no. 4
A counterexample to analyticity in frictional dynamics
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 4, pp. 1437-1449

We consider the motion of a particle acted on by dry friction and a force that is an analytic function of time. We give a counterexample to the claim that such motions are given by analytic functions of time. Several published arguments concerning existence and uniqueness in unilateral dynamics with friction rely on the analyticity of such motions. The counterexample invalidates those arguments for motions in three or more dimensions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1051/m2an/2022033
Classification : 70F40, 49J52, 34A60
Keywords: Unilateral dynamics with friction, frictional dynamical contact problems, uniqueness 
@article{M2AN_2022__56_4_1437_0,
     author = {Dance, Christopher R.},
     title = {A counterexample to analyticity in frictional dynamics},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1437--1449},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {56},
     number = {4},
     doi = {10.1051/m2an/2022033},
     mrnumber = {4444536},
     zbl = {1502.70037},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2022033/}
}
TY  - JOUR
AU  - Dance, Christopher R.
TI  - A counterexample to analyticity in frictional dynamics
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2022
SP  - 1437
EP  - 1449
VL  - 56
IS  - 4
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2022033/
DO  - 10.1051/m2an/2022033
LA  - en
ID  - M2AN_2022__56_4_1437_0
ER  - 
%0 Journal Article
%A Dance, Christopher R.
%T A counterexample to analyticity in frictional dynamics
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2022
%P 1437-1449
%V 56
%N 4
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2022033/
%R 10.1051/m2an/2022033
%G en
%F M2AN_2022__56_4_1437_0
Dance, Christopher R. A counterexample to analyticity in frictional dynamics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 4, pp. 1437-1449. doi: 10.1051/m2an/2022033

[1] P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Ration. Mech. Anal. 154 (2000) 199–274. | MR | Zbl | DOI

[2] P. Ballard and S. Basseville, Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem. ESAIM: M2AN 39 (2005) 59–77. | MR | Zbl | Numdam | DOI

[3] P. Ballard and A. Charles, An overview of the formulation, existence and uniqueness issues for the initial value problem raised by the dynamics of discrete systems with unilateral contact and dry friction. C. R. Méc. 346 (2018) 222–236. | DOI

[4] A. Bressan, Incompatibilità dei teoremi di esistenza e di unicità del moto per un tipo molto comune e regolare di sistemi meccanici. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 14 (1960) 333–348. | MR | Numdam | Zbl

[5] A. Charles and P. Ballard, Existence and uniqueness of solutions to dynamical unilateral contact problems with Coulomb friction: the case of a collection of points. ESAIM: M2AN 48 (2014) 1–25. | MR | Zbl | Numdam | DOI

[6] F. Dubois, V. Acary and M. Jean, The contact dynamics method: a nonsmooth story. C. R. Méc. 346 (2018) 247–262. | DOI

[7] J. Hwangbo, J. Lee and M. Hutter, Per-contact iteration method for solving contact dynamics. IEEE Rob. Autom. Lett. 3 (2018) 895–902. | DOI

[8] S. Kepley and T. Zhang, A constructive proof of the Cauchy-Kovalevskaya theorem for ordinary differential equations. J. Fixed Point Theory App. 23 (2021) 1–23. | MR | Zbl

[9] M. Loday-Richaud, Divergent Series, Summability and Resurgence II. Lecture Notes in Mathematics. Vol. 2154. Springer (2016). | MR | DOI

[10] Z. Manchester and S. Kuindersma, Variational contact-implicit trajectory optimization. In: Robotics Research. Springer (2020) 985–1000. | DOI

[11] M. D. P. Monteiro Marques, Inelastic shocks with or without friction: Existence results. In: Differential Inclusions in Nonsmooth Mechanical Problems. Springer (1993) 72–111. | DOI

[12] J.-J. Moreau, Application of convex analysis to some problems of dry friction. In: Trends in Applications of Pure Mathematics to Mechanics, Pitman (1977) 263–280. | MR | Zbl

[13] D. Percivale, Uniqueness in the elastic bounce problem, II. J. Differ. Equ. 90 (1991) 304–315. | MR | Zbl | DOI

[14] M. Schatzman, A class of nonlinear differential equations of second order in time. Nonlinear Anal. Theory Methods App. 2 (1978) 355–373. | MR | Zbl | DOI

Cité par Sources :

⋆Supplementary Online Material is only available in electronic form at https://doi.org/10.1051/m2an/2022033/olm.