Lagrange Brackets Research Articles

Explicit Krein resolvent identities for singular Sturm-Liouville operators with applications to Bessel operators

We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of the resolvent difference of a pair of self-adjoint realizations of the Bessel expression $-d^2/dx^2+(\nu^2-(1/4))x^{-2}$ on $(0,\infty)$ for values of the parameter $\nu\in[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.

Alternative Set of Nonsingular Quaternionic Orbital Elements

Quaternionic elements in orbital mechanics are usually related to the Kustaanheimo–Stiefel transformation or to the definition of the orbital plane. The new set of regular elements presented in this paper stems from the form of the equations of motion of a rotating solid, which model the evolution of a quaternion defining the orientation of a body-fixed frame and the change in the angular velocity of such frame. By replacing the body-fixed frame with a special orbital frame and accounting for the radial motion separately, an equivalent solution to orbital motion can be constructed. The variation of parameters technique furnishes a new set of elements that is independent from the orbital plane. A second-order Sundman transformation introduces a fictitious time that replaces the physical time as the independent variable. This technique improves the numerical performance of the method and simplifies the derivation. The use of a time element yields an even smoother evolution of the orbital elements under perturbations. Once the Lagrange and Poisson brackets are obtained, the most general nonosculating version of the set of elements is presented. Regarding the performance, numerical experiments show that the method is comparable to other formulations involving similar stabilization and regularization techniques.

케플러궤도운동과 카울라의 인공위성궤도 섭동이론의 상세한 재유도

카울라에 의해 완성되었던 인공위성궤도이론과 케플러의 법칙을 다시 다루었다. 카울라의 원문에서는 생략되었던 부분을 포함한 모든 수식유도과정을 상세히 기술하였다. 특히 15개의 독립적인 라그랑지 괄호들을 계산하는 데에 변환행렬의 직교성을 사용하여, 간결성과 명확성을 이루었다. 여러 중간단계에서 중요한 물리적인 개념들에 대한 설명도 추가하였다. 카울라의 개념적 오류 한 개를 정정하였다. Kaula's theory of satellite orbit and Kepler's law are re-visited. All the mathematical steps of derivation are thoroughly shown including the ones, which had been omitted in Kaula's original publication. Particularly in evaluations of the 15 independent Lagrange brackets, simplicity and clarity are attained by using orthogonal property of transformation matrix. Explanations of important physical concepts are included as well in the midway of derivation. One conceptual blunder of Kaula's is corrected.

First-order partial differential equations in classical dynamics

Carathèodory’s classic work on the calculus of variations explores in depth the connection between ordinary differential equations and first-order partial differential equations. The n second-order ordinary differential equations of a classical dynamical system reduce to a single first-order differential equation in 2n independent variables. The general solution of first-order partial differential equations touches on many concepts central to graduate-level courses in analytical dynamics including the Hamiltonian, Lagrange and Poisson brackets, and the Hamilton–Jacobi equation. For all but the simplest dynamical systems the solution requires one or more of these techniques. Three elementary dynamical problems (uniform acceleration, harmonic motion, and cyclotron motion) can be solved directly from the appropriate first-order partial differential equation without the use of advanced methods. The process offers an unusual perspective on classical dynamics, which is readily accessible to intermediate students who are not yet fully conversant with advanced approaches.

A Modification of the Guiding-Centre Fundamental 1-Form with StrongE×BFlow

A modified guiding-centre fundamental 1-form with strong E × B flow is derived by the phase space Lagrangian Lie perturbation method. Since the symplectic part of the derived 1-form is the same as the standard one without the strong E × B flow, it yields the standard Lagrange and Poisson brackets. Therefore the guiding-centre Hamilton equations keep their general form even when temporal evolution of the E × B flow is allowed. Compensation of keeping the standard symplectic structure is paid by complication of the guiding-centre Hamiltonian. However, it is possible to simplify the Hamiltonian in well localised transport barrier regions like a tokamak edge in a high confinement regime and an internal transport barrier in a reversed shear tokamak. The guiding-centre Vlasov and Poisson equations are derived from the variational principle. The conserved energy of the system is obtained from the Noether's theorem. Correspondence to low-frequency fluid equations is shown.

COVARIANT HAMILTONIAN FIELD THEORY

A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler–Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proven that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noether's theorem. Furthermore, we specify the generating function of an infinitesimal space-time step that conforms to the field equations.

Quantization of contact manifolds and thermodynamics

The physical variables of classical thermodynamics occur in conjugate pairs such as pressure/volume, entropy/temperature, chemical potential/particle number. Nevertheless, and unlike in classical mechanics, there are an odd number of such thermodynamic co-ordinates. We review the formulation of thermodynamics and geometrical optics in terms of contact geometry. The Lagrange bracket provides a generalization of canonical commutation relations. Then we explore the quantization of this algebra by analogy to the quantization of mechanics. The quantum contact algebra is associative, but the constant functions are not represented by multiples of the identity: a reflection of the classical fact that Lagrange brackets satisfy the Jacobi identity but not the Leibnitz identity for derivations. We verify that this ‘quantization’ describes correctly the passage from geometrical to wave optics as well. As an example, we work out the quantum contact geometry of odd-dimensional spheres.

Eigenvalues of the Radially Symmetric p ‐Laplacian in R n

For the p-Laplacian p = div:(| |p–2), p>1, the eigenvalue problem –p + q(|x|)||p–2 = ||p–2 in Rn is considered under the assumption of radial symmetry. For a first class of potentials q(r) as r at a sufficiently fast rate, the existence of a sequence of eigenvalues k if k is shown with eigenfunctions belonging to Lp(Rn). In the case p=2, this corresponds to Weyl's limit point theory. For a second class of power-like potentials q(r)– as r at a sufficiently fast rate, it is shown that, under an additional boundary condition at r=, which generalizes the Lagrange bracket, there exists a doubly infinite sequence of eigenvalues k with k ± if k±. In this case, every solution of the initial value problem belongs to Lp(Rn). For p=2, this situation corresponds to Weyl's limit circle theory.

Geometric Derivation of the Delaunay Variables and Geometric Phases

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T3 on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S1 action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J2-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J2-Hamiltonian is a collective Hamiltonian of the T3 momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J2 effect as geometric phases.

Advanced Dynamics

7R7. Advanced Dynamics. - RB Bhat (Concordia Univ, Montreal, Canada) and RV Dukkipati (Sch of Eng, Fairfield Univ, Fairfield CT 06430-5195). Narosa Publ, New Delhi, India. Distributed in USA by CRC Press LLC, Boca Raton FL. 2001. 395 pp. ISBN 0-8493-1018-0. $99.95.Reviewed by M Pascal (Lab de Modelisation en Mec, Univ Pierre et Marie Curie, Tour 66, 4 Place Jussieu, Paris, 75252 Cedex 05, France).The authors’ aim is to provide a textbook giving insight about the methods used in Analytical Mechanics and which are related to abstract concepts like generalized parameters, virtual displacements, and variational principles. In contrast with other textbooks, the authors did not assume that the basic knowledge of kinematics and dynamics is familiar to the reader; the transition to this abstract presentation of the fundamental laws of mechanics is performed very gradually. The result is a book involving altogether topics related to elementary backgrounds (suitable for first-year graduate students) about dynamics of particles and rigid bodies and advanced theoretical part (useful for undergraduate senior students). The book involves 12 chapters, each of them is followed by a list of references and a great amount of homework problems. The fundamental laws of Mechanics are presented in the first chapter. Only systems of particles or plane motions of rigid bodies are considered in this chapter. Several topics (kinematics of rigid bodies, radius of gyration, center of percussion…) are introduced, but some other basic concepts like Galileen frame, relative and absolute velocities, Coriolis acceleration…are missing and will be presented only in Chapter 4. This reviewer is not convinced that this first chapter is very suitable. For first level students, a more detailed presentation is needed, and for senior students, all these concepts are known. Chapter 2 deals with more interesting topics such as generalized coordinates, classification of constraints, virtual works principle, and generalized forces. However, in this part, only systems of particles are considered, and the fundamental distinction between external and internal forces is not underlined. Several elementary concepts are presented in Chapters 3 and 4, like potential and kinetics energies, linear and angular momentum vectors, relative motions, and transformation of coordinates systems. The Foucault’s pendulum gives an interesting example of these topics. In Chapter 5, an introduction to Kepler’s laws and to orbital motion is presented. Chapter 6 deals with Lagrange equations for holonomic and non-holonomic systems. First integrals are introduced for systems with cyclic coordinates. A great amount of examples are solved. The next two chapters are devoted to basic knowledges about inertia tensors, principal axes of inertia and dynamics of rigid bodies. As an example, the motion of the spinning top is solved. Chapter 9 presents variational methods. Hamilton’s and the least action principle are derived, together with Hamilton’s equations. Canonical transformations and Hamilton-Jacobi equations are studied in Chapter 10; Lagrange and Poisson brackets are also introduced.An interesting classification of dynamics systems is given at the beginning of Chapter 11, devoted to vibrations. Then the linear vibrations of single-degree-of-freedom systems are investigated. The forced vibrations are also considered for harmonic and arbitrary excitation forces. For multi-degree-of-freedom systems, usual modal analysis is presented for undamped and damped cases. The last chapter provides an introduction to numerical methods. Several explicit and implicit numerical schemes are presented for the integration of linear dynamical systems. An example of a non-linear system is then solved by means of these numerical methods in order to underline the problems encountered in this case. In conclusion, this reviewer thinks that this book is without doubt very useful at a time when many students, and even many scientists, think that Analytical Mechanics belongs to the past. Several interesting examples of the use of these methods may be found in this book. In spite of rather numerous printing mistakes in the text, the book is clearly written with a lot of good quality figures. This reviewer thinks that Advanced Dynamics can be used by senior students and also scientists involved in systems dynamics and interested in analytical methods rather than numerical methods.

A FORMULATION OF HAMILTONIAN MECHANICS USING GEOMETRIC ALGEBRA

Geometric Algebra is introduced. The basic concepts of Hamilton mechanics are derived using the Geometric Algebra. Standard Poisson bracket and standard Lagrange bracket is introduced in terms of the Geometric Algebra. The Dirac quantization map is derived in a new view, which is used to derive the plus Poisson bracket, which seems different from the previous forms by other authors.

Lagrange brackets and U(1) fields

The idea of a companion Lagrangian associated with p-branes is extended to include the presence of U(1) fields. The brane Lagrangians are constructed with F ij represented in terms of Lagrange brackets, which make manifest the reparametrisation invariance of the theory; these are replaced by Poisson brackets in the companion Lagrangian, which is now covariant under field redefinition. The ensuing Lagrangians possess a similar formal structure to those in the absence of an anti-symmetric field tensor.

Projective structures and contact forms

In this paper we show that a projective structure on a real simply connected smooth manifold M determines a family of local Lie algebras on M (i.e., a family of Lie brackets on C ~ ( M ) that satisfy the localization condition). As was shown in Kirillov's paper [4], a nondegenerate local Lie algebra is determined by a contact or a symplectic structure on the manifold. The relation between projective and contact geometries has already been mentioned by Cartan (see [9] and also [2]). Many problems of projective differential geometry lead to contact structures. It turns out that this relationship is absolutely direct. Seemingly, the following elementary (and curious) fact remains unnoticed. T h e o r e m 1. Let M be a simply connected manifold (possibly noncompact) of dimension 2k 1. For a projective structure on M to exist it is necessary and sufficient that there be functions f l , . . . , f2k e C ~ ( M ) such that the 1-form k o~ = E ( f 2 i 1 d f 2 i f2idf2i-1) (1) i=1 is contact. Thus, a locally projective manifold of odd dimension is contact. The corresponding local Lie algebra is nondegenerate and is determined by a Lagrange bracket. The choice of the form (1) is not unique. An even-dimensional manifold with a projective structure must not be symplectic. However, this holds if the projective structure is affine. From this viewpoint, there is a sharp difference between even-dimensionai and odd-dimensional projective structures, while there is a strict analogy between "projective-contact" and "affine-symplectic" geometries. T h e o r e m 1'. For a simply connected manifold M of dimension 2k to possess an affine structure it is necessary and sufficient that there be functions f l , . . . , f2k E C¢~(M) such that the 2-form

Canonoid transformations and constants of motion

The necessary and sufficient conditions for a canonoid transformation with respect to a given Hamiltonian are obtained in terms of the Lagrange brackets of the transformation. The relation of these conditions with the constants of motion is discussed.

Hamiltonian magnetic optics

Starting from Hamilton's principle, the imaging properties of charged particle beams in arbitrary static magnetic fields are outlined. The magnetic scalar potential is expressed as a series of multipoles about an arbitrary space curve which need not necessarily coincide with a trajectory of the beam. Appropriate power series expansions are given for the components of the magnetic vector potential where the expansion coefficients are related to the complex curvature of the axis and the strengths of the multipole components of the magnetic scalar potential. The general laws which govern the propagation of the charged particles are discussed by means of Hamilton's characteristic functions. It is shown that the Poisson and the Lagrange brackets are equivalent representations of the symplectic condition. The Poincaré invariant is used for elucidating some characteristic imaging properties of magnetic fields. A perturbation eikonal is introduced which allows a systematic and simultaneous calculation of the geometrical and chromatic aberrations to any order. The method has the advantage that it reveals at the very beginning all interrelations between the aberration coefficients. The iteration procedure starts from the paraxial rays which are supposed to be known. The algorithm is especially suited for numerical calculation of the higher-order aberrations.

Canonical path integral quantization of gauge systems

It is shown that the (possibly explicity time-dependent) gauge fixing conditions z a ( q, p, t ) = 0, invoked for the canonical quantization of gauge theories via the Faddeev path integral formulation, need not satisfy the restrictions { z a , z b } = 0. As a preparation, we reexamine and clarify how the canonical structure on the large (partly unphysical) phasespace Γ induces, through either Poisson brackets or Lagrange brackets or Dirac brackets, the canonical structure on the physical space of observables. The realizability of some particularly convenient coordinate systems on Γ is also proved.

The Two-Sided Factorization of Ordinary Differential Operators

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

On canonical formalism in field theory with derivatives of higher order-canonical transformations

For a canonical formalism with derivatives of higher order, the corresponding theory of canonical transformations is given in the most general case for classical and covariant field theory. Relations with the generating functional, infinitesimal transformations, Hamilton-Jacobi method, Lagrange and Poisson brackets, as well as integral invariants of the first and higher orders and the corresponding Liouville theorem are considered.

The intrinsic properties of rank and nullity of the Lagrange bracket in the one dimensional calculus of variations

This paper establishes the existence of symplectic structure indegeneratevariational problems, i.e. problems whose full development involves a hierarchy of equations of constraint as well as various equations of motion. Any variational problem, degenerate or otherwise, may be calledregularif the equations of the second variation provide a complete description of the infinitesimal relationships subsisting between any orbit and all its infinitesimal neighbour orbits. It is proved that Poincare’s conserved antisymmetric derived bilinear differential form in the orbit manifold of any regular degenerate problem admits no null vectors other than those which represent infinitesimal deviations due to indeterminacy in the evolution of the orbit. Conversely, it is shown how, given any continuous system of orbits endowed with a conserved antisymmetric closed bilinear differential form having this unique property of rank and nullity, one can construct at least one regular variational

On the equinoctial orbit elements

This paper investigates the equinoctial orbit elements for the two-body problem, showing that the associated matrices are free from singularities for zero eccentricities and zero and ninety degree inclinations. The matrix of the partial derivatives of the position and velocity vectors with respect to the orbit elements is given explicitly, together with the matrix of inverse partial derivatives, in order to facilitate construction of the matrizant (state transition matrix) corresponding to these elements. The Lagrange and Poisson bracket matrices are also given. The application of the equinoctial orbit elements to general and special perturbations is discussed.