Computation of the Lagrangian points

Consider an object moving in the plane and subject to a set of forces. Our goal in this application consists in determining the equilibrium positions of this object.

  • Question :
    1. Write the code to represent the following kinds of forces :
      • an elastic force (with zero natural length)
      • a centrifugal force fc:[x y]→[k(x−x0) k(y−y0)]f_c : \left[ \begin{array}{c} x \\\ y \end{array} \right] \rightarrow \left[ \begin{array}{c} k (x-x_0) \\\ k (y-y_0) \end{array} \right]
      • a gravitational force fg:[x y]→[−k⋅x−x0((x−x0)2+(y−y0)2)3/2 −k⋅y−y0((x−x0)2+(y−y0)2)3/2]f_g : \left[ \begin{array}{c} x \\\ y \end{array} \right] \rightarrow \left[ \begin{array}{c} -k \cdot \frac{x-x_0}{((x-x_0)^2+(y-y_0)^2)^{3/2}} \\\ -k \cdot \frac{y-y_0}{((x-x_0)^2+(y-y_0)^2)^{3/2}} \end{array} \right]

Remarks:

  • It is strongly recommended to use functional programming techniques to represent the forces and their Jacobian matrices.
  • For each of these cases, it should be possible to parameterize the force by a constant kk symbolizing its intensity, as well as a central point (x0,y0)(x_0,y_0) from which the force is issued.
  • Question :
    1. Use the Newton-Raphson method to obtain the equilibrium points in the following case :
      • Two gravitational forces with respective coefficients 11 (resp. 0.010.01) originating from [0,0][0,0] (resp. [1,0][1,0]);
      • And a centrifugal force centered on the barycenter of the two masses, with coefficient 11.

As a means of verification, and with the conditions above, at U=[1.5,0]U = [1.5, 0], one gets the following values for the total force and its Jacobian :

f(U)=[1.00565457 0]f(U) = \left[ \begin{array}{c} 1.00565457 \\\ 0 \end{array} \right] and df(U)=[1.752592590. 0.0.6237037]df(U) = \left[ \begin{array}{cc} 1.75259259 & 0. \\\ 0. & 0.6237037 \end{array}\right].