We apply Cartan’s method of equivalence to the case of nonholonomic geometry on three-dimensional... more We apply Cartan’s method of equivalence to the case of nonholonomic geometry on three-dimensional contact manifolds. Our main result is to derive the differential invariants for these structures and give geometric interpretations. We show that the symmetry group of such a structure has dimension at most four. Our motivation is to study the geometry associated with classical mechanical systems with nonholonomic constraints.
Micro-engineering pumping devices without mechani- cal parts appeared "way back" in the... more Micro-engineering pumping devices without mechani- cal parts appeared "way back" in the early 1990's. The working prin- ciple is acoustic streaming. Has Nature "rediscovered" this invention 2.7 Gyr ago? Strands of marine cyanobacteria Synechococcus swim 25 diameters per second without any visible means of propulsion. We show that nanoscale amplitude vibrations on the S-layer (a crys- talline shell outside the outer membrane present in motile strands) and frequencies of the order of 0.5-1.5 MHz (achievable by molecu- lar motors), could produce steady streaming slip velocities outside a (Stokes) boundary layer. Inside this boundary layer the flow pattern is rotational (hence biologically advantageous). In addition to this purported "swimming by singing", we also indicate other possible instantiations of acoustic streaming. Sir James Lighthill has propo- sed that acoustic streaming occurs in the cochlear dynamics, and new findings on the outer hair cel...
In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of... more In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of equivalence? can provide the invariants of nonholonomic systems on a manifold ?? with kinetic lagrangians [29]. Cartan indicated which changes of the metric outside the constraint distribution ?? ? ???? preserve the nonholonomic connection ?????? = Proj?? ?????, ??,?? ? ??, where ????? is the Levi-Civita connection on ?? and Proj?? is the orthogonal projection over ??. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems [30,31,62]. We also discuss an example-a mathematical gem!-found by Oliva and Terra [76]. It implies that there is more freedom (thus more opportunities) using a weaker equivalence, just to preserve the straightest paths: ?????? = 0. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide i...
Certain cyanobacteria, such as open ocean strains of Synechococcus, are able to swim at speeds up... more Certain cyanobacteria, such as open ocean strains of Synechococcus, are able to swim at speeds up to 25 diameters per second, without flagella or visible changes in shape. The means by which Synechococcus generates thrust for self-propulsion is unknown. The only mechanism that has not been ruled out employs tangential waves of surface deformations. In Ehlers et al, the average swimming velocity for this mechanism was estimated using the methods inaugurated by Taylor and Lighthill in the 1950's and revisited in differential geometric language by Shapere and Wilczek in 1989. In this article we propose an entirely different physical principle self propulsion based on acoustic streaming (AS). Micro-pumps in silicon chips, based on AS, have been constructed by engineers since the 1990's, but to the best of our knowledge acoustic streaming as a means of microorganisms locomotion has not been proposed before. Our hypothesis is supported by two recent discoveries: (1) In Samuel, et ...
Hamiltonian Systems and Celestial Mechanics (HAMSYS-98) - Proceedings of the III International Symposium, 2000
We have been interested in nonholonomic motion and in microswimming since two of us (JK and RM) l... more We have been interested in nonholonomic motion and in microswimming since two of us (JK and RM) listened an inspiring talk by Frank Wilczek at Cornell, about ten years ago. Our general program was presented in 3 and in particular, we described the collective" N-body ...
... Montgomery and Bor give an explicit description of the infinitesimal action of G∈ on the ball... more ... Montgomery and Bor give an explicit description of the infinitesimal action of G∈ on the ball-ball system and Zelenko gave a geometric interpretation of the fundamental differential invariant constructed by Cartan. ... SERGEI B. MEDVEDEV1, VLADIMIR ZEITLIN2 ...
Page 1. . Page 2. Spectral methods for Stokes flows: the Lorentz operator ∗ Kurt M.Ehlers†,Joaqui... more Page 1. . Page 2. Spectral methods for Stokes flows: the Lorentz operator ∗ Kurt M.Ehlers†,Joaquin Delgado Fernandez‡, Jair Koiller and Marco Antonio Raupp Laboratório Nacional de Computac˜ao Cientıfica R.Lauro Muller 455, Rio de Janeiro, 22290-160, RJ, Brazil and ...
... Meadows Community College 7000 Dandini Blvd, Reno, NV 89512 USA 2Escola de Matemática Aplicad... more ... Meadows Community College 7000 Dandini Blvd, Reno, NV 89512 USA 2Escola de Matemática Aplicada, Fundaç˜ao Getulio Vargas Praia ... gliding motility on surfaces by individual cells of Myxocococcus xanthus is associated with rotation of a continuous looped helical rotor. ...
Proceedings of the National Academy of Sciences, 1996
Bacteria that swim without the benefit of flagella might do so by generating longitudinal or tran... more Bacteria that swim without the benefit of flagella might do so by generating longitudinal or transverse surface waves. For example, swimming speeds of order 25 microns/s are expected for a spherical cell propagating longitudinal waves of 0.2 micron length, 0.02 micron amplitude, and 160 microns/s speed. This problem was solved earlier by mathematicians who were interested in the locomotion of ciliates and who considered the undulations of the envelope swept out by ciliary tips. A new solution is given for spheres propagating sinusoidal waveforms rather than Legendre polynomials. The earlier work is reviewed and possible experimental tests are suggested.
We apply Cartan’s method of equivalence to the case of nonholonomic geometry on three-dimensional... more We apply Cartan’s method of equivalence to the case of nonholonomic geometry on three-dimensional contact manifolds. Our main result is to derive the differential invariants for these structures and give geometric interpretations. We show that the symmetry group of such a structure has dimension at most four. Our motivation is to study the geometry associated with classical mechanical systems with nonholonomic constraints.
Micro-engineering pumping devices without mechani- cal parts appeared "way back" in the... more Micro-engineering pumping devices without mechani- cal parts appeared "way back" in the early 1990's. The working prin- ciple is acoustic streaming. Has Nature "rediscovered" this invention 2.7 Gyr ago? Strands of marine cyanobacteria Synechococcus swim 25 diameters per second without any visible means of propulsion. We show that nanoscale amplitude vibrations on the S-layer (a crys- talline shell outside the outer membrane present in motile strands) and frequencies of the order of 0.5-1.5 MHz (achievable by molecu- lar motors), could produce steady streaming slip velocities outside a (Stokes) boundary layer. Inside this boundary layer the flow pattern is rotational (hence biologically advantageous). In addition to this purported "swimming by singing", we also indicate other possible instantiations of acoustic streaming. Sir James Lighthill has propo- sed that acoustic streaming occurs in the cochlear dynamics, and new findings on the outer hair cel...
In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of... more In a note at the 1928 International Congress of Mathematicians Cartan outlined how his ?method of equivalence? can provide the invariants of nonholonomic systems on a manifold ?? with kinetic lagrangians [29]. Cartan indicated which changes of the metric outside the constraint distribution ?? ? ???? preserve the nonholonomic connection ?????? = Proj?? ?????, ??,?? ? ??, where ????? is the Levi-Civita connection on ?? and Proj?? is the orthogonal projection over ??. Here we discuss this equivalence problem of nonholonomic connections for Chaplygin systems [30,31,62]. We also discuss an example-a mathematical gem!-found by Oliva and Terra [76]. It implies that there is more freedom (thus more opportunities) using a weaker equivalence, just to preserve the straightest paths: ?????? = 0. However, finding examples that are weakly but not strongly equivalent leads to an over-determined system of equations indicating that such systems should be rare. We show that the two notions coincide i...
Certain cyanobacteria, such as open ocean strains of Synechococcus, are able to swim at speeds up... more Certain cyanobacteria, such as open ocean strains of Synechococcus, are able to swim at speeds up to 25 diameters per second, without flagella or visible changes in shape. The means by which Synechococcus generates thrust for self-propulsion is unknown. The only mechanism that has not been ruled out employs tangential waves of surface deformations. In Ehlers et al, the average swimming velocity for this mechanism was estimated using the methods inaugurated by Taylor and Lighthill in the 1950's and revisited in differential geometric language by Shapere and Wilczek in 1989. In this article we propose an entirely different physical principle self propulsion based on acoustic streaming (AS). Micro-pumps in silicon chips, based on AS, have been constructed by engineers since the 1990's, but to the best of our knowledge acoustic streaming as a means of microorganisms locomotion has not been proposed before. Our hypothesis is supported by two recent discoveries: (1) In Samuel, et ...
Hamiltonian Systems and Celestial Mechanics (HAMSYS-98) - Proceedings of the III International Symposium, 2000
We have been interested in nonholonomic motion and in microswimming since two of us (JK and RM) l... more We have been interested in nonholonomic motion and in microswimming since two of us (JK and RM) listened an inspiring talk by Frank Wilczek at Cornell, about ten years ago. Our general program was presented in 3 and in particular, we described the collective" N-body ...
... Montgomery and Bor give an explicit description of the infinitesimal action of G∈ on the ball... more ... Montgomery and Bor give an explicit description of the infinitesimal action of G∈ on the ball-ball system and Zelenko gave a geometric interpretation of the fundamental differential invariant constructed by Cartan. ... SERGEI B. MEDVEDEV1, VLADIMIR ZEITLIN2 ...
Page 1. . Page 2. Spectral methods for Stokes flows: the Lorentz operator ∗ Kurt M.Ehlers†,Joaqui... more Page 1. . Page 2. Spectral methods for Stokes flows: the Lorentz operator ∗ Kurt M.Ehlers†,Joaquin Delgado Fernandez‡, Jair Koiller and Marco Antonio Raupp Laboratório Nacional de Computac˜ao Cientıfica R.Lauro Muller 455, Rio de Janeiro, 22290-160, RJ, Brazil and ...
... Meadows Community College 7000 Dandini Blvd, Reno, NV 89512 USA 2Escola de Matemática Aplicad... more ... Meadows Community College 7000 Dandini Blvd, Reno, NV 89512 USA 2Escola de Matemática Aplicada, Fundaç˜ao Getulio Vargas Praia ... gliding motility on surfaces by individual cells of Myxocococcus xanthus is associated with rotation of a continuous looped helical rotor. ...
Proceedings of the National Academy of Sciences, 1996
Bacteria that swim without the benefit of flagella might do so by generating longitudinal or tran... more Bacteria that swim without the benefit of flagella might do so by generating longitudinal or transverse surface waves. For example, swimming speeds of order 25 microns/s are expected for a spherical cell propagating longitudinal waves of 0.2 micron length, 0.02 micron amplitude, and 160 microns/s speed. This problem was solved earlier by mathematicians who were interested in the locomotion of ciliates and who considered the undulations of the envelope swept out by ciliary tips. A new solution is given for spheres propagating sinusoidal waveforms rather than Legendre polynomials. The earlier work is reviewed and possible experimental tests are suggested.
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Papers by Kurt Ehlers