“Pendulum and chaos”

“Pendulum and chaos”

SIMPLE PENDULUM │ ├── Dimensions │ ├── 1D (linearized) → small-angle approximation, motion along single axis │ └── 2D (angular) → full motion described by θ (no approximation) │ ├── Phase Space │ ├── Coordinates → (θ , p) = (angle, momentum) │ ├── Trajectories → curves representing motion │ └── Direction → sign of angular velocity shows clockwise/counterclockwise │ ├── Separatrix │ ├── Boundary in phase space │ └── Divides libration (oscillation) and circulation (rotation) │ ├── Stable & Unstable Points │ ├── Stable → θ = 0 (pendulum down) small perturbations = oscillations │ └── Unstable → θ = π (upright) small perturbations = fall away │ ├── Predictability │ ├── Short-term → deterministic equations │ └── Near separatrix → sensitive to initial conditions │ ├── Chaos │ ├── Ideal pendulum → no chaos (integrable system) │ └── Driven/Damped pendulum → chaos emerges │ ├── Librational Motion │ ├── Oscillations around θ = 0 │ └── Phase space → closed loops │ └── Region of Circulation ├── High energy regime → continuous rotation └── Phase space → open trajectories wrapping horizontally