Content
- Plan of the study
- Simple pendulum
- Spherical pendulum
- Inclined pendulum
- Chaotic pendulum
Preliminary study
case of the simple pendulum
We will begin the study of our pendulum by the most simple case we can encounter: the simple pendulum. It is in fact only constituted of a weight m, oscillating without friction in a plane, and suspended by a string of length l and with no mass. This system is subjected to the gravity →g.
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Positions and speeds
Let us begin by seeking the coordinates of the weight regarding the angle θ, and the corresponding speeds: {x=lsinθy=0z=−lcosθ⇒{˙x=l˙θcosθ˙z=l˙θsinθEnergies, Lagrangian and movement equation
We can then find the kinetic and potential energy, and thus the corresponding Lagrangian [ref] L:
Ek=12mv2=12m(√˙x2+˙y2+˙z2)2=12ml2˙θ2Ep=−→W⋅→r=mgz=−mglcosθL=Ek−Ep=12ml2˙θ2+mglcosθ
The Euler-Lagrange equations give us, with θ as a generalized coordinate: ∂L∂θ=ddt∂L∂˙θ⇒¨θ=−glsinθ
Trajectory and phase diagram
We will now integrate the equation above to get the simple pendulum's spatial trajectory and phase diagram.
Since the movement is made in a plan, it is easy to guess that the trajectory will be a part of a circle.
On va maintenant intégrer l'équation obtenue pour obtenir la trajectoire spatiale et le portrait de phase du pendule simple.
Le mouvement étant plan, la trajectoire est aisée à deviner: on obtiendra une portion de cercle.