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In this letter, we study the purely nonlinear oscillator by the method of action-angle variables of Hamiltonian systems. The frequency of the non-isochronous system is obtained, which agrees well with the previously known result. Exact analytic solutions of the system involving generalized trigonometric functions are presented. We also present arguments to show the adiabatic invariance of the action variable for a time-dependent purely nonlinear oscillator., Comment: 17 pages, 7 figures, to appear in IJNLM

The paper is concerned with coordinate representations for rigid parts in multibody dynamics. The discussion is based on the general theory of the dynamics of multibody systems under constraints. General configuration coordinates are introduced and the requirement of their regularity is discussed. The use of Euler angles, quaternions and linear coordinates, where quaternions and linear coordinates require constraint conditions, is analysed in detail. These coordinate systems are all shown to be regular. Equations of motion are formulated, using Lagrange’s as well as Euler’s equations, and they are supplemented by the appropriate constraint conditions in the cases of quaternions and linear coordinates. Mass matrices are derived, and in terms of the Euler angles the mass matrix components are products of trigonometric functions whereas in terms of quaternions the matrix components are quadratic polynomials. Using linear coordinates gives rise to a constant mass matrix. Thus, there is a decreasing degree of complexity, regarding mass matrix components, when going from Euler angles to linear coordinates. This is obtained at the expense of an increasing gross number of degrees of freedom and the necessary introduction of constraint conditions. The different equations of motion obtained are compared with respect to their structural complexity. In all representations the components of the angular velocity are explicitly calculated. This is not always the case in previous investigations of this subject. The paper also gives a new proof of the well-known relation between angular velocity and unit quaternions and their time derivative.

We show that, as distinct from completely integrable Hamiltonian systems, a commutative partially integrable system admits different compatible Poisson structures on a phase manifold that are related by a recursion operator. The existence of action–angle coordinates around an invariant submanifold of such a partially integrable system is proved.

Harmonic coordinates are widely considered to be perfect barycentric coordinates of a polygonal domain due to their attractive mathematical properties. Alas, they have no closed form in general, so must be numerically approximated by solving a large linear equation on a discretization of the domain. The alternatives are a number of other simpler schemes which have closed forms, many designed as a (computationally) cheap approximation to harmonic coordinates. One test of the quality of the approximation is whether the coordinates coincide with the harmonic coordinates for the special case where the polygon is close to a circle (where the harmonic coordinates have a closed form - the celebrated Poisson kernel). Coordinates which pass this test are called "pseudo-harmonic". Another test is how small the differences between the coordinates and the harmonic coordinates are for "real-world" polygons using some natural distance measures.We provide a qualitative and quantitative comparison of a number of popular barycentric coordinate methods. In particular, we study how good an approximation they are to harmonic coordinates. We pay special attention to the Moving-Least-Squares coordinates, provide a closed form for them and their transfinite counterpart (i.e. when the polygon converges to a smooth continuous curve), prove that they are pseudo-harmonic and demonstrate experimentally that they provide a superior approximation to harmonic coordinates. A qualitative and quantitative comparison of popular barycentric coordinate methods.A closed form for MLS coordinates and their transfinite counterpart.Prove MLS coordinates are pseudo-harmonic.Demonstrate MLS coordinates provide a superior approximation to harmonic coordinates.

We consider a feedback control problem for a system of ordinary differential equations in the case when only a part of coordinates of the phase vector are measured and propose a solution algorithm that is stable to perturbations. The algorithm is based on a combination of the theories of dynamical inversion and guaranteed control. It consists of two blocks: a block for the dynamical reconstruction of unmeasured coordinates and a control block.

Barycentric coordinates are commonly used to represent a point inside a polygon as an affine combination of the polygon's vertices and to interpolate data given at these vertices. While unique for triangles, various generalizations to arbitrary simple polygons exist, each satisfying a different set of properties. Some of these generalized barycentric coordinates do not have a closed form and can only be approximated by piecewise linear functions. In this paper we show that subdivision can be used to refine these piecewise linear functions without losing the key barycentric properties. For a wide range of subdivision schemes, this generates a sequence of piecewise linear coordinates which converges to non-negative and C 1 continuous coordinates in the limit. The power of the described approach comes from the possibility of evaluating the C 1 limit coordinates and their derivatives directly. We support our theoretical results with several examples, where we use Loop or Catmull-Clark subdivision to generate C 1 coordinates, which inherit the favourable shape properties of harmonic coordinates or the small support of local barycentric coordinates.

Chaotic dynamics of the dual-spin spacecraft (DSSC) and gyrostats-satellites is analyzed at the presence of the constructional asymmetry and at the action of the internal/external perturbations, including the friction between the coaxial DSSC bodies, electromotors’ torques applied to the rotor-body from the platform-body by internal engines, the counterelectromotive forces/torques in internal engines, internal plyharmonic disturbances and external magnetic perturbations. The heteroclinic chaos suppression tasks are considered basing on the Melnikov–Wiggins formalism. Some alternative chaos suppressing techniques are described.

Two finite-dimensional Lie–Poisson Hamiltonian systems associated with Boussinesq equation are presented. The action-angle variables in the case of systems with non-hyperelliptic spectral curves are obtained by Sklyanin’s method of separation of variables. Moreover, with the help of Hamilton–Jacobi theory for the generating functions of conserved integrals, the Jacobi inversion problems related to the Lie–Poisson Hamiltonian systems and Boussinesq equation are discussed.

A systematical method is presented to derive the state space equations of anisotropic magneto-electro-elastic materials of orthogonal curvilinear coordinates in Hamiltonian system. Based on the three-dimensional theory of magneto-electro-elasticity, the constitutive relations and the Lagrangian function of the total potential are rewritten in terms of the generalized displacements, i.e., displacements, electric potential and magnetic potential, and their derivatives. The Legendre transform is then applied to release the Hamiltonian function and the canonical equations are obtained through variational operation. The canonical equations in curvilinear coordinates, i.e., the desired state equations, can be degenerated readily to the ones of spherical coordinates, cylindrical coordinates and rectangular coordinates. These equations are also deduced conveniently for piezoelectric and elastic materials since these derivation are rendered in terms of matrix. As an example of application, a magneto-electro-elastic cylindrical shell with four simply supported edges is studied when it is bended by sinusoidal load.

The KdV-equation $u_t = -u_{xxx} + 6uu_x$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms $||u||_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order. We further obtain quantitative estimates of the nonlinear Fourier transformed in arbitrary weighted Sobolev spaces., 37 pages

Identification of the collective coordinates that describe rare events in complex molecular transitions such as protein folding has been a key challenge in the theoretical molecular sciences. In the Diffusion Map approach, one assumes that the molecular configurations sampled have been generated by a diffusion process, and one uses the eigenfunctions of the corresponding diffusion operator as reaction coordinates. While diffusion coordinates (DCs) appear to provide a good approximation to the true dynamical reaction coordinates, they are not parametrized using dynamical information. Thus, their approximation quality could not, as yet, be validated, nor could the diffusion map eigenvalues be used to compute relaxation rate constants of the system. Here we combine the Diffusion Map approach with the recently proposed Variational Approach for Conformation Dynamics (VAC). Diffusion Map coordinates are used as a basis set, and their optimal linear combination is sought using the VAC, which employs time-correlation information on the molecular dynamics (MD) trajectories. We have applied this approach to ultra-long MD simulations of the Fip35 WW domain and found that the first DCs are indeed a good approximation to the true reaction coordinates of the system, but they could be further improved using the VAC. Using the Diffusion Map basis, excellent approximations to the relaxation rates of the system are obtained. Finally, we evaluate the quality of different metric spaces and find that pairwise minimal root-mean-square deviation performs poorly, while operating in the recently introduced kinetic maps based on the time-lagged independent component analysis gives the best performance.

Starting from action-angle variables and using a standard asymptotic expansion, we present an original and coincise derivation of the Wave Kinetic equation for a resonant process of the type 2 ↔ 2 . Despite not being more rigorous than others, our procedure has the merit of being straightforward; it allows for a direct control of the random phases and random action of the initial wave field. We show that the Wave Kinetic equation can be derived assuming only initial random phases. The random action approximation has to be taken only after the weak nonlinearity and large box limits are taken. The reason is that the oscillating terms in the evolution equation for the action contain, as an argument, the action-dependent nonlinear corrections which is dropped, using the large box limit. We also show that a discrete version of the Wave Kinetic Equation can be obtained for the Nonlinear Schrödinger equation; this is because the nonlinear frequency correction terms give a zero contribution and the large box limit is not needed. In our calculation we do not make an explicitly use of the Wick selection rule.

The paper addresses the problem of escape of harmonically forced classical particle from asymmetric potential well. Two benchmark models of the potential wells are used – truncated parabolic well with small cubic perturbation, and more common essentially nonlinear quadratic-cubic well. Transient escape dynamics in both models is analyzed in the framework of isolated resonance approximation. Despite substantial difference between both models, the observed escape scenarios are qualitatively similar. In each case, as parameters of the system are modified, the amplitude of the slow phase flow can gradually achieve the escape threshold; this simple mechanism is referred to as "maximum" scenario. The other, potentially more dangerous scenario, involves abrupt transition of the system response from relatively small amplitude to the escape. This pattern is related to passage of the slow-flow phase trajectory through dynamical saddle of the resonant manifold, and thus is referred to as "saddle" scenario. Both escape scenarios are easily observed in full-scale numeric simulations, in complete agreement with the theoretical predictions. The aforementioned resonance escape mechanisms are similar to those which were reviled in the case of symmetric potential well. Also, both mechanisms are generic enough to remain unchanged even in presence of small damping. These findings point on an unexpected universality of the resonance escape mechanisms. Description of the motion outside the well is not included in the current study.

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in [3], where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains., 52 pages, 4 figures. To appear on Annals of Mathematics

In Chap. 3 we showed that choosing the action-angle canonical variables in a one-dimensional Hamiltonian system dramatically simplifies the dynamics: the action remains constant and the angle increases linearly with time. With minor modifications, the same transformation can be applied to the Hamiltonian ( 6.14) that describes betatron oscillations in an accelerator. This yields an invariant of the motion and is also a useful starting point for analyzing more complicated dynamics.

Methods of Hamiltonian dynamics are applied to study the geodesic flow on the resolved conifolds over Sasaki-Einstein space $T^{1,1}$. We construct explicitly the constants of motion and prove complete integrability of geodesics in the five-dimensional Sasaki-Einstein space $T^{1,1}$ and its Calabi-Yau metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. Using the small resolution the geodesic motion on the resolved conifold remains completely integrable. Instead, in the case of the deformation of the conifold the complete integrability is lost., Comment: 12 pages

The phase integral and action-angle variables are the subject of the sixth chapter. The notions of phase integrals, frequency and angular variables are first recalled. Seven exercises are then solved, on a fourth harmonic oscillator, on pendulum small oscillations, on a three-dimensional harmonic oscillator, on energy in a Bohr atom, on Kepler’s classical problem, on Kepler’s relativistic problem, and finally on the advance of Mercury perihelion, for which an approximate solution is found with a classical method.

The problem of two fixed centers was introduced by Euler as early as in 1760. It plays an important role both in celestial mechanics and in the microscopic world. In the present paper we study the spatial problem in the case of arbitrary (both positive and negative) strengths of the centers. Combining techniques from scattering theory and Liouville integrability, we show that this spatial problem has topologically non-trivial scattering dynamics, which we identify as scattering monodromy. The approach that we introduce in this paper applies more generally to scattering systems that are integrable in the Liouville sense.

The present work uses mean value coordinates to construct the shape functions of a hybrid ‘FE-Meshfree’ quadrilateral element, which is named as Quad4-MVC. This Quad4-MVC can be regarded as the development of the ‘FE-Meshfree’ quadrilateral element with radial-polynomial point interpolation (Quad4-RPIM). Similar to Quad4-RPIM, Quad4-MVC has Kronecker delta property on the boundaries of computational domain, so essential boundary conditions can be enforced as conveniently as in the finite element method (FEM). The novelty of the present work is to construct nodal approximations using mean value coordinates, instead of radial basis functions which are used in Quad4-RPIM. Compared to the radial basis functions, mean value coordinates does not utilize any uncertain parameters, which enhances stability of numerical results. Numerical tests in this paper show that the performance of Quad4-RPIM becomes even worse than four-node iso-parametric element (Quad4) when the parameters of radial basis functions are not chosen properly. However, the performance of Quad4-MVC is stably better than Quad4.

In this article we explicitly construct Stackel separable systems in separation coordinates with the help of separation curve as introduced by Sklyanin. Further, we construct explicit transformation between separable and flat coordinates for flat Stackel systems. We also exploit the geometric structure of these systems in the obtained flat coordinates. These coordinates generalize the wellknown generalized elliptic and generalized parabolic coordinates introduced by Jacobi. (C) 2015 Elsevier Inc. All rights reserved. Funding Agencies|Royal Swedish Academy of Sciences [FOA 13Magn-088]

A practical, calculable solution to Vinti’s problem is offered. Vinti’s problem describes the motion of a satellite orbiting an oblate planet, including the point mass and oblateness J2 terms and differing from the actual geopotential only at the higher harmonics. Vinti’s original solution is extended to cover motion over a rotating Earth. An eccentric-anomaly-like transform removes the singularities in the elliptic integrals, rendering them accessible to numerical calculation. The prediction problem requires the simultaneous solution of two “Kepler’s equations.” Reduction to canonical action angle variables can also be accomplished via numerical techniques. This new realization of Vinti’s solution is free of series expansions that limit accuracy and uses only simple well-understood numerical techniques. Possible use of Vinti’s solution for further efforts is discussed.

The motion of a vehicle after blocking the wheels whose slip on a road is described by a model of nonlinear viscous friction with a falling segment of the characteristic is considered. The model in use approximates the dry friction model when the friction of rest exceeds the sliding friction. In this case, the self-oscillations of wheels are observed at some stages of the braking process under the appropriate initial conditions of motion. These self-oscillations generate a periodically varied tangential loading on the road surface, which may cause the appearance of a wavy relief on this surface. Such a wavy relief is most often observed on soil roads at the turns during the intensive braking process. The study of the character of motion is illustrated by numerical examples for the Lagrange equations of the second kind and for the equations derived by the method of averaging the canonical equations written in the action angle variables.

We consider linear elastic complex media the point-body motion of which is described by two vectorial generalized coordinates and the elastic energy contains a term coupling them. All constraints are holonomic and ideal. The dynamics of this continuum is determined by modified Lagrange equations. We specify a class of media whose strain energy depends only on one of the vectorial generalized coordinates, but does not depend on its gradient. We call this generalized coordinate “special.” On the partial frequency of the special generalized coordinates under some conditions for the inertial and elastic characteristics, the medium behaves as a system of independent harmonic oscillators, and under other conditions, only the trivial solution exists. It is shown that independently of the nature of generalized coordinates, if the inertial and elastic tensors satisfy certain requirements, there exists a band gap of frequencies for most of the dispersion curves; i.e., the medium is a single negative acoustic metamaterial in this band of frequencies. The partial frequency of the “special” coordinate limits the band gap. For a specific class of parameters, apart from this, there exists a decreasing part of the dispersion curve; i.e., the medium is also a double negative acoustic metamaterial in a certain domain of frequencies.

Using a calculus and an algebraic approach, the Cartesian coordinates of the Fermat–Torricelli point are deduced for triangles with no internal angle greater than 120°. Although in theory, the deduction of these coordinates could be made ‘by hand’, in practice it is very laborious to obtain them without the aid of mathematical computer software, but with human guidance, since there are mathematical artifices not yet incorporated into the software. It is also shown that these coordinates can be conveniently expressed in terms of the side lengths and the area of the triangle. These coordinates are contrasted with the coordinates of a similar point: one whose sum of the squares of the distances to the vertices of an arbitrary triangle is a minimum.

We extend the positivity-preserving method of Zhang & Shu (2010, JCP, 229, 3091-3120) to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stability-preserving, Runge-Kutta (SSP-RK) time integration. Special care in taken to ensure that the method preserves strict bounds for the phase space distribution function $f$; i.e., $f\in[0,1]$. The combination of suitable CFL conditions and the use of the high-order limiter proposed in Zhang & Shu (2010) is sufficient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergence-free property of the phase space flow. Proofs that highlight the necessary conditions are presented for general curvilinear coordinates, and the details of these conditions are worked out for some commonly used coordinate systems (i.e., spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry, both with spherical momentum coordinates). Results from numerical experiments --- including one example in spherical symmetry adopting the Schwarzschild metric --- demonstrate that the method achieves high-order accuracy and that the distribution function satisfies the maximum principle., Submitted to JCP

Equations of guiding-center motion without the coordinate singularity at the magnetic axis have been derived from the guiding-center Lagrangian. The poloidal magnetic flux is suggested to be included as one of the nonsingular coordinates for the computation of the guiding-center orbit. The numerical results based on different nonsingular coordinates are verified using the GCM code which is based on canonical variables. A comparison of numerical performance among these nonsingular coordinates and canonical coordinates has been carried out by checking the conservation of energy and toroidal canonical momentum. It is found that by using the poloidal magnetic flux in the nonsingular coordinate system, the numerical performance of nonsingular coordinates can be greatly improved and is comparable to that of canonical variables in long-time simulations.

In this work we study a generalization of the well known Fisher equation in cylindrical coordinates. We determine the subclasses of these equations which are nonlinear self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov and the symmetry generators we find conservation laws for these partial differential equations without classical Lagrangians.

We study the problem of inverting rational bilinear mappings, which leads to a one-parameter family of generalized barycentric coordinates for quadrilaterals, including Wachspress coordinates as a special case.

The equations of the diagram equation method are formulated in terms of prolate spheroidal coordinates. Examples of the convergence and stability of numerical algorithms based on the proposed approach are considered. The results of calculations are compared wherever possible to the data obtained by other methods.

It is well-known that conventional continuum theory is not capable of capturing size effects and therefore, many theories have been proposed. Among these theories, strain gradient and modified strain gradient theories have enjoyed great success so far. In the present study, the modified strain gradient theory is in details derived in general curvilinear coordinates. The modified strain gradient theory involves the modified couple stress theory as a special case and therefore, the modified couple stress theory in curvilinear coordinates is readily obtained. The results are then specialized for two practical orthogonal curvilinear coordinates, i.e. cylindrical and spherical coordinates. The expressions including deformation measures, governing equations, boundary conditions and constitutive equations are expressed in terms of physical components which are more common and convenient. The formulations presented here are general and may be appropriately applied to a broad range of problems in which the use of curvilinear coordinates is unavoidable, such as problems of cylindrical and spherical cavity expansion, the analysis of asymptotic crack tip field and the interpretation of micro/nano indentation tests and also bending or twisting tests on small scales.

A kind of multi-link hybrid press mechanism is brought forth. It adopts three motors as actuations to fulfill the single degree of freedom output of end effector. For hybrid press mechanism, the forward kinematics is usually complicated and its numerical solution is computation-intensive and time-consuming. Therefore, the dynamic model is difficult to build up when it needs to analyze the input function’s effect on system dynamic performances. Concept of dependent generalized coordinates is adpoted here and a kind of dynamic modeling method of multi-link hybrid mechanism is presented based on the virtual work principle. The linear and angular displacements of every component can be expressed concisely in the form of dependent generalized coordinates. They are much simpler than that of independent generalized coordinates. Accordingly, linear and angular velocities will be derived by differentiating with respect to displacements. Velocity Jacobian matrix will be simplified under dependent generalized coordinates system and the virtual work principle-based dynamic model will also be simplified accordingly. Then it needs to introduce constraint conditions and multiplier in order to acquire actuation forces. Introduction of the constraints guarantees the real kinetic characteristics of mechanism. In the end, the curves of actuation forces and spherical joints’ inner forces variations are presented using Matlab. By comparing with Adams simulation results, validity of the method has been proved.

There are many tasks where analytical description of the random signals is used. For instance, representation of signals and waves in polar coordinates is widely used in antenna theory and techniques. In this report, we present the results of implementation of the polygaussian approach for describing polar coordinates of random vectors as mixtures of Rayleigh and Rice distributions. Moreover, polygaussian approach gives an opportunity to generalize it to the case of multidimensional vectors.

Goodearl and Launois have shown in Goodearl and Launois (2011) that for a log-canonical Poisson bracket on affine space there is no rational change of coordinates for which the Poisson bracket is constant. Our main result is a proof of a conjecture of Michael Shapiro which states that if affine space is given a log-canonical Poisson bracket, then there does not exist any rational change of coordinates for which the Poisson bracket is linear. Hence, log-canonical coordinates can be thought of as the simplest possible algebraic coordinates for affine space with a log-canonical coordinate system. In proving this conjecture we find certain invariants of log-canonical Poisson brackets on affine space which linear Poisson brackets do not have.

We use the action-angle variables to describe the geodesic motions in the $5$-dimensional Sasaki-Einstein spaces $Y^{p,q}$. This formulation allows us to study thoroughly the complete integrability of the system. We find that the Hamiltonian involves a reduced number of action variables. Therefore one of the fundamental frequency is zero indicating a chaotic behavior when the system is perturbed., 12 pages, to appear in Prog. Theor. Exp. Phys

We describe the construction of the integrals of geodesic motions in the homogeneous Sasaki-Einstein space T 1,1. For this purpose we use the knowledge of the complete set of Killing vectors and Killing tensors of this space. We discuss the integrability of geodesics and construct explicitly the action-angle variables. We find that two pairs of frequencies of the geodesic motions are resonant giving way to chaotic behavior when the integrable Hamiltonian is perturbed by a small nonintegrable piece.

Regular studies of few-nucleon systems reveal various dynamical components, such as three-nucleon force, Coulomb force and relativistic effects, which play an important role in correct description of nuclear interaction. A large set of existing experimental data for 1H(d; pp)n reaction allows for systematic investigations of these dynamical effects, which vary with energy and appear with different strength in certain observables and phase space regions. In order to perform systematic comparisons with precise theoretical calculations, the experimental data are transformed to the variables based on the Lorentz invariants.

A compact, time-explicit, approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft. The rather compact, three-dimensional solution is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear, rather than Cartesian coordinates, and provides analytical expressions for the secular, constant and periodic terms of each coordinate as a function of the initial relative motion conditions or relative orbital elements. Numerical test cases are conducted to show that the approximate solution can be effectively employed to extend the classical linear Clohessy–Wiltshire solution to include non-linear relative motion without significant loss of accuracy up to a limit of 0.4–0.45 in eccentricity and 40–45 $$^\circ $$ in relative inclination for the follower. A very simple, quadratic extension of the classical Clohessy–Wiltshire solution in curvilinear coordinates is also presented.

Here we develop a specific factorization technique for rotations in $\mathbb{R}^3$ into five factors about two or three fixed axes. Although not always providing the most efficient solution, the method allows for avoiding gimbal lock singularities and decouples the dependence on the invariant axis ${\bf n}$ and the angle $\phi$ of the compound rotation. In particular, the solutions in the classical Euler setting are given directly by the angle of rotation $\phi$ and the coordinates of the unit vector $\bf{n}$ without additional calculations. The immediate implementations in rigid body kinematics are also discussed and some generalizations and potential applications in other branches of science and technology are pointed out as well.

The present chapter is intended as a rather in-depth introduction to the classical kinetic energy and the quantum kinetic energy operator for the nuclei, denoted T and \(\hat{T}\), respectively, in the following. The positions of the particles will be described by means of generalized curvilinear coordinates .