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game-physics

Multiple pendulum - simplified equation

Could someone help me with a mathematical formula for N connected pendulums? Some simplified approach, because I know that for 3 pendulums the formula is huge.

Solution

  • Using a Lagrangian formulation:

    enter image description here

    Euler-Lagrange equation for each angle:

    enter image description here

    Various initial derivatives:

    enter image description here

    LHS:

    enter image description here

    RHS:

    enter image description here

    Equating (and cancelling an l):

    enter image description here

    So we arrive at a classic Ax = b type matrix equation that needs to be solved. This does indeed become very unwieldy even at low values of N, so a numerical solution is preferable (e.g. LU decomp.)

    Test for N = 2:

    enter image description here

    Solving these (simply multiply a^-1 with b and rearrange) gives the results on this page and this page, both of which seem to contradict the link you gave.