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.

Hamiltonian formulation for mechanical systems with nonholonomic constraints

2014 ◽  
Vol 11 (03) ◽  
pp. 1450017
Author(s):  
G. F. Torres del Castillo ◽  
O. Sosa-Rodríguez
Keyword(s):  
Finite Number ◽  
Mechanical System ◽  
Infinite Number ◽  
Degrees Of Freedom ◽  
Hamiltonian Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Hamilton Equations

It is shown that for a mechanical system with a finite number of degrees of freedom, subject to nonholonomic constraints, there exists an infinite number of Hamiltonians and symplectic structures such that the equations of motion can be written as the Hamilton equations, with the original constraints incorporated in the Hamiltonian structure.

Reflection symmetric formulation of generalized fractional variational calculus

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Małgorzata Klimek ◽  
Maria Lupa
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Equations Of Motion ◽  
Variational Calculus ◽  
Reflection Symmetry ◽  
Lagrange Equations ◽  
Representation Formulas ◽  
Symmetry Properties ◽  
Symmetric Formulation ◽  
Generalized Fractional Derivatives

AbstractWe define generalized fractional derivatives (GFDs) symmetric and anti-symmetric w.r.t. the reflection symmetry in a finite interval. Arbitrary functions are split into parts with well defined reflection symmetry properties in a hierarchy of intervals [0, b/2m], m ∈ ℕ0. For these parts — [J]-projections of function, we derive the representation formulas for generalized fractional operators (GFOs) and examine integration properties. It appears that GFOs can be reduced to operators determined in subintervals [0, b/2m]. The results are applied in the derivation of Euler-Lagrange equations for action dependent on Riemann-Liouville type GFDs. We show that for Lagrangian being a sum (finite or not) of monomials, the obtained equations of motion can be localized in arbitrary short subinterval [0, b/2m].

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 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.

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.

Fractional Hamilton’s equations of motion in fractional time

Open Physics ◽  
2007 ◽  
Vol 5 (4) ◽  
Author(s):  
Sami Muslih ◽  
Dumitru Baleanu ◽  
Eqab Rabei
Keyword(s):  
Fractional Derivatives ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Mechanical Systems ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

AbstractThe Hamiltonian formulation for mechanical systems containing Riemman-Liouville fractional derivatives are investigated in fractional time. The fractional Hamilton’s equations are obtained and two examples are investigated in detail.

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.

Hamiltonian mechanics

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Configuration Space ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Hamiltonian Mechanics ◽  
Lagrange Equations ◽  
First Order ◽  
Point Transformations ◽  
First Order Differential Equations

This chapter gives a brief overview of Hamiltonian mechanics. The complexity of the Newtonian equations of motion for N interacting bodies led to the development in the late 18th and early 19th centuries of a formalism that reduces these equations to first-order differential equations. This formalism is known as Hamiltonian mechanics. This chapter shows how, given a Lagrangian and having constructed the corresponding Hamiltonian, Hamilton’s equations amount to simply a rewriting of the Euler–Lagrange equations. The feature that makes the Hamiltonian formulation superior is that the dimension of the phase space is double that of the configuration space, so that in addition to point transformations, it is possible to perform more general transformations in order to simplify solving the equations of motion.

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. 

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