3 degree-of-freedom pendulum

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3 degree-of-freedom Pendulum!

Brief Theory

Several definitions:

Number of degrees-of-freedom N: the number of independent parameters that completely define the certain position of system
Generalized coordinates q: array of parameters that completely define the certain position of system
Generalized velocities q': time derivatives of generalized coordinates
Equation of motion: functional relation between accelerations q'' and { coordinates and velocities }

Hamilton's principle of least action is the most common law of mechanics that describes the bodies motion. It says that every mechanical system without dissipation of energy may be characterized by certain function
L(q, q', t) (called Lagrange function)
so that during the system motion the time integral of L has the least possible value.

This law has a corollary that differential equations of motion of system with N degree-of-freedom are:

d  ∂L     ∂L
-- ---- = ----    (i = 1, 2, ..., N)
dt ∂q'_i   ∂q_i

In mechanics theory it is proved that
L = T - U , where
T - kinetic energy
U - potential energy

Let's find Lagrange function for our certain 3 degree-of-freedom Pendulum

The angles between the "rods" and vertical - are the general coordinates.

Every 1 millisecond the applet calculates and every 60 millisecond - redraws the new pendulum position. The calculation of initial problem is based on Runge-Kutta numerical solution of the described differential equations system.

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