scholarly journals Universal local symmetries in classical mechanics and physical degrees of freedom

2012 ◽  
Vol 376 (45) ◽  
pp. 3017-3020 ◽  
Author(s):  
E. Cattaruzza ◽  
E. Gozzi
Keyword(s):  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Local Symmetries

ROTATIONS AND VIBRATIONS OF FULLERENES IN THE MOLECULAR COMPLEX C20@C80

2021 ◽  
pp. 35-48
Author(s):  
M.A. Bubenchikov ◽  
◽  
A.M. Bubenchikov ◽  
D.V. Mamontov ◽  
◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Rotation Frequency ◽  
Dynamic State ◽  
Large Molecules ◽  
Rotational Degrees Of Freedom ◽  
Rotational Motions ◽  
Centers Of Mass ◽  
The Moment

The aim of this work is to apply classical mechanics to a description of the dynamic state of C20@C80 diamond complex. Endohedral rotations of fullerenes are of great interest due to the ability of the materials created on the basis of onion complexes to accumulate energy at rotational degrees of freedom. For such systems, a concept of temperature is not specified. In this paper, a closed description of the rotation of large molecules arranged in diamond shells is obtained in the framework of the classical approach. This description is used for C20@C80 diamond complex. Two different problems of molecular dynamics, distinguished by a fixing method for an outer shell of the considered bimolecular complex, are solved. In all the cases, the fullerene rotation frequency is calculated. Since a class of possible motions for a single carbon body (molecule) consists of rotations and translational displacements, the paper presents the equations determining each of these groups of motions. Dynamic equations for rotational motions of molecules are obtained employing the moment of momentum theorem for relative motions of the system near the fullerenes’ centers of mass. These equations specify the operation of the complex as a molecular pendulum. The equations of motion of the fullerenes’ centers of mass determine vibrations in the system, i.e. the operation of the complex as a molecular oscillator.

Elements of Classical Field Theory

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
Lagrange Equations ◽  
Lie Group Of Transformations ◽  
Infinitesimal Transformations ◽  
Conserved Currents

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

scholarly journals A QUANTUM IS A COMPLEX STRUCTURE ON CLASSICAL PHASE SPACE

2005 ◽  
Vol 02 (04) ◽  
pp. 633-655
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Complex Structure ◽  
Classical Mechanics ◽  
Elementary Excitation ◽  
Differentiable Structure ◽  
Duality Transformations ◽  
Kähler Potential ◽  
Classical Phase Space

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space. For that purpose we consider Kähler phase spaces, endowed with a dynamics whose Hamiltonian equals the local Kähler potential.

Homogeneous variables in classical dynamics

1933 ◽  
Vol 29 (3) ◽  
pp. 389-400 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Degrees Of Freedom ◽  
Scalar Potential ◽  
Rest Mass ◽  
Classical Mechanics ◽  
Hamiltonian Function ◽  
Lagrangian Function ◽  
Practical Case ◽  
Classical Dynamics ◽  
Hamiltonian Method ◽  
Independent Functions

The well-known methods of classical mechanics, based on the use of a Lagrangian or Hamiltonian function, are adequate for the treatment of nearly all dynamical systems met with in practice. There are, however, a few exceptional cases to which the ordinary methods are not immediately applicable. For example, the ordinary Hamiltonian method cannot be used when the momenta pr, defined in terms of the Lagrangian function L by the usual formulae pr = ∂L/∂qr, are not independent functions of the velocities. A practical case of this kind is provided by the electromagnetic field, considered as a dynamical system with an infinite number of degrees of freedom, since the momentum conjugate to the scalar potential at any point vanishes identically. Again, for the very simple example of the relativistic motion of a particle of zero rest-mass in field-free space, the Lagrangian function vanishes and the usual Lagrangian method is not applicable.

Field theory, classical

2018 ◽  
Author(s):  
Mark Wilson
Keyword(s):  
Field Theory ◽  
Twentieth Century ◽  
Electromagnetic Field ◽  
Physical Quantity ◽  
General Framework ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Electrical Strength ◽  
Mass Temperature

A physical quantity (such as mass, temperature or electrical strength) appears as a field if it is distributed continuously and variably throughout a region. In distinction to a ’lumped’ quantity, whose condition at any time can be specified by a finite list of numbers, a complete description of a field requires infinitely many bits of data (it is said to ’possess infinite degrees of freedom’). A field is classical if it fits consistently within the general framework of classical mechanics. By the start of the twentieth century, orthodox mechanics had evolved to a state of ontological dualism, incorporating a worldview where massive matter appears as ’lumped’ points which communicate electrical and magnetic influences to one another through a continuous intervening medium called the electromagnetic field. The problem of consistently describing how matter and fields function together has yet to be fully resolved.

The basis of statistical quantum mechanics

1929 ◽  
Vol 25 (1) ◽  
pp. 62-66 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Dynamical System ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Degrees Of Freedom ◽  
Present Note ◽  
Classical Mechanics ◽  
The Other ◽  
Statistical Nature ◽  
Actual System ◽  
Quantum Treatment

In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.

scholarly journals Classical molecular dynamics simulation of electronically non-adiabatic processes

2016 ◽  
Vol 195 ◽  
pp. 9-30 ◽  
Author(s):  
William H. Miller ◽  
Stephen J. Cotton
Keyword(s):  
Molecular Dynamics ◽  
Potential Energy Surfaces ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Electronic States ◽  
Dynamics Simulation ◽  
Arbitrary Form ◽  
Electronic Density ◽  
Classical Molecular Dynamics ◽  
Adiabatic Processes

Both classical and quantum mechanics (as well as hybrids thereof, i.e., semiclassical approaches) find widespread use in simulating dynamical processes in molecular systems. For large chemical systems, however, which involve potential energy surfaces (PES) of general/arbitrary form, it is usually the case that only classical molecular dynamics (MD) approaches are feasible, and their use is thus ubiquitous nowadays, at least for chemical processes involving dynamics on a single PES (i.e., within a single Born–Oppenheimer electronic state). This paper reviews recent developments in an approach which extends standard classical MD methods to the treatment of electronically non-adiabatic processes, i.e., those that involve transitions between different electronic states. The approach treats nuclear and electronic degrees of freedom (DOF) equivalently (i.e., by classical mechanics, thereby retaining the simplicity of standard MD), and provides “quantization” of the electronic states through a symmetrical quasi-classical (SQC) windowing model. The approach is seen to be capable of treating extreme regimes of strong and weak coupling between the electronic states, as well as accurately describing coherence effects in the electronic DOF (including the de-coherence of such effects caused by coupling to the nuclear DOF). A survey of recent applications is presented to illustrate the performance of the approach. Also described is a newly developed variation on the original SQC model (found universally superior to the original) and a general extension of the SQC model to obtain the full electronic density matrix (at no additional cost/complexity).

REVIEW OF RECENT DEVELOPMENTS IN SUPERSTRING THEORY

1987 ◽  
Vol 02 (03) ◽  
pp. 593-643 ◽  
Author(s):  
JOHN H. SCHWARZ
Keyword(s):  
Degrees Of Freedom ◽  
Classical Solutions ◽  
Type Ii ◽  
World Sheet ◽  
Superstring Theory ◽  
Recent Developments ◽  
Local Symmetries ◽  
Superconformal Theories ◽  
Superstring Theories ◽  
Internal Degrees Of Freedom

After proposing a procedure for classifying string theories, we describe the various local symmetries that can occur on the world sheet with special emphasis on Kac–Moody algebras in superconformal theories. The construction of multiloop amplitudes is briefly reviewed. Then the constraint of modular invariance is analyzed for models in which the internal degrees of freedom are described by fermions. Next we consider the construction of consistent classical solutions. A few examples are presented for both the heterotic and type II superstring theories. A brief description of some recent work in string field theory and other approaches to a nonperturbative formulation of string theory is presented.

Superintegrability in Classical Mechanics: A Contemporary Approach to Bertrand's Theorem

1997 ◽  
Vol 12 (01) ◽  
pp. 271-276 ◽  
Author(s):  
A. L. Salas-Brito ◽  
H. N. Núñez-Yépez ◽  
R. P. Martínez-Y-Romero
Keyword(s):  
Harmonic Oscillator ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Classical System ◽  
Constants Of Motion ◽  
Three Degrees Of Freedom ◽  
Contemporary Approach

Superintegrable Hamiltonians in three degrees of freedom posses more than three functionally independent globally defined and single-valued constants of motion. In this contribution and under the assumption of the existence of only periodic and plane bounded orbits in a classical system we are able to establish the superintegrability of the Hamiltonian. Then, using basic algebraic ideas, we obtain a contemporary proof of Bertrand's theorem. That is, we are able to show that the harmonic oscillator and the Newtonian gravitational potentials are the only 3D potentials whose bounded orbits are all plane and periodic.

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