Invariant Sets of Mechanical Systems
There are discussed problems of the existence and stability of invariant sets of mechanical systems (in particular, steady motions and relative equilibria).
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INVARIANT SETS OF MECHANICAL SYSTEMS
A. V. Karapetyan Russian Academy of Sciences, Moscow, Russia
ABSTRACT There are discussed problems of the existence and stability of invariant sets of mechanical systems (in particular, steady motions and relative equilibria). The existence problern of stable motions (zero-dimensional invariant sets) firstly was investigated in the [1}. Really, the famous Routh theory [1]-[15] gives stability conditions of steady motions of conservative mechanical systems with first integrals as weil as construction method of such steady motions. This method was spreaded for the construction problern of steady motions not only stable [2], [7] and for dissipative system with·first integrals [12]-[15]. Moreover the Routh theory was modified to the existence and stability problems of invariant sets of dynamical systems with an ungrowing function and first integrals [7], [14] (in particular, of conservative and dissipative mechanical systems with symmetry [16]-[19]).
V. V. Rumyantsev et al. (eds.), Modern Methods of Analytical Mechanics and their Applications © Springer-Verlag Wien 1998
154
A.V. Karapetyan
Chapter 1 Introd uction. 1.1
Mechanical systems with cyclic coordinates.
Let us consider a mechanical systems with n degrees of freedom discribed by generalized Coordinates q E Rn and velocities T
qE
Rn ( q =
~~,
t E [0, +oo)). Let
= ~ (A( q)q, q) be kinetic energy, V (q) a potential energy and Q( q, q) nonconser-
2 vative generalized forces ( A( q) E C2 is a positive definite symmetric n x n-matrix, V(q) E C2 :Rn - t R, Q(q, q) E C1 :Rn x Rn - t Rn). If Q(q, 0) := 0, (Q, q) :::; 0 then the generalized forces Q are dissipative. Here and further (., .) is a scalar product. Equations of motion of the system can be written in the Lagrange form
d ßL ßL dt ßq = ßq
(1.1)
+ Q,
where L = T - V is the Lagrangian of the system. If Q are dissipative forces then the equations (1.1) admit the energy equation
d: =
(Q, q) :::; 0,
(1.2)
where H = T + V is the total mechanical energy of the system. In this case H is an ungrowing function. lf (Q,q) 0 (in particular, Q 0) then equations (1.1) admit the energy integral H = h const. (1.3)
=
=
=
In this case the mechanical system is conservative. Assurne that the kinetic energy, the potential energy and generalized dissipative forces do not depend on some generalized coordinates s E Rk and generalized dissipative forces corresponding to these coordinates are absent
T = T(r, r, s),
V= V(r),
Q = Q(ml(r, r) E Rm
(1.4)
155
Invariant Sets of Mechanical Systems ( r E R m, s E R k, m
+k
= n). Coordinates
r and s are essential and cyclic
coordinates respectively. The Lagrange equations (1.1) of the system with cyclic coordinates have the following form .!!__ 8T
_ 8T _ 8V Q(m). ' 8r +
dt 8r - 8r
d 8T
dt
as
=
o.
(1.5)
Obviously, equations (1.5) admit k first integrals
8T
p = 88 =
c=
const
( c E Rk).
(1.6)
Let us separaten x n-matrix Atom X m-matrix Amm' m X k-matrix Amk, k X mmatrix Akm = A~k and k X k-matrix Akk (symbol T de
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