contents:
Engl

2.3.
The analysis of movement of mechanical system
Fig.1 using Lagrange's equations
Lagrange's equations for holonomic systems look like:
where _{ } 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 _{ }and _{ }.
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 _{ }, which for the working period particles, and kinetic energy of system of mobile elements of variable mass join _{ }.
_{ } Kinetic energy of a body _{ }, which for the working period particles of system of mobile elements join:
, 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 _{ } 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.
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 and _{ }, in view of entry conditions:
yields following results:
Value of coordinate _{ } at the moment of time _{ }:
Or, approximately, to within 5 signs:
That in accuracy corresponds to the results received earlier (21) and (23). Value of coordinate _{ } at the moment of time _{ }:
Comparison of expressions (33) and (20) shows, that at , values 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.) 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.
