Differentiability And Fractality In Dynamics Of Physical Systems

Ioan Merches, Maricel Agop

• Publisher: WSPC
• Year: 2015
• eBook ISBN:
  9789814678407
• Topic: Sciences biologiques
• Subtopic: Science générale

Using Cartan's differential 1-forms theory, and assuming that the motion variables depend on Euclidean invariants, certain dynamics of the material point and systems of material points are developed. Within such a frame, the Newtonian force as mass inertial interaction at the intragalactic scale, and the Hubble-type repulsive interaction at intergalactic distances, are developed.

The wave-corpuscle duality implies movements on curves of constant informational energy, which implies both quantizations and dynamics of velocity limits.

Analysis of motion of a charged particle in a combined field which is electromagnetic and with constant magnetism implies fractal trajectories. Mechanics of material points in a fractalic space is constructed, and various applications — fractal atom, potential well, free particle, etc. — are discussed.

Contents:

• Principles of Motion in Invariantive Mechanics
• Inertial Invariantive Motion of the Material Point
• Field Invariantive Theories
• Ondulatory Invariantive Theories. Wave-Corpuscule Duality
• Invariantive Mechanics of Systems of Material Points
• The Photon in Invariantive Ondulatory Theories
• Lagrangian Approach in Invariantive Mechanics
• Considerations on Invariantive Mechanics
• Invariantive Mechanics of Rigid Body
• Covariant Formulation of Conservation Laws in Invariantive Mechanics
• Invariantive Mechanics and Informational Energy
• Chaos Via Fractality in Gravitational Dynamical Systems
• Fractality at Small Scale. Fractal Model of the Atom
• Extended Fractal Hydrodynamic Model with an Arbitrary Fractal Dimension and Its Implications
• Theory of Fractional Scale Relativity and Some Applications

Readership: Master, PhD students and professional researchers in the field of physics.
Key Features:

• Theory of 1-forms is applied to dynamics of material points systems
• Non-differentiability as a possible property of complex systems
• The role played by informational energy in the definition of the constants of motion is discussed

Chapter 1

Principles of Motion in Invariantive Mechanics

1.1.The Euler-Lagrange and Hamilton’s equations obtained by means of exterior forms

Let

be a function of class C², called Lagrangian, where q ≡ (q¹, q², …, qⁿ) are the generalized coordinates,

are the generalized velocities, and t is the time. Let also S be the action, defined by the exterior 1-form

Theorem 1.1. The total derivative of the action with respect to time yields the Hamilton-Jacobi equation [1]

Indeed, if the functional dependence

as well as the definition of generalized momenta

are accepted, the total derivative of S with respect to time is

(N.B. Here and hereafter, the summation symbol and the summation indices shall be omitted).

Since, according to (1.3)

relations (1.7) and (1.8) lead to (1.4), where the Hamiltonian H is defined as

Theorem 1.2. Closeness of the 1-form (1.3)

reduces to Euler-Lagrange equations [1]

and the integral of motion [1]

Written explicitly, relation (1.10) becomes

According to (1.1), we can write

On the other hand, operators d and δ commute, so that

Equation (1.3) can then be written as

Using the Hamiltonian defined by (1.9) and the fact that variations δq and δt are arbitrary, equation (1.16) yields (1.11) and (1.12). If, in particular, L does not explicitly depend on time, then (1.12) reduces to the energy conservation law

As one can see, the 1-form (1.3) can also be written as

where the generalized momentum p is defined by

Theorem 1.3. Closeness of the 1-form (1.18)

reduces to Hamilton’s canonical equations [1]

together with the integral of motion [1]

Explicitating (1.20), we have

Since H = H(p, q, t), one can write

and (1.23), after some simple manipulation, becomes

Equating to zero the coefficients of variations δq, δp, and δt, equations (1.21) and (1.22) follow immediately. In particular, if H does not explicitly depend on time, (1.22) reduces to the energy conservation law (1.17).

1.2.The Cartan motion principle

The previous results allow us to operate with both Lagrange function L and Hamilton’s function H. This means that, as long as the canonical...

Table of contents

• Cover Page
• Title
• Copyright
• Dedication
• Preface
• Contents
• 1. Principles of Motion in Invariantive Mechanics
• 2. Inertial Invariantive Motion of the Material Point
• 3. Field Invariantive Theories
• 4. Ondulatory Invariantive Theories. Wave-Corpuscule Duality
• 5. Invariantive Mechanics of Systems of Material Points
• 6. The Photon in Invariantive Ondulatory Theories
• 7. Lagrangian Approach in Invariantive Mechanics
• 8. Considerations on Invariantive Mechanics
• 9. Invariantive Mechanics of Rigid Body
• 10. Covariant Formulation of Conservation Laws in Invariantive Mechanics
• 11. Invariantive Mechanics and Informational Energy
• 12. Chaos Via Fractality in Gravitational Dynamical Systems
• 13. Fractality at Small Scale. Fractal Model of the Atom
• 14. Extended Fractal Hydrodynamic Model with an Arbitrary Fractal Dimension and its Implications
• 15. Theory of Fractional Scale Relativity and Some Applications
• Index