scholarly journals A generalized model for the classical relativistic spinning particle

2016 ◽  
Vol 31 (07) ◽  
pp. 1650027 ◽  
Author(s):  
Mahdi Hajihashemi ◽  
Ahmad Shirzad
Keyword(s):  
Equations Of Motion ◽  
Oscillatory Solution ◽  
Lagrangian Formulation ◽  
Canonical Structure ◽  
Hamiltonian Approach ◽  
Spinning Particle ◽  
Generalized Model ◽  
Generalized System ◽  
Poincaré Algebra

Following the Poincaré algebra, in the Hamiltonian approach, for a free spinning particle and using the Casimirs of the algebra, we construct systematically a set of Lagrangians for the relativistic spinning particle which includes the Lagrangian given in the literature. We analyze the dynamics of this generalized system in the Lagrangian formulation and show that the equations of motion support an oscillatory solution corresponding to the spinning nature of the system. Then we analyze the canonical structure of the system and present the correct gauge suitable for the spinning motion of the system.

scholarly journals Rendering neuronal state equations compatible with the principle of stationary action

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Erik D. Fagerholm ◽  
W. M. C. Foulkes ◽  
Karl J. Friston ◽  
Rosalyn J. Moran ◽  
Robert Leech
Keyword(s):  
Computational Neuroscience ◽  
Equations Of Motion ◽  
State Equation ◽  
Oscillatory Solution ◽  
Lagrangian Formulation ◽  
Connectivity Matrix ◽  
State Equations ◽  
Stationary Action ◽  
Linear Differential

AbstractThe principle of stationary action is a cornerstone of modern physics, providing a powerful framework for investigating dynamical systems found in classical mechanics through to quantum field theory. However, computational neuroscience, despite its heavy reliance on concepts in physics, is anomalous in this regard as its main equations of motion are not compatible with a Lagrangian formulation and hence with the principle of stationary action. Taking the Dynamic Causal Modelling (DCM) neuronal state equation as an instructive archetype of the first-order linear differential equations commonly found in computational neuroscience, we show that it is possible to make certain modifications to this equation to render it compatible with the principle of stationary action. Specifically, we show that a Lagrangian formulation of the DCM neuronal state equation is facilitated using a complex dependent variable, an oscillatory solution, and a Hermitian intrinsic connectivity matrix. We first demonstrate proof of principle by using Bayesian model inversion to show that both the original and modified models can be correctly identified via in silico data generated directly from their respective equations of motion. We then provide motivation for adopting the modified models in neuroscience by using three different types of publicly available in vivo neuroimaging datasets, together with open source MATLAB code, to show that the modified (oscillatory) model provides a more parsimonious explanation for some of these empirical timeseries. It is our hope that this work will, in combination with existing techniques, allow people to explore the symmetries and associated conservation laws within neural systems – and to exploit the computational expediency facilitated by direct variational techniques.

scholarly journals VARIATIONAL PROBLEM FOR THE FRENKEL AND THE BARGMANN–MICHEL–TELEGDI (BMT) EQUATIONS

2013 ◽  
Vol 28 (01) ◽  
pp. 1250234 ◽  
Author(s):  
A. A. DERIGLAZOV
Keyword(s):  
Abelian Group ◽  
Variational Problem ◽  
Classical Theory ◽  
Equations Of Motion ◽  
Lagrangian Formulation ◽  
Gauge Invariant ◽  
Local Symmetries

We propose Lagrangian formulation for the particle with value of spin fixed within the classical theory. The Lagrangian is invariant under non-Abelian group of local symmetries. On this reason, all the initial spin variables turn out to be unobservable quantities. As the gauge-invariant variables for description of spin we can take either the Frenkel tensor or the Bargmann–Michel–Telegdi (BMT) vector. Fixation of spin within the classical theory implies O(ℏ)-corrections to the corresponding equations of motion.

scholarly journals Recent Progress on the Description of Relativistic Spin: Vector Model of Spinning Particle and Rotating Body with Gravimagnetic Moment in General Relativity

2017 ◽  
Vol 2017 ◽  
pp. 1-49 ◽  
Author(s):  
Alexei A. Deriglazov ◽  
Walberto Guzmán Ramírez
Keyword(s):  
General Relativity ◽  
Relativistic Quantum ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Theoretical Description ◽  
Vector Model ◽  
Relativistic Quantum Mechanics ◽  
Spinning Particle ◽  
Rotating Body ◽  
Relativistic Spin

We review the recent results on development of vector models of spin and apply them to study the influence of spin-field interaction on the trajectory and precession of a spinning particle in external gravitational and electromagnetic fields. The formalism is developed starting from the Lagrangian variational problem, which implies both equations of motion and constraints which should be presented in a model of spinning particle. We present a detailed analysis of the resulting theory and show that it has reasonable properties on both classical and quantum level. We describe a number of applications and show how the vector model clarifies some issues presented in theoretical description of a relativistic spin: (A) one-particle relativistic quantum mechanics with positive energies and its relation with the Dirac equation and with relativistic Zitterbewegung; (B) spin-induced noncommutativity and the problem of covariant formalism; (C) three-dimensional acceleration consistent with coordinate-independence of the speed of light in general relativity and rainbow geometry seen by spinning particle; (D) paradoxical behavior of the Mathisson-Papapetrou-Tulczyjew-Dixon equations of a rotating body in ultrarelativistic limit, and equations with improved behavior.

scholarly journals Lattice vibrations with Rayleigh dissipation

2000 ◽  
Vol 42 (2) ◽  
pp. 244-253 ◽  
Author(s):  
J. N. Boyd ◽  
P. N. Raychowdhury
Keyword(s):  
Natural Frequencies ◽  
Linear Array ◽  
Equations Of Motion ◽  
Dissipation Function ◽  
Lagrangian Formulation ◽  
Harmonic Oscillators ◽  
Lattice Vibrations ◽  
Circular Array ◽  
Coupled Harmonic Oscillators ◽  
Symmetry Coordinates

AbstractWe approximate a linear array of coupled harmonic oscillators as a symmetric circular array of identical masses and springs. The springs are taken to possess mass distributed along their lengths. We give a Lagrangian formulation of the problem of finding the natural frequencies of oscillation for the array. Damping terms are included by means of the Rayleigh dissipation function. A transformation to symmetry coordinates as determined by the group of rotations of the circle uncouples the equations of motion.

NONHOLONOMIC DYNAMICS OF SECOND ORDER AND THE HEISENBERG SPINNING PARTICLE

2012 ◽  
Vol 09 (07) ◽  
pp. 1220010
Author(s):  
MIRCEA CRASMAREANU ◽  
IULIAN STOLERIU
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Spinning Particle ◽  
Nonholonomic Dynamics

The equations of motion for the associated constrained Lagrangian to a nonholonomic Lagrangian of second order are computed. The spinning particle subject to the Heisenberg constraint is treated as example and its dynamics is completely described.

Modeling of Systems With Position-Dependent Mass Revisited: A Port-Hamiltonian Approach

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Dimitri Jeltsema ◽  
Arnau Dòria-Cerezo
Keyword(s):  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Hamiltonian Approach ◽  
Lagrangian And Hamiltonian Approach ◽  
Modeling Framework ◽  
Straightforward Application ◽  
Composite Energy ◽  
Dependent Mass ◽  
Position Dependent Mass ◽  
Classical Lagrangian

It is known that straightforward application of the classical Lagrangian and Hamiltonian formalism to systems with mass varying explicitly with position may lead to discrepancies in the formulation of the equations of motion. Systems with mass varying explicitly with position often arise from situations where the partitioning of a closed system of constant mass leads to open subsystems that exchange mass among themselves. One possible solution is to introduce additional nonconservative generalized forces that account for these effects. However, it remains unclear how to systematically interconnect the Lagrangian or Hamiltonian subsystems. In this note, systems with mass varying explicitly with position and their properties are studied in the port-Hamiltonian modeling framework. The port-Hamiltonian formalism combines the classical Lagrangian and Hamiltonian approach with network modeling and is applicable to various engineering domains. One of the strong aspects of the port-Hamiltonian formalism is that power-preserving interconnections between port-Hamiltonian subsystems results in another port-Hamiltonian system with composite energy and interconnection structure. The motion of a heavy cable being deployed from a reel by the action of gravity is used as an example.

scholarly journals NON-ABELIAN PROCA MODEL BASED ON THE IMPROVED BFT FORMALISM

1998 ◽  
Vol 13 (13) ◽  
pp. 2179-2199 ◽  
Author(s):  
MU-IN PARK ◽  
YOUNG-JAI PARK
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Difficult Problem ◽  
Lagrangian Formulation ◽  
Exponential Form ◽  
Extended Phase ◽  
Physical Variables ◽  
Correction Terms ◽  
Insight Into

We present the newly improved Batalin–Fradkin–Tyutin (BFT) Hamiltonian formalism and the generalization to the Lagrangian formulation, which provide a much more simple and transparent insight into the usual BFT method, with application to the non-Abelian Proca model, which has been a difficult problem in the usual BFT method. The infinite terms of the effectively first class contraints can be made to be the regular power series forms by an ingenious choice of Xαβ and ωαβ matrices. In this new method, the first class Hamiltonian, which also needs infinite correction terms, is obtained simply by replacing the original variables in the original Hamiltonian with the BFT physical variables. Remarkably, all the infinite correction terms can be expressed in the compact exponential form. We also show that in our model the Poisson brackets of the BFT physical variables in the extended phase space have the same structure as the Dirac brackets of the original phase space variables. With the help of both our newly developed Lagrangian formulation and Hamilton's equations of motion, we obtain the desired classical Lagrangian corresponding to the first class Hamiltonian which can be reduced to the generalized Stückelberg Lagrangian which is a nontrivial conjecture in our infinitely many terms involved in the Hamiltonian and the Lagrangian.

A New Lagrangian Formulation of Dynamics for Robot Manipulators

1989 ◽  
Vol 111 (4) ◽  
pp. 559-566 ◽  
Author(s):  
Chang-Jin Li
Keyword(s):  
Control System ◽  
Dynamic Simulation ◽  
Degrees Of Freedom ◽  
Robot Manipulator ◽  
Robot Manipulators ◽  
Equations Of Motion ◽  
Lagrangian Formulation ◽  
Lagrangian Dynamics ◽  
Highly Nonlinear ◽  
Mathematical Operations

In this paper, a new Lagrangian formulation of dynamics for robot manipulators is developed. The formulation results in well structured form equations of motion for robot manipulators. The equations are an explicit set of closed form second order highly nonlinear and coupling differential equations, which can be used for both the design of the control system (or dynamic simulation) and the computation of the joint generalized forces/torques. The mathematical operations of the formulation are so few that it is possible to realize the computation of the Lagrangian dynamics for robot manipulators in real-time on a micro/mini-computer. For a robot manipulator with n degrees-of-freedom, the number of operations of the formulation is at most (6n2 + 107n − 81) multiplications and (4n2 + 102n − 86) additions; for n = 6, about 780 multiplications and 670 additions.

A Hamiltonian structure with contact geometry for the semi-geostrophic equations

1994 ◽  
Vol 272 ◽  
pp. 211-234 ◽  
Author(s):  
I. Roulstone ◽  
J. Norbury
Keyword(s):  
Hamiltonian Structure ◽  
Equations Of Motion ◽  
Contact Geometry ◽  
Hamiltonian Approach ◽  
Symplectic Algorithm ◽  
Fully Nonlinear ◽  
Functional Methods ◽  
Vorticity Advection ◽  
Nonlinear Version

A canonical Hamiltonian structure for the semi-geostrophic equations is presented and from this a reduced non-canonical Hamiltonian structure is derived, providing a fully nonlinear version of the approximate linearized vorticity advection representation. The structure of this model is described naturally within the framework of contact geometry. A Hamiltonian approach leading to a symplectic algorithm for calculating solutions to the equations of motion is formulated. Basic necessary functional methods are introduced and the Lagrangian and Eulerian kinematic structures are discussed, together with their relevance to symplectic integrating algorithms.

Sign in / Sign up

Export Citation Format

Share Document