scholarly journals Quantization of Equations of Motion

10.14311/940 ◽  
2007 ◽  
Vol 47 (2-3) ◽  
Author(s):  
D. Kochan
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Equations Of Motion ◽  
Functional Integral ◽  
Integral Approach ◽  
Classical Dynamics ◽  
World Sheet ◽  
Lagrange Equations ◽  
Standard Quantum ◽  
New Type

The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential). A new type of variational principle employing differential two-form ? is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail. 

scholarly journals DIRECT QUANTIZATION OF EQUATIONS OF MOTION: FROM CLASSICAL DYNAMICS TO TRANSITION AMPLITUDES VIA STRINGS

2010 ◽  
Vol 07 (08) ◽  
pp. 1385-1405
Author(s):  
DENIS KOCHAN
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Equations Of Motion ◽  
Functional Integral ◽  
Classical Dynamics ◽  
World Sheet ◽  
Lagrange Equations ◽  
Standard Quantum ◽  
New Type ◽  
Fundamental Physical

New method of quantization is presented. It is based on classical Newton–Lagrange equations of motion (representing the fundamental physical law of mechanics) rather than on their traditional Lagrangian and/or Hamiltonian precursors. It is shown that classical dynamics is governed by canonical two-form Ω, which embodies kinetic energy and forces acting within the system. New type of variational principle employing differential two-form Ω and "umbilical strings" is introduced. The Feynman path integral over histories of the system is then rearranged to "umbilical world-sheet" functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, world-sheet approach reduces to the standard quantum mechanics. As an example Quantum Mechanics with friction is analyzed in detail.

scholarly journals The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

2016 ◽  
Vol 24 (2) ◽  
pp. 173-193
Author(s):  
Jana Musilová ◽  
Stanislav Hronek
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Conservation Laws ◽  
Calculus Of Variations ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Stationary Points ◽  
General Description ◽  
Lagrange Equations ◽  
Jet Bundles

Abstract As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals.It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-Lagrange equations of an appropriately formulated variational principle. There are several well established geometrical theories providing a general description of variational problems of different kinds. One of the most universal and comprehensive is the calculus of variations on fibred manifolds and their jet prolongations. Among others, it includes a complete general solution of the so-called strong inverse variational problem allowing one not only to decide whether a concrete equation of motion can be obtained from a variational principle, but also to construct a corresponding variational functional. Moreover, conservation laws can be derived from symmetries of the Lagrangian defining this functional, or directly from symmetries of the equations.In this paper we apply the variational theory on jet bundles to tackle some fundamental problems of physics, namely the questions on existence of a Lagrangian and the problem of conservation laws. The aim is to demonstrate that the methods are universal, and easily applicable to distinct physical disciplines: from classical mechanics, through special relativity, waves, classical electrodynamics, to quantum mechanics.

QUANTIZATION OF NON-LAGRANGIAN SYSTEMS

2009 ◽  
Vol 24 (28n29) ◽  
pp. 5319-5340 ◽  
Author(s):  
DENIS KOCHAN
Keyword(s):  
Variational Principle ◽  
Open Systems ◽  
Transition Probability ◽  
Functional Integral ◽  
Quantum Evolution ◽  
Lagrangian Systems ◽  
Classical Dynamics ◽  
Standard Case ◽  
Calculus Of Variation ◽  
Novel Method

A novel method for quantization of non-Lagrangian (open) systems is proposed. It is argued that the essential object, which provides both classical and quantum evolution, is a certain canonical two-form defined in extended velocity space. In this setting classical dynamics is recovered from the stringy-type variational principle, which employs umbilical surfaces instead of histories of the system. Quantization is then accomplished in accordance with the introduced variational principle. The path integral for the transition probability amplitude (propagator) is rearranged to a surface functional integral. In the standard case of closed (Lagrangian) systems the presented method reduces to the standard Feynman's approach. The inverse problem of the calculus of variation, the problem of quantization ambiguity and the quantum mechanics in the presence of friction are analyzed in detail.

scholarly journals COMPLEXIFIED SIGMA-MODEL AND DUALITY

2011 ◽  
Vol 26 (22) ◽  
pp. 1631-1643
Author(s):  
J. A. NIETO
Keyword(s):  
Quantum Mechanics ◽  
Sigma Model ◽  
Equations Of Motion ◽  
Close Relation ◽  
Complex Numbers ◽  
New Type ◽  
Model Action

We show that the equations of motion associated with a complexified sigma-model action do not admit manifest dual SO (n, n) symmetry. In the process we discover new type of numbers which we called "complexoids" in order to emphasize their close relation with both complex numbers and matroids. It turns out that the complexoids allow one to consider the analogue of the complexified sigma-model action but with (1+1)-worldsheet metric, instead of Euclidean-worldsheet metric. Our observations can be useful for further developments of complexified quantum mechanics.

The quantum mechanics of non-holonomic systems

1951 ◽  
Vol 205 (1083) ◽  
pp. 583-595 ◽  
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Classical Dynamics ◽  
Holonomic System ◽  
Holonomic Systems ◽  
Classical Analogue ◽  
Heisenberg Equations ◽  
Subsequent Motion ◽  
Heisenberg Equations Of Motion

Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q (0) r , then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q (0) r .

scholarly journals Modelling of Dynamics of a Wheeled Mobile Robot with Mecanum Wheels with the use of Lagrange Equations of the Second Kind

2017 ◽  
Vol 22 (1) ◽  
pp. 81-99 ◽  
Author(s):  
Z. Hendzel ◽  
Ł. Rykała
Keyword(s):  
Mathematical Model ◽  
Kinetic Energy ◽  
Mobile Robot ◽  
Mobile Robots ◽  
Equations Of Motion ◽  
Wheeled Mobile Robot ◽  
Dynamic Equations ◽  
Lagrange Equations ◽  
Wheeled Mobile Robots ◽  
New Type

Abstract The work presents the dynamic equations of motion of a wheeled mobile robot with mecanum wheels derived with the use of Lagrange equations of the second kind. Mecanum wheels are a new type of wheels used in wheeled mobile robots and they consist of freely rotating rollers attached to the circumference of the wheels. In order to derive dynamic equations of motion of a wheeled mobile robot, the kinetic energy of the system is determined, as well as the generalised forces affecting the system. The resulting mathematical model of a wheeled mobile robot was generated with the use of Maple V software. The results of a solution of inverse and forward problems of dynamics of the discussed object are also published.

Hamiltonian Formulation for Continuous Third-order Systems Using Fractional Derivatives

2021 ◽  
Vol 14 (1) ◽  
pp. 35-47
Keyword(s):  
Variational Principle ◽  
Fractional Derivatives ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Lagrange Equations ◽  
Third Order ◽  
Derivative Operator ◽  
Hamilton Equations ◽  
Liouville Fractional Derivative ◽  
Continuous Field

Abstract: We constructed the Hamiltonian formulation of continuous field systems with third order. A combined Riemann–Liouville fractional derivative operator is defined and a fractional variational principle under this definition is established. The fractional Euler equations and the fractional Hamilton equations are obtained from the fractional variational principle. Besides, it is shown that the Hamilton equations of motion are in agreement with the Euler-Lagrange equations for these systems. We have examined one example to illustrate the formalism. Keywords: Fractional derivatives, Lagrangian formulation, Hamiltonian formulation, Euler-lagrange equations, Third-order lagrangian.

scholarly journals Power and beauty of the Lagrange equations

2020 ◽  
Vol 17 (1 Jan-Jun) ◽  
pp. 47
Author(s):  
Luis De la Peña ◽  
Ana María Cetto ◽  
Andrea Valdés-Hernández
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Integrals Of Motion ◽  
Lagrange Equations ◽  
Holonomic Constraints ◽  
Point Particles ◽  
Lagrange Formulation ◽  
Classical Particles ◽  
Advanced Treatments

The Lagrangian formulation of the equations of motion for point particles isusually presented in classical mechanics as the outcome of a series ofinsightful algebraic transformations or, in more advanced treatments, as theresult of applying a variational principle. In this paper we stress two mainreasons for considering the Lagrange equations as a fundamental descriptionof the dynamics of classical particles. Firstly, their structure can benaturally disclosed from the existence of integrals of motion, in a waythat, though elementary and easy to prove, seems to be less popular--or less frequently made explicit-- than others insupport of the Lagrange formulation. The second reason is that the Lagrangeequations preserve their form in \emph{any} coordinate system --even in moving ones, if required. Their covariant nature makes themparticularly suited to deal with dynamical problems in curved spaces orinvolving (holonomic) constraints. We develop the above and related ideas inclear and simple terms, keeping them throughout at the level of intermediatecourses in classical mechanics. This has the advantage of introducing sometools and concepts that are useful at this stage, while they may also serveas a bridge to more advanced courses.

FRACTIONAL TIME ACTION AND PERTURBED GRAVITY

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 243-247 ◽  
Author(s):  
MADHAT SADALLAH ◽  
SAMI I. MUSLIH ◽  
DUMITRU BALEANU ◽  
EQAB RABEI
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Variational Problems ◽  
Integral Function ◽  
Lagrange Equations ◽  
Action Integral

In this paper, we used the scaling concepts of Mandelbrot of fractals in variational problems of mechanical systems in order to re-write the action integral function as an integration over the fractional time. In addition, by applying the variational principle to this new fractional action, we obtained the modified Euler-Lagrange equations of motion in any fractional time of order 0 < α ≤ 1. Two examples are investigated in detail.

scholarly journals Does a Functional Integral Really Need a Lagrangian?

10.14311/1269 ◽  
2010 ◽  
Vol 50 (5) ◽  
Author(s):  
D. Kochan
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Transition Probability ◽  
Equations Of Motion ◽  
Classical System ◽  
Hamiltonian Function ◽  
Functional Integral ◽  
Integral Formulation ◽  
Path Integral Formulation ◽  
Classical Level

Path integral formulation of quantum mechanics (and also other equivalent formulations) depends on a Lagrangian and/or Hamiltonian function that is chosen to describe the underlying classical system. The arbitrariness presented in this choice leads to a phenomenon called Quantization ambiguity. For example both L1 = ˙q2 and L2 = eq˙ are suitable Lagrangians on a classical level (δL1 = δL2), but quantum mechanically they are diverse. This paper presents a simple rearrangement of the path integral to a surface functional integral. It is shown that the surface functional integral formulation gives transition probability amplitude which is free of any Lagrangian/Hamiltonian and requires just the underlying classical equations of motion. A simple example examining the functionality of the proposed method is considered.

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