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A double pendulum consists of two pendulums attached end to end.. In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. [1]
A double pendulum. The benefits of generalized coordinates become apparent with the analysis of a double pendulum. For the two masses m i (i = 1, 2), let r i = (x i, y i), i = 1, 2 define their two trajectories. These vectors satisfy the two constraint equations,
Pendulum. Inverted pendulum; Double pendulum; Foucault pendulum; Spherical pendulum; Kinematics; Equation of motion; Dynamics (mechanics) Classical mechanics; Isolated physical system. Lagrangian mechanics; Hamiltonian mechanics; Routhian mechanics; Hamilton-Jacobi theory; Appell's equation of motion; Udwadia–Kalaba equation; Celestial ...
Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a configuration space M and a smooth function within that space called a Lagrangian. For many systems, L = T − V , where T and V are the kinetic and potential energy of the system, respectively.
Motion of Swinging Atwood's Machine for M/m = 4.5. The swinging Atwood's machine is a system with two degrees of freedom. We may derive its equations of motion using either Hamiltonian mechanics or Lagrangian mechanics.
"Lagrange" derivation of Coordinates of a simple gravity pendulum. Equation 1 can additionally be obtained through Lagrangian Mechanics . More specifically, using the Euler–Lagrange equations (or Lagrange's equations of the second kind) by identifying the Lagrangian of the system ( L {\displaystyle {\mathcal {L}}} ), the constraints ( q ...
Here, however, pendulum consists of two massive rods. I think, by double pendulum, people usually think about the case with two massless rods and two masses, see most of the references given at the end of the article. Understanding that this case might also be called double pendulum, not even mentioning the more usual meaning is not satisfactory.
The relativistic Lagrangian can be derived in relativistic mechanics to be of the form: = (˙) (, ˙,). Although, unlike non-relativistic mechanics, the relativistic Lagrangian is not expressed as difference of kinetic energy with potential energy, the relativistic Hamiltonian corresponds to total energy in a similar manner but without including rest energy.