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Canonical coordinates

About: Canonical coordinates is a research topic. Over the lifetime, 1597 publications have been published within this topic receiving 38911 citations.


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Book
01 Jan 1974
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Abstract: Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.

11,008 citations

Journal ArticleDOI
Frederick E. Riewe1
TL;DR: In this article, a method was proposed that uses a Lagrangian containing derivatives of fractional order to derive an Euler-Lagrange equation of motion for non-conservative forces such as friction.
Abstract: Traditional Lagrangian and Hamiltonian mechanics cannot be used with nonconservative forces such as friction. A method is proposed that uses a Lagrangian containing derivatives of fractional order. A direct calculation gives an Euler-Lagrange equation of motion for nonconservative forces. Conjugate momenta are defined and Hamilton's equations are derived using generalized classical mechanics with fractional and higher-order derivatives. The method is applied to the case of a classical frictional force proportional to velocity. \textcopyright{} 1996 The American Physical Society.

713 citations

Book
01 Jan 1990
TL;DR: In this article, the authors present a family of Integrable Quartic Potentials related to Symmetric Spaces, which they call Symplectic Non-Kahlerian Manifolds.
Abstract: 1. Preliminaries.- 1.1 A Simple Example: Motion in a Potential Field.- 1.2 Poisson Structure and Hamiltonian Systems.- 1.3 Symplectic Manifolds.- 1.4 Homogeneous Symplectic Spaces.- 1.5 The Moment Map.- 1.6 Hamiltonian Systems with Symmetry.- 1.7 Reduction of Hamiltonian Systems with Symmetry.- 1.8 Integrable Hamiltonian Systems.- 1.9 The Projection Method.- 1.10 The Isospectral Deformation Method.- 1.11 Hamiltonian Systems on Coadjoint Orbits of Lie Groups.- 1.12 Constructions of Hamiltonian Systems with Large Families of Integrals of Motion.- 1.13 Completeness of Involutive Systems.- 1.14 Hamiltonian Systems and Algebraic Curves.- 2. Simplest Systems.- 2.1 Systems with One Degree of Freedom.- 2.2 Systems with Two Degrees of Freedom.- 2.3 Separation of Variables.- 2.4 Systems with Quadratic Integrals of Motion.- 2.5 Motion in a Central Field.- 2.6 Systems with Closed Trajectories.- 2.7 The Harmonic Oscillator.- 2.8 The Kepler Problem.- 2.9 Motion in Coupled Newtonian and Homogeneous Fields.- 2.10 Motion in the Field of Two Newtonian Centers.- 3. Many-Body Systems.- 3.1 Lax Representation for Many-Body Systems.- 3.2 Completely Integrable Many-Body Systems.- 3.3 Explicit Integration of the Equations of Motion for Systems of Type I and V via the Projection Method.- 3.4 Relationship Between the Solutions of the Equations of Motion for Systems of Type I and V.- 3.5 Explicit Integration of the Equations of Motion for Systems of Type II and III.- 3.6 Integration of the Equations of Motion for Systems with Two Types of Particles.- 3.7 Many-Body Systems as Reduced Systems.- 3.8 Generalizations of Many-Body Systems of Type I-III to the Case of the Root Systems of Simple Lie Algebras.- 3.9 Complete Integrability of the Systems of Section 3.8.- 3.10 Anisotropic Harmonic Oscillator in the Field of a Quartic Central Potential (the Garnier System).- 3.11 A Family of Integrable Quartic Potentials Related to Symmetric Spaces.- 4. The Toda Lattice.- 4.1 The Ordinary Toda Lattice. Lax Representation. Complete Integrability.- 4.2 The Toda Lattice as a Dynamical System on a Coadjoint Orbit of the Group of Triangular Matrices.- 4.3 Explicit Integration of the Equations of Motion for the Ordinary Nonperiodic Toda Lattice.- 4.4 The Toda Lattice as a Reduced System.- 4.5 Generalized Nonperiodic Toda Lattices Related to Simple Lie Algebras.- 4.6 Toda-like Systems on Coadjoint Orbits of Borel Subgroups.- 4.7 Canonical Coordinates for Systems of Toda Type.- 4.8 Integrability of Toda-like Systems on Generic Orbits.- 5. Miscellanea.- 5.1 Equilibrium Configurations and Small Oscillations of Some Integrable Hamiltonian Systems.- 5.2 Motion of the Poles of Solutions of Nonlinear Evolution Equations and Related Many-Body Problems.- 5.3 Motion of the Zeros of Solutions of Linear Evolution Equations and Related Many-Body Problems.- 5.4 Concluding Remarks.- Appendix A.- Examples of Symplectic Non-Kahlerian Manifolds.- Appendix B.- Solution of the Functional Equation (3.1.9).- Appendix C.- Semisimple Lie Algebras and Root Systems.- Appendix D.- Symmetric Spaces.- References.

697 citations

Journal ArticleDOI
TL;DR: In this article, a general rule for writing down the coefficients of the transformation to symmetry coordinates is derived together with a method of obtaining the kinetic energy reciprocal matrix (G) in terms of symmetry coordinates with a minimum of algebra.
Abstract: Developments which reduce the labor of calculating the vibration frequencies of complex molecules are described. In particular a vectorial scheme is given for obtaining the reciprocal of the matrix of the kinetic energy in terms of valence‐type coordinates. A general rule for writing down the coefficients of the transformation to symmetry coordinates is derived together with a method of obtaining the kinetic energy reciprocal matrix (G) in terms of symmetry coordinates with a minimum of algebra. A treatment of redundant coordinates is developed. In addition, reduction of the secular equation by the splitting out of high frequencies, a new type of isotope product rule, and the determination of normal coordinates are discussed. The molecule CH3Cl is worked out as an illustration.

685 citations

Journal ArticleDOI
TL;DR: In this article, a Lagrangian variational formulation of twist maps is proposed to compute the flux escaping from regions bounded by partial barriers formed from minimizing orbits, which form a scaffold in the phase space and constrain the motion of remaining orbits.
Abstract: Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamics-from simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the flux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions.

627 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20234
202214
202126
202037
201933
201832