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In recent years, a wide range of generalized barycentric coordinates has been suggested. However, all of them lack control over derivatives. We show how the notion of barycentric coordinates can be extended to specify derivatives at control points. This is also known as Hermite interpolation. We introduce a method to modify existing barycentric coordinates to higher order barycentric coordinates and demonstrate, using higher order mean value coordinates, that our method, although conceptually simple and easy to implement, can be used to give easy and intuitive control at interactive frame rates over local space deformations such as rotations.

A generalized coordinates molecular Hamiltonian for polyatomic molecules has been obtained. Using this Hamiltonian in Radau coordinates as the orthogonal coordinates, the molecular Hamiltonian is derived for tri-atomic molecules. An exact derivation is presented of the molecular Hamiltonian in different axis embedding such as bond embedding, bisector embedding, and the linear combination of Radau vector coordinates. In addition, the symmetrized Radau coordinates are considered as an example for this general formulation. This new approach is greatly simplified. Also presented are simple and explicit expressions without cross-terms in Radau coordinates as vibrational variables.

In this paper, the dynamic simulation of constrained mechanical systems that are interconnected of rigid bodies is studied using projection recursive algorithm. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. Closed-chain system is transformed to open-chain by cutting suitable kinematical joints and introducing cut-joint constraints. For the resulting open-chain system, the equations of motion are generated recursively along the serial chains. An example is chosen to demonstrate the generality and simplicity of the developed formulation.

We propose a simple generalization of Shephard's interpolation to piecewise smooth, convex closed curves that yields a family of boundary interpolants with linear precision. Two instances of this family reduce to previously known interpolants: one based on a generalization of Wachspress coordinates to smooth curves and the other an integral version of mean value coordinates for smooth curves. A third instance of this family yields a previously unknown generalization of discrete harmonic coordinates to smooth curves. For closed, piecewise linear curves, we prove that our interpolant reproduces a general family of barycentric coordinates considered by Floater, Hormann and Kos that includes Wachspress coordinates, mean value coordinates and discrete harmonic coordinates.

We propose a method for solving frictional contact problems which is based on including the generalized coordinates of absolutely rigid bodies in the degrees of freedom of the system under study and on varying the functional of the variational problem with respect to these coordinates. As a result, one can include the generalized coordinates or the energy-conjugate generalized forces directly in the right-hand side of the resolving system of equations, which permits easily taking into account any laws of motion or loading of absolutely rigid bodies.

The MIC-Kepler system is studied via the Milshtein–Strakhovenko variant of the so(2,1) Lie algebra. Green's function is constructed in spherical coordinates, with the help of the Kustaanheimo–Stiefel variables and the generators of the SO(2,1) group. The energy spectrum and the normalized wavefunctions of the bound states are obtained.

The paper presents a method of modeling dynamics of multibody systems with open and closed kinematic chains. The joint coordinates and homogeneous transformations are applied in order to formulate the equations of motion of a rigid body. In this method, constraint equations are introduced only in the case when closed subchains are considered or when the joint reactions have to be calculated. This allows the number of generalized coordinates in the system to be reduced in comparison to the case when absolute coordinates are applied. It is shown how the method can be applied to modeling of vehicle dynamics. The calculation results are compared with those obtained when the ADAMS/Car package is used. Experimental verification has been performed and is reported in the paper, as well. In both cases, a good correspondence of results has been achieved. Final remarks concerning advantages and disadvantages of the method are formulated at the end of the paper.

A new exact quantum mechanical rovibrational Hamiltonian operator for molecules exhibiting large amplitude inversion and torsion motions is derived. The derivation is based on a division of a molecule into two parts: a frame and a top. The nuclei of the frame only are used to construct a molecular system of axes. The inversion motion of the frame is described in the umbrella-like coordinates, whereas the torsion motion of the top is described by the nonstandard torsion angle defined in terms of the nuclear vectors and one of the molecular axes. The internal coordinates chosen take into account the properties of the inversion and torsion motions. Vibrational s and rotational Omega vectors obtained for the introduced internal coordinates determine the rovibrational tensor G defined by simple scalar products of these vectors. The Jacobian of the transformation from the Cartesian to the internal coordinates considered and the G tensor specify the rovibrational Hamiltonian. As a result, the Hamiltonian for penta-atomic molecules like NH2OH with one inverter is presented and a complete set of the formulas necessary to write down the Hamiltonian of more complex molecules, like NH2NH2 with two inverters, is reported. The approach considered is essentially general and sufficiently simple, as demonstrated by derivation of a polyatomic molecule Hamiltonian in polyspherical coordinates, obtained by other methods with much greater efforts.

Dynamic ray tracing plays an important role in paraxial ray methods. In this paper, dynamic ray tracing systems for inhomogeneous anisotropic media, consisting of four linear ordinary differential equations of the first order along the reference ray, are studied. The main attention is devoted to systems expressed in a particularly simple choice of ray-centered coordinates, here referred to as the standard ray-centered coordinates, and in wavefront orthonormal coordinates. These two systems, known from the literature, were derived independently and were given in different forms. In this paper it is proved that both systems are fully equivalent. Consequently, the dynamic ray tracing system, consisting of four equations in wavefront orthonormal coordinates, can also be used if we work in ray-centered coordinates, and vice versa.

The isotropic coordinate system of Schwarzschild spacetime has several attractive properties similar with the Painleve–Gullstrand coordinates. The purpose for us to choose the isotropic coordinates is to resolve the ambiguities of the tunneling picture in Hawking radiation. Based on energy conservation, we investigate Hawking radiation as massless particles tunneling across the event horizon of the Schwarzschild black hole in the isotropic coordinates. Because the amplitude for a black hole to emit particles is related to the amplitude for it to absorb, we must take into account the contribution of ingoing solution to the action, Im S = Im S out − Im S in . It will be shown that the imaginary part of action for ingoing particles is zero ( Im S in = 0 ) in the Painleve–Gullstrand coordinates, so the equation Im S = Im S out − Im S in is valid in both the isotropic coordinates and the Painleve–Gullstrand coordinates.

In the early eighties, the author and co-workers created and further developed the natural coordinates to describe the motion of 2-D and 3-D multibody systems. Natural coordinates do not need angles or angular parameters to define orientation, leading to constant inertia matrices and to the simplest form of the constraint equations. Natural coordinates are composed by the Cartesian coordinates of some points and the Cartesian components of some unit vectors distributed on the different bodies of the system. The points and vectors can be located in the joints, being shared by contiguous bodies, decreasing or even eliminating the need to set joint constraints and reducing the total number of variables. However, other authors prefer not to share variables in order to get even simpler equations and to keep a bigger decoupling of equations, which is preferable in some cases. In this paper the history of natural coordinates is reviewed, as well as the main contributions coming from other research groups. In the second part of the paper some application areas in which natural coordinates can be particularly advantageous are examined.

This paper discusses the relationship between natural coordinates in fluid mechanics and orthogonal curvilinear coordinates. Since orthogonal curvilinear coordinates have some excellent mathematical properties, natural coordinates can be applied more widely if they can be transformed to orthogonal curvilinear coordinates. Frenet formulas which describe the differential properties of natural coordinates were compared with the derivative formulas of orthogonal curvilinear coordinates to show that natural coordinates are not generally orthogonal curvilinear coordinates. A method was introduced to transform natural coordinates into orthogonal curvilinear coordinates by rotating the normal planes of the natural coordinates about the streamlines. The transformation is true as long as the natural coordinates satisfy several equations. Vorticity decomposition in the natural coordinates is used to show that these conditional equations are satisfied only if the streamlines are perpendicular to the vortexlines on every given point in the flow field. These equations apply in both planar flows and axisymmetric flows without a circumferential velocity component, but do not apply in some 3-D flows such as Beltrami flow.

In this paper we present and analyze the most essential aspects of reduced masses along generalized internal coordinates. The definition of reduced masses in the internal coordinate formalism is established through the Wilson G-matrix concept and includes sophisticated relations between internal and Cartesian coordinates. Moreover, reduced masses in internal coordinates are, in general, no longer constant but coordinate-dependent. Based on the approach presented earlier [Stare, J.; Balint-Kurti, G. G. J. Phys. Chem. A 2003, 107, 7204-7214] and on our experience with reduced masses discussed in this paper, we have developed a robust program for the calculation of Wilson G-matrix elements and their functional coordinate dependence. The approach is based on the first principles and can be used in virtually any (internal) coordinate set. Since the program allows for projection of any kind of nuclear motion on the selected internal coordinates, the method is particularly suitable for ab initio or DFT potential energy functions calculated by partial geometry optimization. Moreover, reduced masses obtained by this program can be used as a decision tool for selecting the most appropriate internal coordinates for the considered vibrational problem and for the inclusion or omission of the kinetic coupling terms in the vibrational Hamiltonian.

A tractable method is presented for obtaining transformations to pseudo-Cartesian coordinates in locally flat pseudo-Riemannian spaces. The procedure is based on the properties of parallel covector fields. As an illustration, the method is applied to obtain certain transformations that arise in the Hamilton–Jacobi theory of separation of variables.

In [1–3], equations describing the deformation of an ideal rigidly plastic medium in curvilinear coordinates are given. In [4], it is shown that, in the case of plane strain, these equations have simple solutions, and the stresses depend only on the metric coordinates.

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.

Some results on existence of global Chebyshev coordinates on a Riemannian manifold or, more generally, on Aleksandrov surface are proved. For instance, if the positive and the negative parts of integral curvature of a Riemannian manifold M are less than 2\pi each, then there exist global Chebyshev coordinates on M. These conditions are optimal. Such coordinates help to get bi-Lipschitz maps between surfaces.}, Comment: This is a corrected version of our previous submisson

The finite element method is used to investigate steady-state vibrations of a curved beam with geometrical nonlinearity. The nonlinear strain-displacement relation is employed and the in-plane displacement in middle plane is included in the model. Equations of motion are derived by applying the principle of virtual work. A reduced-order model is derived by transforming the equations of motion from the physical coordinates to the modal coordinates on the pseudo-mode space. We can obtain only a few d.o.f. equations of motion in the modal coordinates, and present numerical results are compared with results obtained by applying a numerical integration directly to the equations of motion in the physical coordinates.

Triangular form of Newton equations is a strong property. Together with the existence of a single quadratic with respect to velocities integral of motion, it usally implies existence of further n - 1 integrals that are also quadratic. These integrals make the triangular system separable in new type of coordinates. The separation coordinates are built of quadric surfaces that are nonorthogonal and noconfocal and can intersect along lower dimensional singular manifolds. We present here separability theory for n-dimensional triangular systems and analyze the structure of separation coordinates in two and three dimensions. © 2007 by the Massachusetts Institute of Technology.

A method is presented to derive the state equations of the free vibration of piezoelectric materials under cylindrical coordinates in Hamiltonian system. Based on the 3D theory of piezoelectricity, the constitutive relations and the Lagrangian function of the generalized strain energy are rewritten in terms of the generalized displacements, i.e., displacements and electric potential and their derivatives. The Legendre transform is then applied to release the Hamiltonian function and the canonical equations are obtained through the variational operation. The canonical equations, i.e., the desired state equations, in cylindrical coordinates are obtained readily and could be used in the further analysis.

It is shown that the system of unit vectors corresponding to the internal coordinates is non-orthogonal generally. The deduction starts with the well-known orthonormality of unit vectors of the Cartesian coordinates. The crucial point of the GDIIS method is discussed regarding a “partially isomorphic” relationship between two vector spaces. Some features of the pseudoinverse of the Eliashevich–Wilsonian matrix B are deduced and discussed: these are analogous to the conditions formulated originally for the elements of the B-matrix.

Methods of construction of the composition function, left- and right-invariant vector fields and differential 1-forms of a Lie group from the structure constants of the associated Lie algebra are proposed. It is shown that in the second canonical coordinates these problems are reduced to the matrix inversions and matrix exponentiations, and the composition function can be represented in quadratures. Moreover, it is proven that the transition function from the first canonical coordinates to the second canonical coordinates can be found by quadratures.

Mathematical models based on nonlinear differential equations for dynamic systems of classical mechanics and biophysics are considered. The research concentrates on these equations integration features, their solutions properties, stability and behavior nearby an equilibrium point. Factors which can change the solution stability such as a form of a problem definition, a choice of the generalized coordinates and equations describing the process, the presence of small periodic or random perturbation are taken into account in the research.

We study the space of periodic solutions of the elliptic sinh -Gordon equation by means of spectral data consisting of a Riemann surface Y and a divisor D and prove the existence of certain Darboux coordinates.

We will give an overview on the use of Fatou coordinates in the study of (semi-)parabolic fixed points and their bifurcations.

Here we prove the existence of local action angle coordinates for a Liouville integrable Hamiltonian system near a compact connected fiber of its integral mapping.

Integration of Hamiltonian systems by reduction to action-angle variables has proven to be a successful approach. However, when the solution depends on elliptic functions the transformation to action-angle variables may need to remain in implicit form. This is exactly the case of the simple pendulum, where in order to make explicit the transformation to action-angle variables one needs to resort to nontrivial expansions of special functions and series reversion. Alternatively, it is shown that the explicit expansion of the transformation to action-angle variables can be constructed directly, and that this direct construction leads naturally to the Lie transforms method, in this way avoiding the intricacies related to the traditional expansion of elliptic functions., Comment: 19 pages, 3 figures

International audience; In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (regular NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which always exist in the neighborhood of a compact connected component of the regular fibers of the momentum map, global action-angle variables rarely exist. The fact that there are obstructions to the existence of global action-angle variables was first observed and analyzed by Duistermaat in the case of Liouville integrable systems on symplectic manifolds and later by Dazord-Delzant in the case of non-commutative integrable systems on symplectic manifolds. In our more general case where phase space is an arbitrary Poisson manifold, there are more obstructions. Our approach makes use of a few new features which we introduce: the action bundle and the action lattice bundle of the NCI system (these bundles are canonically defined) and three foliations (the action, angle and transverse foliation), whose existence is also subject to obstructions, often of a cohomological nature.

The dynamic equations of multibody railroad vehicle systems can be formulated using different sets of generalized coordinates; examples of these sets of coordinates are the absolute Cartesian and trajectory coordinates. The absolute coordinate based formulations do not require introducing an intermediate track coordinate system since all the absolute coordinates are defined in the global system. On the other hand, when the trajectory coordinates are used, a track coordinate system that follows the motion of a body in the railroad vehicle system is introduced. This track coordinate system is defined by the track geometry and the distance traveled by the body along the track centerline. The configuration of the body with respect to the track coordinate system is defined using five coordinates; two translations and three Euler angles. In this paper, the formulations based on the absolute and trajectory coordinates are compared. It is shown that these two sets of coordinates require different degrees of differentiability and smoothness. When an elastic contact formulation is used to study the wheel/rail dynamic interaction, there are significant differences in the order of the derivatives required in both formulations. In fact, as demonstrated in this study, in the absence of a contact constraint formulation, higher order derivatives with respect to geometric parameters are still required when the equations are formulated using the trajectory coordinates. The formulation of the constraints used in the analysis of the wheel/rail contact is discussed and it is shown that when the absolute coordinates are used, only third order derivatives need to be evaluated. The relationship between the track frame used in railroad vehicle dynamics and the Frenet frame used in the theory of curves to describe the curve geometry is also discussed in this paper. Based on the analysis presented in this paper, the advantages and drawbacks of a hybrid method which employs both the absolute and trajectory coordinates and planar contact conditions in order to reduce the number of contact constraints and relax the differentiability requirements are discussed. In this method, the absolute coordinates are used to formulate the equations of motion of the railroad vehicle system. The absolute coordinate solution can be used to determine the trajectory coordinates and their time derivatives. Using the trajectory coordinates, the motion of the body in the vehicle with respect to the track coordinate system can be predicted and used in the formulation of the planar contact model.

In this correspondence, our aim is to establish a connection between low-rank detection, low-rank estimation, and canonical coordinates. The key to this connection is the observation that Gauss-Gauss detectors and estimators share canonical coordinates in the case where the underlying model is a signal-plus-noise model. We show that in Gauss-Gauss detection J-divergence is a function of squared canonical correlations, and hence is invariant to nonsingular transformations of the data channels. Further, we show that J-divergence has a special decomposition in canonical coordinates, impelling their use for rank-reduction. Canonical coordinates have been found earlier to be fundamental for low-rank estimation, as they decompose three important performance measures, namely the relative volume of error concentration ellipse, processing gain, and information rate. This correspondence shows that canonical coordinates are also fundamental for low-rank detection, making them more useful in signal processing and communication problems when low-rank modeling is required to achieve computational efficiency or robustness against noise and model uncertainties

The new approach to the analysis of four-body systems and computation of various four-body integrals is proposed. The approach is based on the use of six perimetric coordinates which can be introduced for an arbitrary four-body system. The proper (i.e. non-conflicting) definition of the four-body perimetric coordinates is given for an arbitrary four-body system. It is shown that these six internal perimetric coordinates describe all possible configurations in an arbitrary four-body system and can be used to simplify computations of many four-body integrals written in the relative coordinates r12, r13, r23, r14, r24 and r34. In addition to this, a number of new, very effective procedures for variational computation of different four-body systems can now be developed.

We present an automated formulation that generates multibody kinematic and dynamic equations in terms of indirect coordinates, a general set of motion variables that can be used to represent the relative motion of any two bodies in a system. This new formulation is achieved by combining classical mechanics with linear graph theory and the concept of a “virtual joint” between any two bodies. By selecting this joint into the spanning tree of a graph-theoretic representation of the system, the governing equations are systematically generated in terms of the indirect coordinates corresponding to the virtual joint. Different trees may be selected for translational and rotational motions, which can result in fewer equations to be solved than those obtained using joint coordinates. As shown in the examples, the use of indirect coordinates can also lead to a reduction in equation complexity and an increase in the computational efficiency of numerical simulations.

Based on the three-order Lagrangian equations, Hamilton's function of acceleration H* and generalized acceleration momentum Pα* are defined, and pseudo-Hamilton canonical equations corresponding to three-order Lagrangian equations are obtained. The equations are similar to Hamilton's canonical equations of analytical mechanics in form.

A general method for solving the problem of quantization of a top is proposed that allows for finding the Hamiltonian eigenfunctions in the form of polynomials of various degree n in the Cartesian coordinates with the use of the Lame functions. All three coordinates x, y, and z are equivalently involved in computations, as well as their relations with ellipsoidal coordinates, which makes the computations symmetric.

Whereas the ray-centred coordinates for isotropic media by Popov and Psencik are uniquely defined by the selection of the basis vectors at one point along the ray, there is considerable freedom in selecting the ray-centred coordinates for anisotropic media. We describe the properties common to all ray-centred coordinate systems for anisotropic media and general conditions, which may be imposed on the basis vectors. We then discuss six different particular choices of ray-centred coordinates in an anisotropic medium. This overview may be useful in choosing the ray-centred coordinates best suited for a particular application. The equations are derived for a general homogeneous Hamiltonian of an arbitrary degree and are thus applicable both to the anisotropic-ray-theory rays and anisotropic common S-wave rays.

It is shown that a new set of internal coordinates called the equivalent delocalized internal coordinates can be defined through a linear combination of conventional equivalent internal coordinates. The equivalent internal coordinates are isolated in the delocalized internal coordinates by a Gram-Schmidt orthogonalization. Then, the new coordinates are obtained from the linear combinations of the delocalized internal coordinates. With this approach an unified scheme for the calculation of equivalent and constant delocalized internal coordinates is obtained. It is shown that the new set of coordinates permits a better description of the concerted movement of atoms thereby allowing a more efficient geometry optimization in complex systems. Examples for the application of equivalent and constant delocalized internal coordinates are given.

Barycentric coordinates are unique for triangles, but there are many possible generalizations to convex polygons. In this paper we derive sharp upper and lower bounds on all barycentric coordinates over convex polygons and use them to show that all such coordinates have the same continuous extension to the boundary. We then present a general approach for constructing such coordinates and use it to show that the Wachspress, mean value, and discrete harmonic coordinates all belong to a unifying one-parameter family of smooth three-point coordinates. We show that the only members of this family that are positive, and therefore barycentric, are the Wachspress and mean value ones. However, our general approach allows us to construct several sets of smooth five-point coordinates, which are positive and therefore barycentric.

We present a new scheme for the geometry optimization of equilibrium and transition state structures that can be used for both strong and weak coordinates. We use a screening function that depends on atom-pair distances to differentiate strong coordinates from weak coordinates. This differentiation significantly accelerates the optimization of these coordinates, and thus of the overall geometry. An adapted version of the delocalized coordinates setup is used to generate automatically a set of internal coordinates that is shown to perform well for the geometry optimization of systems with weak and strong coordinates. For the Baker test set of 30 molecules, we need only 173 geometry cycles with PW91/TZ2P calculations, which compares well with the best previous attempts reported in literature. For the localization of transition state structures, we generate the initial Hessian matrix, using appropriate force constants from a database. In this way, one avoids the explicit computation of the Hessian matrix. © 2006 Wiley Periodicals, Inc.