Doppelpendel

Doppelpendel, or double pendulum, is a planar mechanical system consisting of two rigid rods connected by a frictionless hinge. The first rod is attached at a fixed pivot and supports a second rod at its far end. With gravity acting downward, the motion of the two connected pendulums depends on four generalized coordinates (the two angles) and exhibits nonlinear coupling, making the system rich in dynamical behavior.

In the standard planar model, the first rod of length l1 and mass m1 carries the second rod of length l2 and mass m2. Let θ1 and θ2 be the angles of the first and second rods from the vertical. The Lagrangian L = T − V yields a pair of coupled nonlinear equations of motion. A convenient form is obtained by writing the kinetic energy T and potential energy V, then applying the Euler–Lagrange equations. The equations involve the angle difference Δ = θ1 − θ2 and include terms with θ̇1, θ̇2, θ̈1, θ̈2, and sin and cos of Δ. Explicit expressions are widely tabulated, for example:

(m1 + m2) l1 θ̈1 + m2 l2 θ̈2 cos Δ + (1/2) m2 l2 θ̇2^2 sin Δ + (m1 + m2) g sin θ1 = 0

m2 l2 θ̈2 + m2 l1 θ̈1 cos Δ − m2 l1 θ̇1^2 sin Δ + m2 g sin θ2 = 0

These equations describe how the two angles evolve in time for given initial conditions.

Doppelpendel is a classic example of a chaotic, nonlinear dynamical system. For certain energy levels and initial states, its motion is highly sensitive to initial conditions, leading to complex, aperiodic trajectories. It serves as a standard teaching and research model in studies of chaos, nonlinear dynamics, and energy transfer in coupled mechanical systems. Variants include unequal lengths or masses, damping, or three or more linked pendulums, each enriching the behavior further.