Uncorrect Problems in Mechanics

The concept of uncorrect statement of initial and boundary value problems for partial differential equations (PDE) was introduced by Hadamard. He presented the first example of an uncorrect statement of the problem for a specific PDE. Meanwhile, examples of uncorrect statement of the problem exist in all branches of mechanics. Hadamard and some of his followers believed that an uncorrect statement of the problem does not make physical sense and such problems should not be solved. This paper gives some examples of uncorrect statements of mechanics problems. It also shows that if a problem has an applied character, the overcoming of uncorrectness in mathematical sense can help to improve the design in practice. The latter fact may justify the studying of uncorrect problems.

References

[1] Adamar Zh. Zadacha Koshi dlya lineynykh uravneniy s chastnymi proizvodnymi giperbolicheskogo tipa [Cauchy problem for linear partial differential equations of hyperbolical nature]. Moscow, Nauka Publ., 1978. 351 p.

[2] Vladimirov V.S. Uravneniya matematicheskoy fiziki [Mathematical physics equations]. Moscow, Nauka Publ., 1967. 436 p.

[3] Neymark Yu.I., Fufaev N.A. Paradoksy Penleve i dinamika tormoznoy kolodki. Painleve paradoxes and brake shoe dynamics. Prikladnaya Matematika i Mekhanika, 1995, vol. 59, no. 3, pp. 366-375 (in Russ.).

[4] Gladwell Gr.M.L. Inverse problems in vibration. Springer Netherlands, 2005. 457 p. (Russ. ed.: Obratnye zadachi teorii kolebaniy. Moscow-Izhevsk, NITs RKhD Publ., 2008. 607 p.).

[5] Medvedev F.A. Rannyaya istoriya aksiomy vybora [Early history of selection axiom]. Moscow, Nauka Publ., 1982. 305 p.

[6] Koriolis G. Matematicheskaya teoriya yavleniy billiardnoy igry [Mathematical theory of billiards game events]. Moscow, LKI Publ., 2007. 240 p.

[7] Neymark Yu.I., Fufaev N.A. Dinamika negolonomnykh system [Nonholonomic systems dynamics]. Moscow, Nauka Publ., 1967. 519 p.

[8] Zhuravlev V.F. Dynamics of a heavy homogeneous ball on a rough plane. Izv. RAN. MTT. [Mechanics of Solids], 2006, no. 6, pp. 1-5 (in Russ.). Available at: http://mtt.ipmnet.ru/ru/Issues.php?n=6&p=3&y=2006

[9] Maylybaev A.A., Seyranyan A.P. Mnogoparametricheskie zadachi ustoychivosti [Multiparameter buckling problems]. Moscow, Fizmatlit Publ., 2009. 399 p.

[10] Panovko Ya.G., Gubanova I.I. Ustoychivost’ i kolebaniya uprugikh system [Stability and oscillations of elastic system]. Moscow, Nauka Publ., 1967. 420 p.