Articulo Periodic Oscillations of The Relativistic Pendulum With Friction
Articulo Periodic Oscillations of The Relativistic Pendulum With Friction
Articulo Periodic Oscillations of The Relativistic Pendulum With Friction
_
1
x
2
c
2
_
+kx
+kx
+kx
+a sinx = p(t),
where : ]c, c[ R is given by
(u) =
u
_
1
u
2
c
2
.
Of course, the inverse
1
is a bounded operator. The rst step
of the proof is to make the change of variables x =arcsin y. Then
Eq. (1.1) is equivalent to
_
y
_
1 y
2
_
+k
y
_
1 y
2
+ay = p(t). (2.1)
The second step is to write the problem of nding a T -periodic
solution of (2.1) as a xed point problem for a suitable operator.
A rst integration of the equation gives
_
y
_
1 y
2
_
+k arcsin y =
t
_
0
_
p(s) ay(s)
_
ds +C,
where C is a constant to be xed later. For convenience, let us
dene the operator
F [y](t) =
t
_
0
_
p(s) ay(s)
_
ds k arcsin y.
Then, we get
y
=
_
1 y
2
1
_
F [y](t) +C
_
.
Finally, a new integration gives
y(t) =
t
_
0
_
1 y
2
1
_
F [y](s) +C
_
ds + D.
Lemma 1. For any y
C
T
, there exists a unique choice of C
y
, D
y
such
that
T [y](t)
t
_
0
_
1 y
2
1
_
F [y](s) +C
y
_
ds + D
y
C
T
. (2.2)
Proof. Periodicity is equivalent to
T
_
0
_
1 y
2
1
_
F [y](s) +C
y
_
ds =0.
As a function of C
y
, the left-hand side of this equation is contin-
uous and increasing, so the existence of a unique solution C
y
for
such equation follows from a basic application of the Mean Value
Theorem. Once C
y
is xed, D
y
is given by
D
y
=
1
T
T
_
0
t
_
0
_
1 y(s)
2
1
_
F [y](s) +C
y
_
ds dt,
which is the unique choice such that T [y](t)
C
T
. 2
Therefore, we have a well-dened functional T :
C
T
C
T
. Let
us dene the closed and convex set
K =
_
y
C
T
: y
2cT
_
.
The operator T is well dened, continuous and compact on K.
Take y K. Note that
1
[y]
T [y](t)
t
_
0
_
1 y(s)
2
1
_
F [y](s) +C
y
_
ds
<2cT
for all t. In consequence, by the Schauders xed point Theo-
rem there exists a T -periodic solution y of (2.1). The hypothesis
2cT 1 enables to invert the change and hence x = arcsin y is a
T -periodic solution of the original equation (1.1).
As a nal note, let us remark that the same proof works if the
linear friction term kx
is replaced by h(x)x