Modelling
of systems and inverse problems of dynamics
Peoples
Friendship University of Russia
117198
Moscow, M.-Maklaya str., 6
The construction method of the physical systems dynamics equations,
providing constraints stabilization, is discussed. The problem of corresponding
constraints reactions or determination of control actions is reduced to the
construction of the system of differential equations, assuming that the partial
integrals are given. The conditions of asymptotic stability an integral
manifold's and constraints on the stabilization for numerically solution are
defined. The solution of the inverse problem of the qualitative theory of
differential equations and its application to the problem of a trajectory
design is proposed. The inverse problems of dynamics and the control problem of
the dynamical systems are considered. The results of the studies are applied to
the inverse problems of rigid body dynamics.
Nowadays, when the computerization filled all of the spheres of human activity,
the meaning of mathematics has grown considerably. Mathematics, itself, has
changed radically. It is known that the formation of mathematics was closely
connected with studying of the natural phenomenon and the development of
manufacture. So, to the well known Newton's law of gravitation preceded 3
Kepler's laws of motion of celestial bodies, which themselves were based on the
thorough study of results of astronomer Tiho Brage.
Modern mathematics has two main trends: pure
mathematics and applied mathematics. It is considered, that pure mathematics
what is allowed and how it is needed, and the applied mathematics does what is
needed and how it is allowed. The applied mathematics is searching for the
methods of finding solution of the problem with given precision. That is why
the engineers say that they believe the theorems of existence, but all they
need are methods of finding solutions. We think that analytical dynamics has to
do what is needed and how it is needed, it means that it possesses all the
necessary mathematical proofs, but at the same time it gives methods of finding
solutions of the applied problems with needed precision.
The main stages of finding solution for the problems of modelling and the inverse problems of dynamics.
Even in the middle of the last century, it was
mentioned that the modelling problems and the inverse problems of dynamics are
among the main important directions of development of the applied mathematics
and that a general approach to finding their solutions can be worked out. As
the number of constraints' equations usually turns to be fewer then the number
of variables, which describe the system's condition, then the construction of
equations of dynamics and finding the solution of the inverse problems of
dynamics amount to the solution of the system of linear algebraic equations
with rectangular matrix of coefficients. The system of differential equations
of the first order obtained by the transformation of dynamical equations of the
investigated system, describes the motion of the representative point on the
manifold in the space of variables. This motion must be stable with respect to
manifold and it must provide the constraints' stabilization while searching for
numerical solution of the system of algebraic-differential equations,
determined by the constraints' equations and by the system's equations of
dynamics. The construction of the system of differential equations, which
solutions possess the given properties, begins with finding the solution of the
system of linear algebraic equations with rectangular matrix of coefficients.
The problem of modelling and the inverse problems of dynamics can be divided
into following stages:
1)
definition
of solution of the systems of linear-algebraic equations with rectangular
matrix of coefficients ;
2)
proposition
of method of construction of the systems of differential equations with given
partial integrals;
3)
definition
of stability conditions of the integral manifold, corresponding to the partial
integrals;
4)
construction
of equations of non-holonomic constraints, using the solution of the inverse
problems of the qualitative theory of differential equations;
5)
construction
of the numerical methods of solution for the systems of differential-algebraic
equations;
6)
construction
of equations of dynamics, that provide the stabilization of constraints;
7)
the
inverse problems of dynamics;
8)
the
applied problems.
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