# Conditions d'existence d'une solution non triviale à l'équation du pendule simple ou double

Abstract : Our work deals with the existence of non constant periodic solutions for the equations of the simple and double penduli with constant forcing terms. For the simple pendulum, we consider the case when the forcing term is constant. We begin by establish a necessary condition of the existence of a non constant periodic solution. Then, by two different methods (namely, an analysis in phase plane, and a variational method of construction of critical points) we prove that for a period of oscillation sufficiently large, the simple pendulum equation with constant forcing term has always a non constant periodic solution. The advantage of the variational metod is that we can use it in the case of the double (or even multiple) pendulum equation. We then consider the case of a double pendulum with two constant forcing terms. Again, we prove that, under for a period of oscillation sufficiently large, the double pendulum equation with constant forcing terms, has a non constant periodic solution. In fact, we prove that the functional of the corresponding variational problem satisfies a modified Palais-Smale condition. Using the Ambrosetti-Rabinowitz mountainpass theorem, we prove that this functional has critical values. Then, by using the result of H. Hofer on the Morse-index of the mountainpass-type critical points, we prove that at least one of those critical value corresponds to a non trivial critical point which is the expected solution.
Mots-clés :
Document type :
Theses
File URL :
http://docnum.univ-lorraine.fr/prive/SCD_T_1997_0028_TAGNI.pdf

https://hal.univ-lorraine.fr/tel-01747315
Contributor : Thèses Ul <>
Submitted on : Thursday, March 29, 2018 - 10:57:07 AM
Last modification on : Friday, March 30, 2018 - 1:30:59 AM

### Identifiers

• HAL Id : tel-01747315, version 1

### Citation

Sandrine Kaméni Tagni. Conditions d'existence d'une solution non triviale à l'équation du pendule simple ou double. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 1997. Français. ⟨NNT : 1997NAN10028⟩. ⟨tel-01747315⟩

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