A general approach to the derivation of equations of motion of
as holonomic, as nonholonomic systems with the constraints of any
order is suggested. The system of equations of motion in the
generalized coordinates is regarded as a one vector relation,
represented in a space tangential to a manifold of all possible
positions of system at given instant. The tangential space is
partitioned by the equations of constraints into two orthogonal
subspaces. In one of them for the constraints up to the second
order, the motion low is given by the equations of constraints and
in the other one for ideal constraints, it is described by the
vector equation without reactions of connections. In the whole
space the motion low involves Lagrangian multipliers. It is shown
that for the holonomic and nonholonomic constraints up to the
second order, these multipliers can be found as the function of
time, positions of system, and its velocities. The application of
Lagrangian multipliers for holonomic systems permits us to
construct a new method for determining the eigenfrequencies and
eigenforms of oscillations of elastic systems and also to suggest a
special form of equations for describing the system of motion of
rigid bodies. The nonholonomic constraints, the order of which is
greater than two, are regarded as programming constraints such that
their validity is provided due to the existence of generalized
control forces, which are determined as the functions of time. The
closed system of differential equations, which makes it possible to
find as these control forces, as the generalized Lagrange
coordinates, is compound. The theory suggested is illustrated by
the examples of a spacecraft motion. The book is primarily
addressed to specialists in analytic mechanics.
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