Beyond an Input/Output Paradigm for Systems: Design Systems by Intrinsic Geometry
<p>The Cartesian reference has no defects and local geometry is the same of the global geometry. After the transformation, we have another reference that has the same properties of the original Cartesian coordinates (Definition: A system is compatible if the local geometry is the same as the global geometry).</p> "> Figure 2
<p>Intrinsic geometry of circles which derivative is <span class="html-fig-inline" id="systems-02-00661-i008"> <img alt="Systems 02 00661 i008" src="/systems/systems-02-00661/article_deploy/html/images/systems-02-00661-i008.png"/></span> .</p> "> Figure 3
<p>The basis of the new reference is a function of the position q and time t in the Euclidean space.</p> "> Figure 4
<p>Spherical intrinsic geometry. Locally, the space is flat but globally we have a curvature for which the basis moving on the sphere is not commutative.</p> "> Figure 5
<p>Geodesic triangle and geodesic trajectories. On the surface of the intrinsic geometry (spherical geometry), the geodesics are straight lines without curvature and so are stress-free.</p> "> Figure 6
<p>Moving reference on the sphere.</p> "> Figure 7
<p>Moving reference in the electrical circuit.</p> "> Figure 8
<p>Angular displacement q = θ and mind control of initial and final positions.</p> "> Figure 9
<p>Rotation and torsion geometry.</p> "> Figure 10
<p>Change of reference or crystal medium by curved system where the center is a singularity or defect in the disclination.</p> "> Figure 11
<p>Torsion as defects in translation.</p> "> Figure 12
<p>Defects in translation or crystal dislocation.</p> "> Figure 13
<p>From compatible medium on the left, there is incompatible medium on the screen.</p> "> Figure 14
<p>Active noise control (ANC) by the algorithm or DSP for S*.</p> ">
Abstract
:1. Introduction
2. Local Intrinsic Geometry Used to Map Global Intrinsic Geometry
2.1. Change of Intrinsic Geometry by Moving Reference
2.2. Electrical Circuit and Moving Reference
2.3. Deformation and Displacement in Media with Defects for Rotation (Disclination) and Translation (Dislocation)
- (1)
- The deformed elements fit perfectly or they do not. In the latter case, we must apply a further deformation to re-compact the body. In the first case, we speak of a compatible deformation.
- (2)
- In the second case, we have an incompatible deformation. Let us imagine that during the deformation the coordinates are dragged with the medium. In the compatible deformation, the internal or intrinsic observer cannot see any difference as the Galileo internal observer for inertial system. In the incompatible deformation, the internal observer notices a change in the number of particles along a cycle in the medium as excess of holes or particles. The internal point of view is useful to find an incompatible deformation, due to the presence of defects. Mathematically, an incompatible deformation corresponds to the non-integrability of the differential form dsj where sj is the displacement. The non-integrability means that the displacement field sj (x) is multivalued, and thus discontinuities or defects arise when passing from one point to another. This fact is expressed by,
3. Incompatible Condition for Commutators and Wave Field Control by Active Secondary Sources
4. Schrödinger and Maxwell Equations Commutators and Incompatible Equations
5. Dynamic Equations with Torsion in Non-Conservative Gravity Maxwell-Like Equations
6. Conclusions
- (a)
- The description of a suitable substratum and its global and local properties on invariance;
- (b)
- The field potentials are compensative fields defined by a gauge covariant derivative. They share the global invariance properties with the substratum;
- (c)
- The calculation of the commutators of the covariant derivatives in (b) provides the relations between the field strength and the field potentials;
- (d)
- The Jacobi identity applied to commutators provides the dynamic equations satisfied by the field strength and the field potentials;
- (e)
- The commutator between the covariant derivatives (b) and the commutator (c) (triple Jacobian commutator) fixes the relations between field strength and field currents.
Acknowledgments
Author Contributions
Conflicts of Interest
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Appendix A
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Resconi, G.; Licata, I. Beyond an Input/Output Paradigm for Systems: Design Systems by Intrinsic Geometry. Systems 2014, 2, 661-686. https://doi.org/10.3390/systems2040661
Resconi G, Licata I. Beyond an Input/Output Paradigm for Systems: Design Systems by Intrinsic Geometry. Systems. 2014; 2(4):661-686. https://doi.org/10.3390/systems2040661
Chicago/Turabian StyleResconi, Germano, and Ignazio Licata. 2014. "Beyond an Input/Output Paradigm for Systems: Design Systems by Intrinsic Geometry" Systems 2, no. 4: 661-686. https://doi.org/10.3390/systems2040661