Geodesic
Construction of analytic solutions of the Cauchy problem for a two-dimensional Hamiltonian system
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 131-141 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider a two-dimensional Hamiltonian system whose Hamiltonian is the support function of a flat smooth convex compact set that contains the origin in its interior. Closed trajectories of this system are similar to polar curves of the original convex compact set (level lines of the support function). The introduction of generalized polar coordinates reduces the two-dimensional Cauchy problem to a one-dimensional problem, and its solution in some cases can be presented in analytic form. The interest in this topic is connected with the analysis of Chaplygin's generalized problem. We use the technique of support functions; its efficiency in optimal control problems was noted in a number of the authors' papers. Examples are illustrated by graphs.
Keywords: convex sets, support and distance functions, hamiltonian system, Pontryagin maximum principle. 
@article{TIMM_2013_19_4_a13,
     author = {Yu. N. Kiselev and S. N. Avvakumov},
     title = {Construction of analytic solutions of the {Cauchy} problem for a two-dimensional {Hamiltonian} system},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {131--141},
     year = {2013},
     volume = {19},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a13/}
}
TY  - JOUR
AU  - Yu. N. Kiselev
AU  - S. N. Avvakumov
TI  - Construction of analytic solutions of the Cauchy problem for a two-dimensional Hamiltonian system
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2013
SP  - 131
EP  - 141
VL  - 19
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a13/
LA  - ru
ID  - TIMM_2013_19_4_a13
ER  - 
%0 Journal Article
%A Yu. N. Kiselev
%A S. N. Avvakumov
%T Construction of analytic solutions of the Cauchy problem for a two-dimensional Hamiltonian system
%J Trudy Instituta matematiki i mehaniki
%D 2013
%P 131-141
%V 19
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a13/
%G ru
%F TIMM_2013_19_4_a13
Yu. N. Kiselev; S. N. Avvakumov. Construction of analytic solutions of the Cauchy problem for a two-dimensional Hamiltonian system. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 131-141. http://geodesic.mathdoc.fr/item/TIMM_2013_19_4_a13/

[1] Avvakumov S. N., Kiselëv Yu.N., Orlov M. V., “Metody resheniya zadach optimalnogo upravleniya na osnove printsipa maksimuma Pontryagina”, Tr. MIAN, 211, 1995, 3–31 | MR | Zbl

[2] Kiselëv Yu. N., Avvakumov S. N., Orlov M. V., Optimalnoe upravlenie. Lineinaya teoriya i prilozheniya, MAKS Press, M., 2007, 270 pp.

[3] Kiselëv Yu. N., “Upravlyaemye sistemy s integralnym invariantom”, Differents. uravneniya, 32:4 (1996), 44–51 | Zbl

[4] Kiselëv Yu. N., “Postroenie tochnykh reshenii dlya nelineinykh zadach bystrodeistviya spetsialnogo vida”, Fundament. i prikl. matematika, 3:3 (1997), 847–868 | MR | Zbl

[5] Kiselëv Yu. N., “Metody resheniya gladkoi lineinoi zadachi bystrodeistviya”, Tr. MIAN, 185, 1988, 106–115 | MR

[6] Lavrentev M. A., Lyusternik L. A., Kurs variatsionnogo ischisleniya, GONTI-NKTI, M.–L., 1938, 192 pp.

[7] Pontryagin L. S., Obyknovennye differentsialnye uravneniya, Nauka, M., 1961, 311 pp. | MR | Zbl

[8] Kiselev Yu. N., “Generalized Chaplygin's problem”, Proc. of the Tenth Crimean Autumn Math. School, v. 10, 2000, 160–163 | MR