scholarly journals A Differential Geometry Associated with Dissipative Systems

1965 ◽  
Vol 8 (4) ◽  
pp. 433-451 ◽  
Author(s):  
M. A. McKiernan
Keyword(s):  
Differential Geometry ◽  
Calculus Of Variations ◽  
Dissipative Systems ◽  
Image Position ◽  
Problem Of Lagrange

Consider the following problem of Lagrange in the calculus of variations: relative to differentiable curves xi(t) satisfying xi(t0) = xi0 and xi(t1) = xi1 find a curve minimizing1

scholarly journals THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

2018 ◽  
Vol 83 (04) ◽  
pp. 1667-1679
Author(s):  
MATÍAS MENNI
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Double Negation ◽  
Synthetic Differential Geometry ◽  
Simplicial Sets ◽  
Image Position

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

A General Hamilton-Jacobi Equation and Associated Problem of Lagrange

1965 ◽  
Vol 8 (3) ◽  
pp. 317-321 ◽  
Author(s):  
M. A. McKiernan
Keyword(s):  
Variational Problem ◽  
Jacobi Equation ◽  
Image Position ◽  
Hamilton Jacobi Equation ◽  
Problem Of Lagrange

It is well known [1] that the variational problem of minimizing1where F is positive homogeneous of degree one in (henceforth abbreviated to "plus-one" in ) leads to a Hamiltonian H{xi, pi} and corresponding Hamilton-Jacobi equation2

The general double integral problem in the calculus of variations

1933 ◽  
Vol 29 (2) ◽  
pp. 207-211
Author(s):  
R. P. Gillespie
Keyword(s):  
Calculus Of Variations ◽  
General Problem ◽  
Necessary Condition ◽  
Double Integral ◽  
Integral Problem ◽  
Image Position

In a previous paper in these Proceedings the problem of the double integralwas discussed when the function F had the formwhereIt is proposed in the present paper to extend the method to the general problem, where F may have any form provided only that it satisfies the necessary condition of being homogeneous of the first degree in A, B, C.

scholarly journals Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jacky Cresson ◽  
Fernando Jiménez ◽  
Sina Ober-Blöbaum
Keyword(s):  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Fractional Derivatives ◽  
Dissipative Systems ◽  
Alternative Formulation ◽  
Conserved Quantities ◽  
Lagrange Equations ◽  
Discrete Counterpart ◽  
Fractional Calculus Of Variations

<p style='text-indent:20px;'>We prove a Noether's theorem of the first kind for the so-called <i>restricted fractional Euler-Lagrange equations</i> and their discrete counterpart, introduced in [<xref ref-type="bibr" rid="b26">26</xref>,<xref ref-type="bibr" rid="b27">27</xref>], based in previous results [<xref ref-type="bibr" rid="b11">11</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. Prior, we compare the restricted fractional calculus of variations to the <i>asymmetric fractional calculus of variations</i>, introduced in [<xref ref-type="bibr" rid="b14">14</xref>], and formulate the restricted calculus of variations using the <i>discrete embedding</i> approach [<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b18">18</xref>]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.</p>

VIII.—The Invariant Theory of the Quaternary Quadratic Complex. I. The Prepared System

1929 ◽  
Vol 48 ◽  
pp. 70-91
Author(s):  
H. W. Turnbull
Keyword(s):  
Differential Geometry ◽  
Quadratic Form ◽  
Invariant Theory ◽  
Complex I ◽  
Quadratic Complex ◽  
Image Position ◽  
Quaternary Quadratic Form ◽  
Associated Variables

Projective and differential geometry are in close touch at two places, once because of the fundamental rôle played by a quaternary quadratic form in each,and again through the quadratic in six associated variables,where

THE FRACTIONAL CALCULUS OF VARIATIONS FROM EXTENDED ERDÉLYI-KOBER OPERATOR

2009 ◽  
Vol 23 (16) ◽  
pp. 3349-3361 ◽  
Author(s):  
AHMAD RAMI EL-NABULSI
Keyword(s):  
Fractional Calculus ◽  
Harmonic Oscillator ◽  
Calculus Of Variations ◽  
Fractional Derivatives ◽  
Time Dependent ◽  
Dissipative Systems ◽  
Hamilton Equations ◽  
Different Types ◽  
Fractional Calculus Of Variations ◽  
Dependent Mass

Fractional calculus of variations (FCV) has recently attracted considerable attention as it is deeply related to the fractional quantization procedure. In this work, the FCV from extended Erdélyi-Kober fractional integral is constructed. Our main goal is to exhibit a general treatment for dissipative systems, in particular the harmonic oscillator (HO) that has time-dependent mass and time-dependent frequency. The general linear equation of damped Erdélyi-Kober harmonic oscillator is constructed from which a time-dependent mass generalized law was derived exhibiting different types of behavior. This relatively new time-dependent mass law permits us to point out several possible cases simultaneously in contrast to many models discussed in the literature and without making use of any types of fractional derivatives. Some results on Hamiltonian part, namely Hamilton equations for the damped HO were obtained and discussed in detail.

Sign in / Sign up

Export Citation Format

Share Document