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2024 | OriginalPaper | Chapter

Radial Laplacian on Rotation Groups

Author : Pierre Degond

Published in: From Particle Systems to Partial Differential Equations

Publisher: Springer International Publishing

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Abstract

The Laplacian on the rotation group is invariant by conjugation. Hence, it maps class functions to class functions. A maximal torus consists of block diagonal matrices whose blocks are planar rotations. Class functions are determined by their values on this maximal torus. Hence, the Laplacian induces a second order operator on the maximal torus called the radial Laplacian. In this paper, we derive the expression of the radial Laplacian. Then, we use it to find the eigenvalues of the Laplacian, using that characters are class functions whose expressions are given by the Weyl character formula. Although this material is familiar to Lie-group experts, we gather it here in a synthetic and accessible way which may be useful to non experts who need to work with these concepts.

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Adams, J.F.: Lectures on Lie Groups. University of Chicago Press (1982)

Ahn, H., Ha, S.-Y., Shim, W.: Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds. Kinet. Relat. Models 14(2), 323 (2021)MathSciNetCrossRef

Bröcker, T., Tom Dieck, T.: Representations of Compact Lie Groups. vol. 98. Springer Science & Business Media (2013)

Bump, D.: Lie Groups, vol. 8. Springer (2004)

Chevalley, C.: Theory of Lie Groups. Courier Dover Publications (2018)

Degond, P., Diez, A., Frouvelle, A.: Body-attitude coordination in arbitrary dimension, not yet published (2021). arXiv:2111.05614

Degond, P., Diez, A., Frouvelle, A., Merino-Aceituno, S.: Phase transitions and macroscopic limits in a BGK model of body-attitude coordination. J. Nonlinear Sci. 30, 2671–2736 (2020)MathSciNetCrossRef

Degond, P., Diez, A., Walczak, A.: Topological states and continuum model for swarmalators without force reciprocity. Anal. Appl. 20, 1215–1270 (2022)MathSciNetCrossRef

Degond, P., Frouvelle, A.: Macroscopic limit of a Fokker-Planck model of swarming rigid bodies. European J. Appl. Math., appeared online (2024). https://doi.org/10.1017/S0956792524000111

10.

Degond, P., Frouvelle, A., Merino-Aceituno, S.: A new flocking model through body attitude coordination. Math. Models Methods Appl. Sci. 27(06), 1005–1049 (2017)MathSciNetCrossRef

11.

Degond, P., Frouvelle, A., Merino-Aceituno, S., Trescases, A.: Alignment of self-propelled rigid bodies: from particle systems to macroscopic equations. In: International Workshop on Stochastic Dynamics out of Equilibrium, pp. 28–66. Springer (2017)

12.

Degond, P., Frouvelle, A., Merino-Aceituno, S., Trescases, A.: Quaternions in collective dynamics. Multiscale Model. Simul. 16(1), 28–77 (2018)MathSciNetCrossRef

13.

Degond, P., Motsch, S.: A macroscopic model for a system of swarming agents using curvature control. J. Stat. Phys. 143, 685–714 (2011)MathSciNetCrossRef

14.

Do Carmo, M.:.Riemannian Geometry. Springer (1992)

15.

Duistermaat, J.J., Kolk, J.A.: Lie Groups. Springer Science & Business Media (2012)

16.

Faraut, J.: Analysis on Lie Groups, An Introduction. Cambridge University Press (2008)

17.

Fetecau, R.C., Ha, S.-Y., Park, H.: Emergent behaviors of rotation matrix flocks. SIAM J. Appl. Dyn. Syst. 21(2), 1382–1425 (2022)MathSciNetCrossRef

18.

Fetecau, R.C., Zhang, B.: Self-organization on Riemannian manifolds. J. Geom. Mech. 11(3), 397–426 (2019)MathSciNetCrossRef

19.

Fulton, W., Harris, J.: Representation Theory: A First Course. Springer (2013)

20.

Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry. Springer (1990)

21.

Golse, F., Ha, S.-Y.: A mean-field limit of the Lohe matrix model and emergent dynamics. Arch. Ration. Mech. Anal. 234(3), 1445–1491 (2019)MathSciNetCrossRef

22.

Gurarie, D.: Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications. Courier Corporation (2007)

23.

Ha, S.-Y., Ko, D., Ryoo, S.W.: Emergent dynamics of a generalized Lohe model on some class of Lie groups. J. Stat. Phys. 168(1), 171–207 (2017)MathSciNetCrossRef

24.

Hall, B.C.: Lie Groups, Lie Algebras, and Representations. Springer (2013)

25.

Helgason, S.: Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, vol. 83. American Mathematical Society (2022)

26.

Hsu, E.P.: Stochastic Analysis on Manifolds, vol. 38. American Mathematical Society (2002)

27.

Humphreys, J.E.: Introduction to Lie Algebras And Representation Theory, vol. 9. Springer Science & Business Media (2012)

28.

Knapp, A.W.: Lie Groups Beyond an Introduction, vol. 140. Springer (1996)

29.

Sepanski, M.R.: Compact Lie Groups. Springer (2007)

30.

Simon, B.: Representations of Finite and Compact Groups. Graduate Studies in Mathematics, vol. 10. American Mathematical Society (1996)

31.

Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226 (1995)MathSciNetCrossRef

32.

Warner, F.W.: Foundations Of Differentiable Manifolds and Lie Groups. Springer (1983)

33.

Weyl, H.: The Classical Groups: Their Invariants and Representations, vol. 1. Princeton University Press (1946)

Title: Radial Laplacian on Rotation Groups
Author: Pierre Degond
Publisher: Springer International Publishing
Book: From Particle Systems to Partial Differential Equations
Print ISBN: 978-3-031-65194-6

Electronic ISBN: 978-3-031-65195-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-65195-3_2

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