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**The inverse problem of the calculus of variations for ordinary differential equations.**
*(English)*
Zbl 0760.49021

Mem. Am. Math. Soc. 98, No. 473, 110 p. (1992).

Authors’ abstract: “The inverse problem of the calculus of variations is the problem of finding variational principles for systems of differential equations. By using the general theory of the variational bicomplex, it is shown that the inverse problem for ordinary differential equations is equivalent to the problem of finding differential two forms, with certain prescribed algebraic properties, which are closed on the prolonged equation manifold. For second order ordinary differential equations, this approach leads directly to the fundamental equations of J. Douglas. An algorithm for the analysis of these equations based upon the Cartan- Kähler theorem for exterior differential systems is presented, the inverse problem for higher order ordinary differential equations is solved, and a variety of new examples are considered in detail.”

The section headings are: 1. Introduction; 2. The variational bicomplex for ordinary differential equations; 3. First integrals and the inverse problem for second order ordinary differential equations; 4. The inverse problem for fourth order ordinary differential equations; 5. Exterior differential systems and the inverse problem for second order ordinary differential equations; 6. Examples; 7. The inverse problem for two- dimensional sprays; 8. References.

The section headings are: 1. Introduction; 2. The variational bicomplex for ordinary differential equations; 3. First integrals and the inverse problem for second order ordinary differential equations; 4. The inverse problem for fourth order ordinary differential equations; 5. Exterior differential systems and the inverse problem for second order ordinary differential equations; 6. Examples; 7. The inverse problem for two- dimensional sprays; 8. References.

Reviewer: O.Cârjá (Iaşi)

### MSC:

49N45 | Inverse problems in optimal control |

58E30 | Variational principles in infinite-dimensional spaces |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |