A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (II)

2011 ◽  
Vol 08 (05) ◽  
pp. 969-983
Author(s):  
AUREL BEJANCU
Keyword(s):  
Mechanical System ◽  
Riemannian Geometry ◽  
Classical Mechanics ◽  
Linear Connection ◽  
Nonholonomic Mechanics ◽  
Riemannian Connection ◽  
The One

We show that the sub-Riemannian connection ∇, constructed in the first part of our study [2], is the one which enables us to express the Lagrange–d'Alembert equations for a nonholonomic mechanical system in a form that is similar to the Newton's equations from classical mechanics (cf. (2.21)). Also, we define the generalized Chaplygin systems and show that the Lagrange–d'Alembert equations for these systems depend only on some horizontal variables. Two examples are given to illustrate our theory in comparison with other studies.

A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (I)

2011 ◽  
Vol 08 (04) ◽  
pp. 725-752 ◽  
Author(s):  
AUREL BEJANCU
Keyword(s):  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Ricci Tensor ◽  
Differential Operators ◽  
Linear Connection ◽  
Riemannian Metric ◽  
Riemannian Connection ◽  
Schur Theorem ◽  
Horizontal Curvature ◽  
Horizontal And Vertical Distribution

We study the geometry of a sub-Riemannian manifold (M, HM, VM, g), where HM and VM are the horizontal and vertical distribution respectively, and g is a Riemannian extension of the Riemannian metric on HM. First, without the assumption that HM and VM are orthogonal, we construct a sub- Riemannian connection ▽ on HM and prove some Bianchi identities for ▽. Then, we introduce the horizontal sectional curvature, prove a Schur theorem for sub-Riemannian geometry and find a class of sub-Riemannian manifolds of constant horizontal curvature. Finally, we define the horizontal Ricci tensor and scalar curvature, and some sub-Riemannian differential operators (gradient, divergence, Laplacian), extending some results from geometry to the sub-Riemannian setting.

scholarly journals Torsion and the second fundamental form for distributions

2016 ◽  
Vol 24 (1) ◽  
pp. 23-28
Author(s):  
Geoff Prince
Keyword(s):  
Fundamental Form ◽  
Bilinear Form ◽  
Riemannian Geometry ◽  
Linear Connection ◽  
Shape Operator ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Codimension One ◽  
Integrable Distribution

Abstract The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one case generalises the traditional shape operator of Riemannian geometry.

Identity in Physics

Philosophy ◽  
2014 ◽  
Author(s):  
Décio Krause ◽  
Jonas R. B. Arenhart
Keyword(s):  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Structural Realism ◽  
Exclusion Principle ◽  
Qualitative Properties ◽  
Identity Of Indiscernibles ◽  
Quantum Particles ◽  
Controversial Topic ◽  
The Difference ◽  
The One

Traditionally, the problem of identity is closely associated with the problem of individuality: What is it that makes something being what it is? Approaches to the problem may be classified into two classes: reductionism and transcendental identity. The first group tries to reduce identity to some qualitative feature of the entities dealt with, while the second either grounds identity on some feature other than qualitative properties or else take it to be primitive. The debate is generally centred on the validity of the Principle of the Identity of Indiscernibles (PII), which states that qualitative indiscernibility amounts to numerical identity. If PII is valid, then reductionism concerning identity is at least viable; if PII is invalid, then reductionism seems less plausible and some form of transcendental identity seems required. It is common to say that objects in classical mechanics are individuals. This fact is exhibited by postulating that physical objects obey Maxwell-Boltzmann statistics; if we have containers A and B to accommodate two objects a and b, there are four equiprobable situations: (1) both objects in A, (2) both in B, (3) a in A and b in B, and finally (4) a in B and b in A. Since situations (3) and (4) differ, there may be something that makes the difference—a transcendental individuality or some quality. In quantum mechanics, assuming that we have two containers A and B to accommodate objects a and b, there are just three equiprobable situations for bosons: (1) both objects in A, (2) both in B, (3) one object in A and one in B. It makes no sense to say that it is a or b that is in A: Switching them makes no difference. For fermions we have only one possibility due to the exclusion principle: (1) one object in A and one in B. Again, switching them makes no difference whatsoever. The dispute in quantum mechanics concerns non-individuality on the one side and individuality (be it reductionism or transcendental individuality) on the other. That distinction was grounded on the fact that quantum particles may be qualitatively indiscernible, and, as the statistics show, permutations are unobservable. The actual debate concerns whether some form of reductionism may survive in quantum mechanics or whether some form of transcendental identity should be adopted on the one hand and whether non-individuality is a viable option. Furthermore, a third option, Ontic Structural Realism (OSR), proposes that we transcend the debate and choose a metaphysics of structures and relations, leaving the controversial topic individuals × non-individuals behind.

On the quantum-mechanical treatment of the optics of crystal lattices

1949 ◽  
Vol 45 (3) ◽  
pp. 452-462 ◽  
Author(s):  
Kun Huang
Keyword(s):  
Electromagnetic Field ◽  
Local Field ◽  
Mechanical Treatment ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Electric Moment ◽  
Crystal Lattices ◽  
Optical Wave ◽  
Quantum Mechanical Treatment ◽  
The One

In Ewald's theory of crystal optics, based on classical principles, an optical wave through a crystal lattice of polarizable atoms (idealized as isotropic oscillators) is a self-sustaining system of vibrations so constituted that, on the one hand, the electric moment of each atom is caused to oscillate by the electromagnetic field and, on the other hand, the electromagnetic field is itself the resultant field due to the superposition of the dipole waves produced by the lattice atoms. Born extended the theory to the case of movable lattice ions and showed that in an optical wave the electromagnetic field is so coupled to the lattice vibrations that each lattice ion vibrates in phase with the local field which, as in Ewald's case, is itself produced by the vibrating ions. In Born's original theory, the motion of the lattice particles had to be treated by classical mechanics. It is shown that the results of the quantum-mechanical treatment of the lattice motion agrees exactly with the classical theory. Not only is the induced current at each lattice point in phase with the local field, but the magnitude of the current is also identical with the classical value, completely independent of whichever vibrational state the crystal might be in.

DEFORMATION OF THE QUANTUM MECHANICAL PHASE SPACE WITH BOSONIC OR FERMIONIC COORDINATES

1991 ◽  
Vol 06 (13) ◽  
pp. 1225-1235 ◽  
Author(s):  
BRUNO ZUMINO
Keyword(s):  
Phase Space ◽  
Quantum Group ◽  
Mechanical System ◽  
Field Theories ◽  
Quantum Mechanical ◽  
Quantum Mechanical System ◽  
The Many ◽  
The One ◽  
Parameter Case ◽  
Parameter Deformation

We describe the one-parameter deformation of the phase space of a quantum mechanical system and show that this twisted phase space is covariant under the action of the symplectic quantum group. The analogous case of a system with fermionic coordinates is also considered and the phase space is shown to be covariant under the action of the orthogonal quantum group. Twisted commutation relations occur in the description of deformed spaces or superspaces as well as in the formulation of field theories with generalized statistics. The many-parameter case is briefly discussed.

Performance Comparison of Typical Electrothermal Actuators

2013 ◽  
Vol 562-565 ◽  
pp. 528-533
Author(s):  
Ling Zhou ◽  
Xue Jin Shen ◽  
Zhen Lu Wang ◽  
Yi Hu
Keyword(s):  
Mechanical System ◽  
Mechanical Performance ◽  
Beam Width ◽  
Micro Electro Mechanical System ◽  
Performance Comparison ◽  
Secondary Beam ◽  
Electrothermal Actuators ◽  
Micro Actuator ◽  
Mechanical Models ◽  
The One

As the power producer, the micro actuator is an important part of the micro electro- mechanical system, which is used as the conversion of energy, motion and force producing. In this paper, the mechanical models of three bent beam electrothermal actuators are summarized to make comparison with each other and analyze mechanical performance. For Π-shaped actuators, the tip displacements are proportional to temperature increment and length, having no relationship with the beam width and thickness. However, the displacement of Π-shaped actuators is smaller than the one of V-shaped actuators with the same parameters. For V-shaped actuators, the tip displacements of cascaded V-shaped actuators are larger than the ones of single V-shaped actuators with the same parameters. Especially, cascaded V-shaped actuator with secondary beam having no current has the largest displacement.

Polyrhythmic multifrequency synchronization in coupled oscillators with exactly solvable attractors

2021 ◽  
Vol 35 (03) ◽  
pp. 2150047
Author(s):  
Till D. Frank
Keyword(s):  
Limit Cycle ◽  
Activity Patterns ◽  
Music Performance ◽  
Classical Mechanics ◽  
The Other ◽  
Dissipative Systems ◽  
Other Hand ◽  
Generic Bifurcation ◽  
The One ◽  
Systems Framework

While stable polyrhythmic multifrequency [Formula: see text] dynamics has traditionally been an important element in music performance, recently, this type of dynamics has been discovered in the human brain in terms of elementary temporal neural activity patterns. In this context, the canonical-dissipative systems framework is a promising modeling approach due to its two key features to bridge the gap between classical mechanics and life sciences, on the one hand, and to provide analytical or semi-analytical solutions, on the other hand. Within this framework, a family of testbed models is constructed that exhibit [Formula: see text] multifrequency limit cycle attractors describing two components oscillating with frequencies at [Formula: see text] ratios and stable polyrhythmic phase relationships. The attractors are super-integrable due to the existence of third invariants of motion for all [Formula: see text] ratios. Strikingly, all [Formula: see text] attractors models satisfy the same generic bifurcation diagram. The study generalizes earlier work on super-integrable systems, on the one hand, and canonical-dissipative limit cycle oscillators, on the other hand. Explicit worked-out models for 1[Formula: see text]:[Formula: see text]4 and 2[Formula: see text]:[Formula: see text]3 frequency ratios are presented.

scholarly journals TIME AND GEOMETRIC QUANTIZATION

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2347-2354 ◽  
Author(s):  
A. A. ABRIKOSOV ◽  
E. GOZZI ◽  
D. MAURO
Keyword(s):  
Geometric Quantization ◽  
Classical Mechanics ◽  
Von Neumann ◽  
Functional Version ◽  
The One

In this paper we briefly review the functional version of the Koopman-von Neumann operatorial approach to classical mechanics. We then show that its quantization can be achieved by freezing to zero two Grassmannian partners of time. This method of quantization presents many similarities with the one known as Geometric Quantization.

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