**Full paper in compressed Postscript *.ps.gz
**

**Mechanics as a science used computer-aided symbolic calculations since
their appearance and indirectly helped to develop computer algebra in its
modern form. It is possible to think that mutual interest of these
disciplines is partially explained by the fact that Newton's dynamics
for long period of its development offered a huge diversity of problems,
methods and algorithms. The fields of authors interest presented in this
paper are qualitative theory of ordinary differential equations, stability
of motion and dynamics of solid body. Using the methods of qualitative
analysis requires constructing and possible transformation of motion
equations of complex mechanical systems in the Lagrange, Hamilton,
Euler-Lagrange, Routh and other forms. It is labour-intensive
but well algorithmizable job.
**

**In many problems of gyroscopia, space dynamics and robotics
the representation of motion equations in symbolic form allows to avoid
unreasonably great
numerical experiments, stimulates effective usage and development of
qualitative methods. In these areas computer algebra systems usually offer
the significant help both on modelling stage (construction of kinetic energy
and force function of mechanical system, construction and transformation
of motion equations) and during qualitative analysis of obtained equations.
This help is significant even for objects of moderate dimension.
**

**Authors of the report together with the colleagues have developed and use
software package based on computer algebra system Mathematica. This package
implements modelling of motion of solid bodies systems with holonomic links
in Lagrange's formalism; construction of kinetic energy and force functions
for certain force fields, construction of differential equations in the Lagrange,
Hamilton and Routh forms; qualitative analysis of the constructed equations;
construction and analysis of stationary solutions stability by second
Lyapunov's method (including the first approximation methods that use the
Routh-Hurwitz method and its generalizations);
finding of first integrals and investigation of system spectrum, bifurcation sets of
stationary manifolds, etc. with the help of these integrals. The algorithms of
mechanical systems modelling and qualitative analysis of their motion
equations are published in [1,2]
**

**Let us discuss several applications of the above mentioned software
package.
**

**1. Modelling of solid body (satellite) with a point of mass ***m*_{2}**attached to main body with visco-elastic spring, moving in Newton's central
force field (Fig.1).
**

**Motion of holonomical mechanical system with **** degrees of
freedom in potential force field is described by Lagrange's function
**

**To calculate Lagrange's function of the mechanical system in Fig.1,
we must supply the description of the system geometry to program, as it
shown below. We use three coordinate systems for this description:
inertial ****, orbital **** and **** connected to the point
***m*_{2}**.
**

**Program input**.

**The description of motion of the coordinate system **** relative to ****:
**

**mass = 0,
**

*r*_{O1}^{0}** = **
**,
**

*r*_{C1}^{1}** = **
**,
**

*v*_{O1}^{0}** =
**
**,
**

**,
**

**the numbers of rotation axes = **
**,
**

**the rotation angles = **
**,
**

**the list of coordinates = **
**.
**

**The description of the satellite motion in the coordinate system ****:
**

**mass = ***m*_{1}**,
**

*r*_{O1}^{1}** = **
**,
**

*r*_{C1}^{1}** = **
**,
**

*v*_{O1}^{1}** = **
**,
**

**,
**

**the numbers of rotation axes = **
**,
**

**the rotation angles = **
**,
**

**the list of coordinates = **
**.
**

**The description of the coordinate system **** motion relative to ****:
**

**mass = 0,
**

*r*_{O2}^{1}** = **
**,
**

*r*_{C2}^{2}** = **
**,
**

*v*_{O2}^{1}** = **
**,
**

**,
**

**the numbers of rotation axes = **
**,
**

**the rotation angles = **
**,
**

**the list of coordinates = ****.
**

**The description of motion of the point ***m*_{2}** in the coordinate system ****:
**

**mass = ***m*_{2}**,
**

*r*_{O2}^{2}** = **
**,
**

*r*_{C2}^{2}** = **
**,
**

*v*_{O2}^{2}** = **
**,
**

**,
**

**the numbers of rotation axes = **
**,
**

**the rotation angles = **
**,
**

**the list of coordinates = ****,
**

**Rayleigh's function = **
**,
**

**the potential energy of the spring = **
**.
**

**Here ***r*_{Oi}^{j}** is the radius-vector of attachment point ***O*_{i}** of
***i***-th and ***j***-th bodies in coordinate system ****;
***r*_{Ci}^{i}** is the radius-vector of mass center of ***i***-th body in coordinate
system ****;
***v*_{Oi}^{j}** is the relative speed of point ***O*_{i}** in projections on axes of
****;
***J*^{Oi}** is the inertia tenzor of solid body;
***O*_{1}** is the inertia center of the satellite;
**
** is the angle velocity of mass center ***O*_{1}**;
***m*_{1}** is the mass of the satellite;
***A*, *B*, *C*** are the main central inertia moments of the satellite;
**
**; **
**;
***z*** is the extension of the spring; ***k*_{1}, *k*_{2}** are the viscosity and
the elasticity coefficients of the spring.
**

**Program output.**

**Lagrange's function:
**

**Rayleigh's function: **

**The list of generalized coordinates: **

**Elapsed time: 0 h. 0 min. 43:1 sec.
**

**2. Modelling of RLC-circuits.
**

**Authors tried to apply the methods of modelling and analysis of
mechanical systems to electrical ones. The results of computer-aided
modelling and analysis of linear circuits are presented in [3]. Here
nonlinear circuits are discussed.
**

**According to [5,6], the state of nonlinear RLC-circuit can be described
by the equations system:
**

**The function ***P*(*i*,*v*)** can be used not just for the construction of
circuit equations, but also for qualitative analysis of them.
In latter case it could be treated as an analogue for Lyapunov's function.
**

**In [5] it is discussed the construction of the mixed potential for the
class of complete circuits. The complete means
that a circuit is described by the complete set of variables. The
complete set of variables is chosen independently without leading to a
violation of Kirchhoff's laws and determines current or voltage
in any branch of circuit.
**

**The set of currents in inductors and voltages across capacitors
is complete. If a circuit is not complete then it is transformed to the
complete one by adding inductors in series and capacitors in
parallel. The original circuit is considered as a limiting case of the new
one.
**

**Let us describe the procedure of constructing the mixed potential
for complete circuits. It consists of the following steps.
**

**Choose in the graph a maximal tree **** with
the set of links ****.
**

**Choose in **** a subtree **
** with
the set of links **
**. **
** is all branches
which connect two nodes of **
** and form
a loop with branches of **
** only.
**

**According to [5], the currents **
** in
**
** and the voltages **
**in **
** form the complete set of variables. This is correct also
for **

**Calculate the mixed potential by the formula:
**

** is the set of branches where the currents
can be determined from **
** by Kirchhoff's node law;
**** is the set of branches where the voltages can be determined from
**
** by Kirchhoff's loop law;
**
** is a loop;
**** is a loop current; **
** is the voltage of branches of
**
** in **
**.
**

**To illustrate the procedure of constructing the mixed potential
we will consider the circuit in Fig.2 and its graph in Fig.3.
**

**Here the nonlinear elements are resistors **
**.
**

**The maximal tree **** consists of branches **
**,
**** = **
**, **
** = **
**,
**
** =**
**, **
** = **
**.
So the currents **
*i*_{1},*i*_{2},*i*_{3},*i*_{4}** in the inductors **
*L*_{1},*L*_{2},*L*_{3},*L*_{4}**and the voltages **
*v*_{5},*v*_{6},*v*_{7},*v*_{8}** across the capacitors **
*C*_{1},*C*_{2},*C*_{3},*C*_{4}**are the complete set of variables.
**

**Now we can construct the mixed potential by the formula (1).
**

** = **
**. According to (2),
**
*F*(*i*_{1},*i*_{2},*i*_{3},*i*_{4})** is calculated only for the resistive branches
**
**.
**

**To calculate **
** we construct the loops
**
** and consider their
intersections with **
**.
**

** = **
**, **** = **
**,
**** = **
**, **** = **
**.
**
** = ****,
**
** = ****,
**
** = ****,
**
** = ****.
**

**According to (2), **
** =
**
*i*_{1} (*v*_{5} - *v*_{8}) + *i*_{2}(*v*_{6} - *v*_{5}) + *i*_{3}( -*v*_{6} - *v*_{7}) +
*i*_{4} (*v*_{7} + *v*_{8})** and the mixed potential
**

**Ability to apply the analytical methods of mechanical systems to
electrical ones allows to formulate a problem of modelling of mechanical
systems with circuits. This problem can be applied, for example, to analysis
of transient states of mechanical systems after losing stability near the
boundaries of stationary motions family. The number of these problems could
be increased. It is important that parameters of circuits can be measured
easier than parameters of mechanical ones.
**

**3. Finally, we present the results of investigation of one generic
mechanical system with first integrals. Let us discuss the system of absolutely
solid bodies, connected by spherical and cylindrical hinges to single carrying
body, that has a fixed point. The positions of carried bodies we will define as
generalized coordinates **
*q*_{1},*q*_{2},...,*q*_{n}**.
Let us suppose that forces acting on the system are potential and could be
determined by the force function **
*U*(*q*_{1},*q*_{2},...,*q*_{n})**.
The full mechanical energy of such system looks like
**

**The differential equations of described system motion have the following
form:
**

**the integral of kinetic moment projection to vertical axis:
**

**For the entire class of problems that are described by the above differential
equations we can make qualitative analysis of phase space by selecting a set
of invariant manifolds that provide stationary values for different first
integrals from algebra of first integrals with basis formed by the integrals
**
*H*,*V*_{1},*V*_{2},*V*_{3}.** Also we can study stability and bifurcations of
selected invariant manifolds of stationary motions using Lyapunov and
Poincaré methods. For example, let us discuss bundle of the first integrals
**

**According to the theorem of [3] it is possible to state that the elements of
manifolds family
**

**Using this approach we can obtain stability conditions for the elements of
found family of invariant manifolds of stationary motions (IMSM) by
second Lyapunov's method. Non-writing the equations of disturbed motion
in the neighbourhood of our IMSM, let us consider the first integral ***K***as Lyapunov's function. Let us write it in the terms of deviation
from the manifold in question, i.e (in our case) in variables
**
**.
The direct calculation gives:
**

**
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**

**
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**

**
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**

**
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**

**
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**

**
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**

**
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**