MECHANICAL SYSTEMS, CLASSICAL MODELS
366
are verified for
1, 2, 3i = , where T is a time constant independent of
i
v and t , while
τ is a time constant equal to unity, then it exists a unique solution ()
ii
xxt= ,
()
ii
vvt= of the system (6.1.30) which satisfies the initial conditions (6.1.30') and is
defined on the interval
00
tTttT−≤≤+, where
00
0
min , , ,
ii
XV
TTτ
⎛⎞
≤
⎜⎟
⎝⎠
T
VV
,
max ,
i
i
F
v
m
τ=∈
VD.
The continuity of functions
i
v and
i
F on the interval D ensures the existence of the
solution, after the theorem of Peano. For the uniqueness of the solution, the conditions
of Lipschitz must be fulfilled too; the latter conditions may be replaced by other less
restrictive conditions, in conformity to which the partial derivatives of first order of
functions
i
v and
i
F , 1, 2, 3i = , must exist and be bounded in absolute value on the
interval
D. Besides, the conditions in Theorem 6.1.2 are sufficient conditions of
existence and uniqueness which are not necessary.
We notice that the existence and the uniqueness of the solution have been put in
evidence only on the time interval
00
,tTtT
+ , in the neighbourhood of the initial
moment
0
t (besides, the moment
0
t must not be – necessarily – the initial moment, but
may be a moment arbitrarily chosen); taking, for instance,
0
tT
as initial moment, it
is possible, respecting the above reasoning, to extend the solution on an interval of
length
1
2T a.s.o., if – certainly – the sufficient conditions of existence and uniqueness
of the theorem hold in the neighbourhood of the new initial moment. Thus, we can
prolong the solution for
12
,ttt
, corresponding to an interval of time in which
takes place the considered mechanical phenomenon (even for
(,)t
−∞ ∞ ). Often,
even the solution of the boundary value problem is not unique from a mathematical
point of view, the principle of inertia may bring the necessary precision for the searched
solution, which becomes unique from a mechanical point of vies, as it was shown by V.
Vâlcovici (see Chap. 1, Subsec. 1.2.1).
Other theorems emphasize some important properties of the solution; thus, we state
Theorem 6.1.3 (on the continuous dependence of the solution on a parameter). If the
functions
123
(,, ;,)
i
vxxxtμ ,
123123
(, , ,,,;,)
i
Fx x x vv v tμ are continuous with respect
to the parameter
[
12
,μμμ∈ and satisfy the conditions of the existence and
uniqueness theorem, while the constant
T of Lipschitz does not depend on μ , then the
solution
(, )
i
xtμ , (, )
i
vtμ , 1, 2, 3i
, of the system (6.1.30), which satisfies the
conditions (6.1.30'), depends continuously on
μ .
Analogously, one may state theorems concerning the continuous dependence of the
solution on the initial conditions (that allows to get an approximate solution, fulfilling
the initial conditions with a certain approximation) or on several parameters.
Concerning the analyticity problem, we mention
Theorem 6.1.4 (on the analytical dependence of the solution on a parameter;
Poincaré). The solution
(, )
i
xtμ , (, )
i
vtμ , 1, 2, 3i
, of the system (6.1.30), which