Pith. sign in

REVIEW 1 minor 1 cited by

Nonreciprocal slender bodies form a shape-space flow whose activity instabilities produce rigid, swimming, and chaotic motion.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-06-27 08:16 UTC pith:Q4QUJ4BN

load-bearing objection The paper derives a shape-space formulation for nonreciprocal slender bodies that frames their elastohydrodynamics as a geometric flow with activity-driven instabilities, but the abstract supplies no equations to inspect.

arxiv 2606.11807 v1 pith:Q4QUJ4BN submitted 2026-06-10 cond-mat.soft

Shape-space dynamics and geometric pattern formation in nonreciprocal slender bodies

Bal\'azs N\'emeth , Mohamed Warda , Ronojoy Adhikari This is my paper
classification cond-mat.soft
keywords nonreciprocal interactionsslender bodieselastohydrodynamicsshape-space dynamicsgeometric pattern formationactive matterreaction-advection-diffusionactivity-driven instabilities
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a shape-space formulation for the elastohydrodynamics of nonreciprocal slender bodies by assuming invariance under Euclidean symmetries, which separates intrinsic deformation from rigid motion. This yields a nonlinear reaction-advection-diffusion system that functions as a geometric flow. Activity-driven instabilities in the flow then generate steady, oscillatory, and chaotic patterns that appear as rigid, swimming, and chaotic motion. A sympathetic reader would care because the work unifies observations of active slender structures by connecting nonreciprocal interactions directly to geometric pattern formation.

Core claim

Assuming invariance under Euclidean symmetries, the authors derive a shape-space formulation for the elastohydrodynamics of nonreciprocal slender bodies that separates intrinsic deformation from rigid motion. The resulting nonlinear reaction-advection-diffusion system represents a geometric flow whose activity-driven instabilities generate steady, oscillatory, and chaotic patterns. These manifest as rigid, swimming, and chaotic motion, linking nonreciprocal elastohydrodynamics to geometric pattern formation and unifying recent observations in slender active structures.

What carries the argument

The shape-space formulation derived from Euclidean symmetry invariance, which separates intrinsic deformation from rigid motion and produces the nonlinear reaction-advection-diffusion system as a geometric flow.

Load-bearing premise

The system remains invariant under Euclidean symmetries, allowing separation of intrinsic deformation from rigid motion in the shape-space description.

What would settle it

Direct numerical simulation or experiment on nonreciprocal slender filaments showing whether the predicted steady, oscillatory, and chaotic patterns appear under the derived reaction-advection-diffusion equations.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Activity-driven instabilities produce steady patterns that correspond to rigid motion of the bodies.
  • Oscillatory instabilities generate swimming motion in the nonreciprocal slender structures.
  • Chaotic patterns in the geometric flow lead to chaotic motion.
  • The formulation unifies observations across different slender active structures through the common geometric flow mechanism.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same shape-space reduction could be tested on active filaments of varying lengths or stiffness to check pattern robustness.
  • Tuning the nonreciprocal strength parameter might allow controlled switching between rigid, swimming, and chaotic regimes in applications.
  • The geometric flow description may connect to pattern formation in other nonreciprocal systems such as active sheets or networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The manuscript derives a shape-space formulation for the elastohydrodynamics of nonreciprocal slender bodies by assuming invariance under Euclidean symmetries. This separates intrinsic deformation from rigid motion and yields a nonlinear reaction-advection-diffusion system. The authors argue that activity-driven instabilities in this geometric flow generate steady, oscillatory, and chaotic patterns that manifest as rigid, swimming, and chaotic motion, thereby linking nonreciprocal elastohydrodynamics to geometric pattern formation and unifying observations in slender active structures.

Significance. If the central derivation holds, the work supplies a geometric-mechanics framework that connects nonreciprocal force laws to pattern-forming instabilities in filaments. The explicit reduction to a reaction-advection-diffusion system on shape space and the classification of resulting rigid/swimming/chaotic regimes constitute a concrete advance that could organize existing experimental reports on active slender bodies.

minor comments (1)
  1. The abstract is information-dense; expanding the introduction to include a brief schematic of the shape-space reduction (e.g., the decomposition into intrinsic and rigid components) would improve accessibility without altering the technical content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the central derivation, and recommendation to accept. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from the standard assumption of Euclidean invariance to separate shape dynamics from rigid motion, which is an independent physical premise not derived from the target result. The abstract and available text present the nonlinear reaction-advection-diffusion system as a consequence of this assumption followed by analysis of its instabilities, without any quoted reduction of predictions to fitted inputs, self-definitional loops, or load-bearing self-citations. The central claims remain independent of the outputs they describe.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Euclidean-invariance assumption used to separate deformation from rigid motion and on the subsequent derivation of the reaction-advection-diffusion system; no free parameters, invented entities, or additional axioms are visible in the abstract.

axioms (1)
  • domain assumption Invariance under Euclidean symmetries
    Invoked to derive the shape-space formulation separating intrinsic deformation from rigid motion.

pith-pipeline@v0.9.1-grok · 5621 in / 1237 out tokens · 22302 ms · 2026-06-27T08:16:49.719856+00:00 · methodology

0 comments
read the original abstract

Nonreciprocal interactions in active solids violate action-reaction symmetry and produce a net response to strain. Assuming invariance under Euclidean symmetries, we derive a shape-space formulation for the elastohydrodynamics of nonreciprocal slender bodies that separates intrinsic deformation from rigid motion. The resulting nonlinear reaction-advection-diffusion system represents a geometric flow whose activity-driven instabilities generate steady, oscillatory, and chaotic patterns. These manifest as rigid, swimming, and chaotic motion, linking nonreciprocal elastohydrodynamics to geometric pattern formation and unifying recent observations in slender active structures.

Figures

Figures reproduced from arXiv: 2606.11807 by Bal\'azs N\'emeth, Mohamed Warda, Ronojoy Adhikari.

Figure 1
Figure 1. Figure 1: Active force and torque densities (6) and their sym [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Fixed points of shape dynamics (see Movie 1 of the SM [57] for animations of the fixed point behavior). (a1)–(a2): [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Curvature evolution of a polar, achiral rod with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Chaotic dynamics for δ = 15, β = 3.5. (a) Temporal evolution of the curvature of the midpoint of the rod. (b) Curvature kymograph of the rod. (c) Evolution of the shape of the rod at the midpoint in Π2 −Π3 space. (d) Chaotic attractor formed from scatter plot of dynamics projected onto the amplitudes of the first three dominant eigenmodes obtained from PCA. (e1)–(e4) Snapshots of chaotic rod dynamics. See … view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Tuning nonlinear waves in nonreciprocal active filaments

    cond-mat.soft 2026-07 unverdicted novelty 5.0

    A geometrically exact theory of nonreciprocal filaments shows nonreciprocity coupled to inertia or pre-stress amplifies and advects curvature variations, allowing selection of one-way shape morphing patterns via dissi...

Reference graph

Works this paper leans on

75 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    Gray and G

    J. Gray and G. J. Hancock, The Propulsion of Sea-Urchin Spermatozoa, Journal of Experimental Biology32, 802 (1955)

    1955

  2. [2]

    Audoly and Y

    B. Audoly and Y. Pomeau, eds.,Elasticity and geome- try: from hair curls to the non-linear response of shells (Oxford University Press, 2010)

    2010

  3. [3]

    Audoly and S

    B. Audoly and S. Neukirch, Fragmentation of Rods by Cascading Cracks: Why Spaghetti Does Not Break in Half, Physical Review Letters95, 095505 (2005)

    2005

  4. [4]

    W. F. Baker and A. McRobie, eds.,The Geometry of Equilibrium: James Clerk Maxwell and 21st-Century Structural Mechanics, 1st ed. (Cambridge University Press, 2025)

    2025

  5. [5]

    Euler,Methodus inveniendi lineas curvas maximi minive proprietate gaudentes(Bousquet, Lausanne & Geneva, 1744)

    L. Euler,Methodus inveniendi lineas curvas maximi minive proprietate gaudentes(Bousquet, Lausanne & Geneva, 1744)

  6. [6]

    Kirchhoff, Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, Journal für die reine und angewandte Mathematik56, 285 (1859)

    G. Kirchhoff, Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, Journal für die reine und angewandte Mathematik56, 285 (1859)

  7. [7]

    Timoshenko, LXVI

    S. Timoshenko, LXVI. On the correction for shear of the differentialequationfortransversevibrationsofprismatic bars, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science41, 744 (1921)

    1921

  8. [8]

    J. L. Ericksen and C. Truesdell, Exact theory of stress and strain in rods and shells, Archive for Rational Me- chanics and Analysis1, 295 (1957)

    1957

  9. [9]

    Goriely and M

    A. Goriely and M. Tabor, The Nonlinear Dynamics of Filaments, Nonlinear Dynamics21, 101 (2000)

    2000

  10. [10]

    Antman,Nonlinear Problems of Elasticity(Springer, New York, 2004)

    S. Antman,Nonlinear Problems of Elasticity(Springer, New York, 2004)

    2004

  11. [11]

    D. E. Moulton, H. Oliveri, and A. Goriely, Multiscale in- tegration of environmental stimuli in plant tropism pro- duces complex behaviors, Proceedings of the National Academy of Sciences117, 32226 (2020)

    2020

  12. [12]

    Sartori, V

    P. Sartori, V. F. Geyer, J. Howard, and F. Jülicher, Cur- vature regulation of the ciliary beat through axonemal twist, Physical Review E94, 042426 (2016)

    2016

  13. [13]

    Chakrabarti and D

    B. Chakrabarti and D. Saintillan, Spontaneous oscilla- tions, beating patterns, and hydrodynamics of active mi- crofilaments, Physical Review Fluids4, 043102 (2019)

    2019

  14. [14]

    J. M. Rieser, B. Chong, C. Gong, H. C. Astley, P. E. Schiebel, K. Diaz, C. J. Pierce, H. Lu, R. L. Hatton, H. Choset, and D. I. Goldman, Geometric phase pre- dicts locomotion performance in undulating living sys- tems across scales, Proceedings of the National Academy of Sciences121, e2320517121 (2024)

    2024

  15. [15]

    Kaeser, J

    C. Kaeser, J. Kwon, E. Challita, H. Tuazon, R. J. Wood, S. Bhamla, and J. Werfel, Individual and Collec- tive Behaviors in Soft Robot Worms Inspired by Living Worm Blobs, in2025 IEEE International Conference on Robotics and Automation (ICRA)(IEEE, Atlanta, GA, USA, 2025) pp. 2577–2583

    2025

  16. [16]

    Lighthill, Flagellar Hydrodynamics, SIAM Review18, 161 (1976)

    J. Lighthill, Flagellar Hydrodynamics, SIAM Review18, 161 (1976)

    1976

  17. [17]

    C. H. Wiggins and R. E. Goldstein, Flexive and Propul- sive Dynamics of Elastica at Low Reynolds Number, Physical Review Letters80, 3879 (1998)

    1998

  18. [18]

    Lauga, Floppy swimming: Viscous locomotion of ac- tuated elastica, Physical Review E75, 041916 (2007)

    E. Lauga, Floppy swimming: Viscous locomotion of ac- tuated elastica, Physical Review E75, 041916 (2007)

    2007

  19. [19]

    M. D. Butler, B. J. Walker, T. D. Montenegro- Johnson, and P. Katsamba, Elastohydrodynamics of three-dimensional chemically active filaments, Journal of Fluid Mechanics1029, A42 (2026)

    2026

  20. [20]

    M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody- namics of soft active matter, Reviews of Modern Physics 85, 1143 (2013)

    2013

  21. [21]

    U.Makanga, A.Varma,andP.Katsamba,Instabilityand self-propulsion of flexible autophoretic filaments, Physi- cal Review Fluids11, 053101 (2026)

    2026

  22. [22]

    Maitra and S

    A. Maitra and S. Ramaswamy, Oriented Active Solids, Physical Review Letters123, 238001 (2019)

    2019

  23. [23]

    Brauns, M

    F. Brauns, M. O’Leary, A. Hernandez, M. J. Bowick, and M. C. Marchetti, Active Solids: Topological Defect Self- Propulsion Without Flow, Physical Review Letters136, 058302 (2026)

    2026

  24. [24]

    Elgeti, R

    J. Elgeti, R. G. Winkler, and G. Gompper, Physics of mi- croswimmers—single particle motion and collective be- havior: a review, Reports on Progress in Physics78, 056601 (2015)

    2015

  25. [25]

    Parthasarathy, F

    T. Parthasarathy, F. K. Chan, and M. Gazzola, Streaming-enhanced flow-mediated transport, Journal of Fluid Mechanics878, 647 (2019)

    2019

  26. [26]

    Ranzani, M

    T. Ranzani, M. Cianchetti, G. Gerboni, I. D. Falco, and A. Menciassi, A Soft Modular Manipulator for Minimally InvasiveSurgery: DesignandCharacterizationofaSingle Module, IEEE Transactions on Robotics32, 187 (2016)

    2016

  27. [27]

    Tekinalp, N

    A. Tekinalp, N. Naughton, S. H. Kim, U. Halder, R. Gillette, P. G. Mehta, W. Kier, and M. Gazzola, Topology, dynamics, and control of a muscle-architected soft arm, Proceedings of the National Academy of Sci- ences121, e2318769121 (2024)

    2024

  28. [28]

    Gopal Subramaniam, M

    A. Gopal Subramaniam, M. Kumar, S. Thutupalli, and R. Singh, Rigid flocks, undulatory gaits, and chiral foldamers in a chemically active polymer, New Journal of Physics26, 083009 (2024)

    2024

  29. [29]

    X. Chao, K. Skipper, C. P. Royall, S. Henkes, and T. B. Liverpool, Traveling Strings of Active Dipolar Colloids, Physical Review Letters134, 018302 (2025)

    2025

  30. [30]

    Wei and D

    M. Wei and D. J. Kraft, Life-like behavior emerging in active and flexible microstructures, Proceedings of the National Academy of Sciences123, e2531743123 (2026)

    2026

  31. [31]

    Brandenbourger, X

    M. Brandenbourger, X. Locsin, E. Lerner, and C.Coulais,Non-reciprocalroboticmetamaterials,Nature Communications10, 4608 (2019)

    2019

  32. [32]

    Y. Chen, X. Li, C. Scheibner, V. Vitelli, and G. Huang, Realization of active metamaterials with odd micropolar elasticity, Nature Communications12, 5935 (2021)

    2021

  33. [33]

    Gelvan, A

    M. Gelvan, A. Chirko, J. Kirpitch, Y. Lavie, N. Israel, and N. Oppenheimer, Hydrodynamic spin-pairing and active polymerization of oppositely spinning rotors, Na- ture Communications16, 10368 (2025)

    2025

  34. [34]

    Souslov, V

    J.Veenstra, C.Scheibner, M.Brandenbourger, J.Binysh, A. Souslov, V. Vitelli, and C. Coulais, Adaptive locomo- 7 tion of active solids, Nature639, 935 (2025)

    2025

  35. [35]

    Jayaraman, S

    G. Jayaraman, S. Ramachandran, S. Ghose, A. Laskar, M. S. Bhamla, P. B. S. Kumar, and R. Adhikari, Au- tonomous Motility of Active Filaments due to Sponta- neousFlow-SymmetryBreaking,PhysicalReviewLetters 109, 158302 (2012)

    2012

  36. [36]

    R. G. Winkler and G. Gompper, The physics of active polymers and filaments, The Journal of Chemical Physics 153, 040901 (2020)

    2020

  37. [37]

    K. R. Prathyusha, F. Ziebert, and R. Golestanian, Emer- gent conformational properties of end-tailored trans- versely propelling polymers, Soft Matter18, 2928 (2022)

    2022

  38. [38]

    Lough, D

    W. Lough, D. B. Weibel, and S. E. Spagnolie, Self- buckling and self-writhing of semi-flexible microorgan- isms, Soft Matter19, 7349 (2023)

    2023

  39. [39]

    Scheibner, A

    C. Scheibner, A. Souslov, D. Banerjee, P. Surówka, W. T. M. Irvine, and V. Vitelli, Odd elasticity, Nature Physics16, 475 (2020)

    2020

  40. [40]

    S. C. Al-Izzi, Y. Du, J. Veenstra, R. G. Morris, A. Souslov, A. Carlson, C. Coulais, and J. Binysh, Nonre- ciprocal buckling makes active filaments polyfunctional, Proceedings of the National Academy of Sciences123, e2531723123 (2026)

    2026

  41. [41]

    Ishimoto, C

    K. Ishimoto, C. Moreau, and K. Yasuda, Self-organized swimming with odd elasticity, Physical Review E105, 064603 (2022)

    2022

  42. [42]

    G.DeCanio, E.Lauga,andR.E.Goldstein,Spontaneous oscillations of elastic filaments induced by molecular mo- tors, Journal of The Royal Society Interface14, 20170491 (2017)

    2017

  43. [43]

    F. Ling, H. Guo, and E. Kanso, Instability-driven oscil- lations of elastic microfilaments, Journal of The Royal Society Interface15, 20180594 (2018)

    2018

  44. [44]

    Clarke, Y

    B. Clarke, Y. Hwang, and E. E. Keaveny, Bifurcations andnonlineardynamicsofthefollowerforcemodelforac- tive filaments, Physical Review Fluids9, 073101 (2024)

    2024

  45. [45]

    Warda and R

    M. Warda and R. Adhikari, Elastohydrodynamic insta- bilities of a soft robotic arm in a viscous fluid, Physical Review Research8, 013229 (2026)

    2026

  46. [46]

    J. J. Blum and M. Hines, Biophysics of flagellar motility, Quarterly Reviews of Biophysics12, 103 (1979)

    1979

  47. [47]

    Mondal, R

    D. Mondal, R. Adhikari, and P. Sharma, Internal fric- tion controls active ciliary oscillations near the instability threshold, Science Advances6, eabb0503 (2020)

    2020

  48. [48]

    Poncet and D

    A. Poncet and D. Bartolo, When Soft Crystals Defy Newton’s Third Law: Nonreciprocal Mechanics and Dis- location Motility, Physical Review Letters128, 048002 (2022)

    2022

  49. [49]

    Németh, T

    B. Németh, T. Kobayashi, and R. Adhikari, Nonrecipro- cal constitutive laws for oriented active solids, New Jour- nal of Physics28, 034401 (2026)

    2026

  50. [50]

    R. E. Goldstein and S. A. Langer, Nonlinear Dynamics of Stiff Polymers, Physical Review Letters75, 1094 (1995)

    1995

  51. [51]

    J. F. Cass and H. Bloomfield-Gadêlha, The reaction- diffusion basis of animatedpatterns in eukaryoticflagella, Nature Communications14, 5638 (2023)

    2023

  52. [52]

    M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Reviews of Modern Physics65, 851 (1993)

    1993

  53. [53]

    Cross and H

    M. Cross and H. Greenside,Pattern Formation and Dy- namics in Nonequilibrium Systems, 1st ed. (Cambridge University Press, 2009)

    2009

  54. [54]

    Garg and A

    M. Garg and A. Kumar, A slender body theory for the motionofspecialCosseratfilamentsinStokesflow,Math- ematics and Mechanics of Solids28, 692 (2023)

    2023

  55. [55]

    Aditi Simha and S

    R. Aditi Simha and S. Ramaswamy, Hydrodynamic Fluc- tuations and Instabilities in Ordered Suspensions of Self- Propelled Particles, Physical Review Letters89, 058101 (2002)

    2002

  56. [56]

    G. P. Alexander, S. J. Kole, A. Maitra, and S. Ra- maswamy, Screw symmetry, chiral hydrodynamics, and odd instability in active cholesterics, Physical Review E 112, 055424 (2025)

    2025

  57. [57]

    See supplemental material at [url will be inserted by pub- lisher] for animations of rod dynamics and derivations of equations, which includes ref. [74]

  58. [58]

    Kumar, A

    M. Kumar, A. Murali, A. G. Subramaniam, R. Singh, and S. Thutupalli, Emergent dynamics due to chemo- hydrodynamic self-interactions in active polymers, Na- ture Communications15, 4903 (2024)

    2024

  59. [59]

    L. D. Landau, L. P. Pitaevskii, E. M. Lifshitz, and A. M. Kosevich,Theory of elasticity, 3rd ed. (Butterworth- Heinemann, Oxford, England, 1984)

    1984

  60. [60]

    R. E. Goldstein, T. R. Powers, and C. H. Wiggins, Vis- cous Nonlinear Dynamics of Twist and Writhe, Physical Review Letters80, 5232 (1998)

    1998

  61. [61]

    Fruchart, R

    M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature592, 363 (2021)

    2021

  62. [62]

    Shapere and F

    A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number, Journal of Fluid Mechanics 198, 557 (1989)

    1989

  63. [63]

    Kanso, Swimming due to transverse shape deforma- tions, Journal of Fluid Mechanics631, 127 (2009)

    E. Kanso, Swimming due to transverse shape deforma- tions, Journal of Fluid Mechanics631, 127 (2009)

    2009

  64. [64]

    Montgomery, Gauge Theory of the Falling Cat, inDy- namics and Control of Mechanical Systems, edited by M.Enos(AmericanMathematicalSociety,1993)pp.193– 218

    R. Montgomery, Gauge Theory of the Falling Cat, inDy- namics and Control of Mechanical Systems, edited by M.Enos(AmericanMathematicalSociety,1993)pp.193– 218

    1993

  65. [65]

    Krishnamurthy and M

    D. Krishnamurthy and M. Prakash, Emergent pro- grammable behavior and chaos in dynamically driven ac- tive filaments, Proceedings of the National Academy of Sciences120, e2304981120 (2023)

    2023

  66. [66]

    Sarkar, B

    S. Sarkar, B. Ash, Y. Wu, N. Boechler, S. Shankar, and X. Mao, Mechanochemical Feedback Drives Complex In- ertial Dynamics in Active Solids, Physical Review Letters 135, 258301 (2025)

    2025

  67. [67]

    Bonacci, B

    F. Bonacci, B. Chakrabarti, O. d. Roure, A. Lind- ner, and D. Saintillan, Reversibility, Chaos, and Attrac- tors in Periodically Sheared Elastic Filaments (2026), arXiv:2601.00643 [cond-mat.soft]

    work page arXiv 2026

  68. [68]

    Kuramoto and T

    Y. Kuramoto and T. Tsuzuki, Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium, Progress of Theoretical Physics 55, 356 (1976)

    1976

  69. [69]

    Sivashinsky, Nonlinear analysis of hydrodynamic in- stability in laminar flames—I

    G. Sivashinsky, Nonlinear analysis of hydrodynamic in- stability in laminar flames—I. Derivation of basic equa- tions, Acta Astronautica4, 1177 (1977)

    1977

  70. [70]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Advances in Physics69, 249 (2020)

    2020

  71. [71]

    Qiao and R

    L. Qiao and R. Kapral, Control of Active Polymeric Fila- ments by Chemically Powered Nanomotors, Physical Re- view Applied18, 024051 (2022)

    2022

  72. [72]

    Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics51, 477 (1972)

    H. Hasimoto, A soliton on a vortex filament, Journal of Fluid Mechanics51, 477 (1972)

    1972

  73. [73]

    A. E. Cohen, A. D. Hastewell, S. Pradhan, S. W. Flavell, and J. Dunkel, Schrödinger Dynamics and Berry Phase of Undulatory Locomotion, Physical Review Letters130, 258402 (2023). 8

    2023

  74. [74]

    A. Müller, Review of the exponential and Cayley map on SE(3) as relevant for Lie group integration of the gener- alized Poisson equation and flexible multibody systems, Proceedings of the Royal Society A: Mathematical, Phys- ical and Engineering Sciences477, 20210303 (2021). SUPPLEMENT AR Y MA TERIAL I. GEOMETRIC FLOW AND SHAPE SP ACE EQUA TIONS In this s...

    2021

  75. [75]

    Z 1 1/2 Π3du # −sin

    By symmetry, even modes can only translate with a constant velocity, while odd modes rotate about their geometric center. Shapes were computed by solving the BVP Eq. (S35) using the routinesolve_bvpinscipy. forA, B, C, Dconstants. The boundary conditions lead to the following system of equations: B+D= 0, ξA+C= 0, Asinξ+Bcosξ+C+D= 0, Aξcosξ−Bξsinξ+C= 0. El...