On the Mayer problem I. General principles

journal article - other

English

Given an underdetermined system of ordinary differential equations (i.e., the Monge system, the optimal control system) expressed by Pfaffian equations $\omega\equiv 0 \ (\omega\in\Omega)$ where $\Omega$ is a~module of differential 1--forms on a~space $\bf{M}$, we determine submodules $\breve\Omega\subset\Omega$ which satisfy the congruence $d\breve\Omega\simeq 0$ ($\mbox{mod}\,\breve\Omega, \Omega\wedge\Omega$) along a~certain special subspace $\mathbf{E}\subset\mathbf{M}$ of the total space $\mathbf{M}$. Then $\breve\Omega$ and $\mathbf{E}$ may be interpreted in terms of Poincar\'e--Cartan forms and Euler--Lagrange equations for various Mayer problems that belong to the given Monge system. They yield a~universal canonical formalism including the Weierstrass--Hilbert extremality theory. The occurences of uncertain coefficients (Lagrange multipliers, adjoint variables) are suppressed and occasionally eliminated (e.g., for all Mayer problems arising from a~Lagrange problem), the degenerate cases are not excluded.

diffiety, Mayer problem, Poincaré-Cartan module, Euler-Lagrange subspace, Hamilton--Jacobi equation

CHRASTINOVÁ, V.; TRYHUK, V.

2002

1. 1. 2002

SAV

Bratislava

0139-9918

Mathematica Slovaca

52

5

Slovak Republic

555

570

16