system is likely due to the Kozai mechanism. The mechanism is thought to affect the growth of central black holes in dense star clusters . It also drives the evolution of certain classes of binary black holes and may play a role in enabling black hole mergers . The effect was first described in 1909 by the Swedish astronomer Hugo von Zeipel in his work on

1927-404: The problem and make it more tractable computationally, the hierarchical three-body Hamiltonian can be secularised , that is, averaged over the rapidly varying mean anomalies of the two orbits. Through this process, the problem is reduced to that of two interacting massive wire loops. The simplest treatment of the von Zeipel-Lidov–Kozai mechanism assumes that one of the inner binary's components,

Kozai - Misplaced Pages Continue

1974-431: The rather abstract notion of the tautological one-form .) For a robotic arm consisting of numerous rigid linkages, the configuration space consists of the location of each linkage (taken to be a rigid body, as in the section above), subject to the constraints of how the linkages are attached to each other, and their allowed range of motion. Thus, for n {\displaystyle n} linkages, one might consider

2021-406: The same term [REDACTED] This disambiguation page lists articles associated with the title Kozai . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Kozai&oldid=826364691 " Category : Disambiguation pages Hidden categories: Short description

2068-528: The secondary have a conserved quantity : the component of the secondary's orbital angular momentum parallel to the angular momentum of the primary / perturber orbit. This conserved quantity can be expressed in terms of the secondary's eccentricity e and inclination i relative to the plane of the outer binary: Conservation of L z means that orbital eccentricity can be "traded for" inclination. Thus, near-circular, highly inclined orbits can become very eccentric. Since increasing eccentricity while keeping

2115-399: The set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system. Notice that this is a notion of "unrestricted" configuration space, i.e. in which different point particles may occupy the same position. In mathematics, in particular in topology, a notion of "restricted" configuration space

2162-410: The total space [ R 3 × S O ( 3 ) ] n {\displaystyle \left[\mathbb {R} ^{3}\times \mathrm {SO} (3)\right]^{n}} except that all of the various attachments and constraints mean that not every point in this space is reachable. Thus, the configuration space Q {\displaystyle Q} is necessarily

2209-419: Was first published in a Russian-language journal in 1961. The Japanese astronomer Yoshihide Kozai was among the 1961 conference participants. Kozai published the same result in a widely read English-language journal in 1962, using the result to analyze orbits of asteroids perturbed by Jupiter . Since Lidov was the first to publish, many authors use the term Lidov–Kozai mechanism. Others, however, name it as

#607392