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contents:
Engl
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 2.3.
The analysis of movement of mechanical system
Fig.1 using Lagrange's equations
Lagrange's equations for holonomic systems look like:
where
For the system represented on
Fig.1, for the generalized
coordinates it is possible to accept coordinates of the center of mass of
a body
Kinetic
energy of system develops of kinetic energy of a body
Kinetic
energy of a body
, where :
Kinetic energy of system of mobile elements develops of kinetic energy of progress of the center of mass and rotary concerning the center of mass.
Speed
of the center of mass of system of mobile elements develops
of own speed
On a condition of a task, on our mechanical system external forces do not operate. Therefore:
The
decision of system of the differential equations of the second
order rather
yields following results:
Value of coordinate
Or, approximately, to within 5 signs:
That in accuracy corresponds to the results received earlier (21) and (23).
Value
of coordinate
Comparison
of expressions (33) and
(20) shows, that at
values
Conclusion: The analysis of movement of mechanical system by means of Lagrange's equations also shows an opportunity of moving of the closed mechanical system without external influence.
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2. 
The generalized coordinates, which number it is equal
to number n degrees of freedom of system,
generalized speeds,
generalized forces, T
The kinetic energy of system expressed through
.
, which for the working period particles, and kinetic
energy of system of mobile elements of variable mass join
. 
, which for the working period particles of system of
mobile elements join: 
and portable
. 
and
according to a projection of speed of the
center of mass of system of mobile elements to axes
and
. 
The moment of inertia of system of mobile
elements concerning the center of mass of system.
and
, in view of entry conditions: 
at the moment of time
: 
at the moment of time
: 
,
are practically equal. (at
the mistake makes
<0.5%
. Probably, it is connected with a method of the decision of
the differential equations. In the further calculations the
coordinate x in general is excluded. About
it is hardly below.) 