Secondly, being free from gimbal lock allows an all attitude operation. We shall recall and explain this idea of cayley and its developments by klein. Thermodynamic properties of quantum models based on the. The cayleyklein parameters may be viewed as parameters of a matrix denoted q for its close relationship with quaternions which characterizes the transformations 14 15. In fact, rotation may be described as the motion of a point on a 3d nonlinear manifold, the lie group of special orthogonal transformations ofthe 3d space. Also pointed out is the link between the cayleyklein parameters and the weinorman ordering functions, exploited in. Cayley klein parameters can be used for the description of rigid body with one point fixed. A set of four complex numbers used to specify the orientation of a body, or equivalently, the rotation r which produces that orientation, starting from some reference orientation.
Rigid body motion fundamental theorem on rigid body motion, inertia tensor, euler equations, euler angles, cayley klein parameters su2, dynamics, rotating coordinate systems. The construction was developed in further detail by felix klein in papers in 1871 and 1873, and subsequent books and papers. One or more of the following statements may affect this document. The parameters,, and which, like the three euler angles, provide a way to uniquely characterize the orientation of a solid body. Analytical mechanics of aerospace systems hanspeter schaub and john l. The cayleyklein parameters are a set of parameters for representing rotations, either classical or spinor. Of these representations, homogenous transformations based on 4x4 real matrices orthonormal matrices have been used most often in robot. Cayleyklein parameters are an alternative to euler parameters for describing the spherical motion group.
It is this isomorphism that underlies the equivalent utility of the cayleyklein parameters or the er parameters in the classical description of rotations. The paper focuses on the use of cayley klein ck parameters in the description of rigidbody kinematics. Any proper motion of the euclidean space decomposes to. To these we add a fourth, the rotation vector, which has many of the benefits of both euler angles and quaternions, but neither the singularities of the former, nor the quadratic. In this paper the time behavior of quantum states ruled by hamiltonians linear in the su2, su1,1, and su3 generators in terms of the cayleyklein parameters, originally introduced in classical mechanics is analyzed. Cayleyklein parameters can be used for the description of rigid body with one point fixed. Klein tools has provided the correct stub height on each bender head. The general commutative rules of multiplication for these matrices is obtained. Caytoniales article about caytoniales by the free dictionary. The ideal situation is the fully automatic algorithm, which requires no interaction. Thirdly, because of the absence of mechanical platform gimbals, the system 1. Although physical motions with a fixed point are an important case such as ones described in the centerofmass frame, or motions of a joint, this approach creates a knowledge about all motions. The fundamental idea of angular velocity is straight from the lie algebra theory.
Applications of poisson geometry to physical problems 227 examples. These isotropic vectors z are seen also via pauli matrices as 2 x 2 traceless 2nilpotent matrices z z and natu. Caytoniales an order of fossil gymnospermous plants, which were widespread between the. General disclaimer one or more of the following statements. We can show this by parametrizing su2 via the cayleyklein parameters, i. Assume initial conditions so that body axes are the same as the external axes at t 0. As an example, to bend 34 emt conduit have a free end height of 8. Cory abstractnuclear magnetic resonance nmr spectroscopy has proven to be a facile means of achieving smallscale demonstrations of quantum information processing. Nawratil institute of discrete mathematics and geometry, vienna university of technology, austria, email. By multiplying both sides by the complex unit i and by introducing the cayley klein ck parameters.
New formulas for first and second time derivatives of 2. They were discovered in the nineteenth century before the advent of quantum. Cayleyklein parameters, pauli spin matrices, axis and angle, orthonormal matrices, and hamilton s quaternions. The schematic representation of forward and inverse kinematics. On the conduit, measure from the free end to be bent up the calculated number and mark the conduit.
We present the three main mathematical constructs used to represent the attitude of a rigid body in threedimensional space. Thanks for contributing an answer to physics stack exchange. Purchased from american institute of aeronautics and astronautics. Cayleyklein parameters and evolution of two and threelevel systems and squeezed states article pdf available in journal of mathematical physics 311 january 1990 with 396 reads. The numbers y1, y2, y3, y4 are known as cayleyklein parameters. As time progresses, the orientation of the body changes a at time dependence is obtained by solving lagranges eqtns. The titles below are free to download from a university computer. We use the morrisshore transformation to express the propagator in terms of morrisshore basis states and cayleyklein parameters, which allows us to connect properties of laser pulses to hilbertspace motion.
Encoding 5 5 a forest of trees 7 1 introduction in this paper, i will outline the basics of graph theory in an attempt to explore cayleys formula. Because a general rotation can be completely specified by three free parameters e. Purchased from american institute of aeronautics and. Oct 25, 2007 in this paper we complete the study of the evolution of multilevel systems in terms of a geometrical picture based on the socalled cayley klein parameters introduced to describe rotations in classical mechanics. The cayley klein parameters are a set of parameters for representing rotations, either classical or spinor. They were discovered in the nineteenth century before the advent of quantum mechanics, and were originally intended for use in classical problems of rigid body motion. On the hydrodynamical representation of nonrelativistic. Cayley klein parameters the group is parameterized by two. It is shown that these parameters and their identification in terms of euler angles are helpful and insightful tools to clarify the dynamical features of two and threelevel atoms. In the following, however, we shall stick to the familiar notions of vectors and. Euler angles, cayley klein parameters su2, dynamics, rotating coordinate systems. We solve hamiltonian for each of models and find the eigenvalues for them by nu and susy qm methods, in terms of the contraction parameter the thermodynamic properties for two systems are discussed and the results are reported. Kotsyuba soviet applied mechanics volume 7, pages 734 739 1971cite this article. Mar 17, 2017 the numbers y1, y2, y3, y4 are known as cayley klein parameters.
Warranty department 349 n vinewood street escondido ca 92029. These independent transformations to unitary cayleyklein groups maintain the quadratic form of invariance. Theory and applications article pdf available in mechanism and machine theory 106. Rotation formalisms are focused on proper orientationpreserving motions of the euclidean space with one fixed point, that a rotation refers to. Prove cayleys formula for proper orthonormal matrices. In this paper new matrices of the cayleyklein parameters are used to represent composition of several consecutive finite rotations of rigid body. This is the main theme of kleins two papers on the socalled noneuclidean geometry, i and ii 32 34 of klein. In this paper new matrices of the cayley klein parameters are used to represent composition of several consecutive finite rotations of rigid body. In this paper we complete the study of the evolution of multilevel systems in terms of a geometrical picture based on the socalled cayley klein parameters introduced to describe rotations in classical mechanics. Pdf cayleyklein parameters and evolution of two and three. Cayleyklein parameters, 119 classical rodrigues parameters, 104 euler angles, 81 euler parameters, 96 modified rodrigues parameters, 110 principal rotation vector, 90 directrix, 383 disturbance function, 461 disturbing potential function, 483 dualgimbal cmc, 353 duffing oscillator, 341 eccentric anomaly, 386 eccentricity parameter, 383. In mathematics, a cayleyklein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a crossratio. Cayley klein parameters, 119 classical rodrigues parameters, 104 euler angles, 81 euler parameters, 96 modified rodrigues parameters, 110 principal rotation vector, 90 directrix, 383 disturbance function, 461 disturbing potential function, 483 dualgimbal cmc, 353 duffing oscillator, 341 eccentric anomaly, 386 eccentricity parameter, 383. In this paper we complete the study of the evolution of multilevel systems in terms of a geometrical picture based on the socalled cayleyklein parameters introduced to describe rotations in classical mechanics.
The cayley klein parameters may be viewed as parameters of a matrix denoted q for its close relationship with quaternions which characterizes the transformations 14 15. In particular, it explains the dynamics of rotating, spinning and rolling rigid bodies from a. Forward and inverse kinematics serdar kucuk and zafer bingul 1. Caytoniales an order of fossil gymnospermous plants, which were widespread between the second half of the triassic period and the end of the cretaceous period. By multiplying both sides by the complex unit i and by introducing the cayleykleinck parameters. Applications of poisson geometry to physical problems. The cayleyklein parameters we now consider the matter of the cayleyklein parameters, which are discussed briefly by sakurai. The parameters,, and which, like the three euler angles, provide a way to uniquely characterize the orientation of a. Institute of discrete mathematics and geometry, vienna. Cayley klein parameters the group is parameterized by two complex numbers a and from phy 3063 at university of florida. They can be expressed in terms of the euler angles. Integrable hamiltonians of the unitary groups of rank one is studied.
Two main solution techniques for the inverse kinematics problem are analytical and numerical methods. Floquet solution for a spinning symmetric rigid body with. The leaves were petiolate and compound palmate, usually of four lanceolate leaflets with pinnate venation. In the su2 representation, a general rotation matrix q can be written. Cayley klein parameters the parameters,, and which, like the three euler angles, provide a way to uniquely characterize the orientation of a solid body.
This was in large part made possible by the sophisticated methods of quantum. These are 1 the rotation matrix, 2 a triple of euler angles, and 3 the unit quaternion. The four matrix elements appearing in this relation are the socalled cayley klein parameters. Pdf cayleyklein parameters and evolution of two and. The cayleyklein parameters and geometrical picture of the. The rotations form a lie group which underlies all of rigid body mechanics.
Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. The rotation angleaxis \\phi,\hat\bfn\ and the euler angle. Parallelmanipulatorsintermsof parameters cayleyklein dual. The transformation matrix is given in terms of the cayley klein parameters by 12 goldstein 1960, p. Chapter 2 studies rigid body motions, angular velocity, and the physical concepts of angular momentum and kinetic energy. Pdf dual cayleyklein parameters and mobius transform. Analytic solutions eulers equations of motion the spin of a rigid body is controlled by eulers. Kleins construction was motivated by an idea of cayley. The construction originated with arthur cayleys essay on the theory of distance where he calls the quadric the absolute. We show that the ensuing standard eigenvalue problem, solved numerically, can provide an arbitrarily accurate solution for all possible motion. Application of cayleyklein parameters in the investigation of the quasiprecessional movement of a free solid body. But avoid asking for help, clarification, or responding to other answers. Application of cayley klein parameters in the investigation of the quasiprecessional movement of a free solid body. This textbook introduces modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering.
Cayleyklein parameters 3 independent quantities are needed to specify a rigid body orientation. Graph theory and cayleys formula university of chicago. After a brief historical overview, an introduction to this concept is offered. The transformation matrix is given in terms of the cayleyklein parameters by 12 goldstein 1960, p. It is a general property of the matrices of the algebra a 2, that they can be represented either in terms of components or in terms of matrix elements. Furthermore, by using these matrices it is convenient to prove the theorem on finite. It is nice to have a physical copy, though, especially to do problems since they often require flipping through the book for information and formulas, and it is difficult to flip around in a book using a pdf copy.
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