d1,2=士i号V2-√27B-a2-28 Because the mass of the Sun is much greater than the mass of the Earth,B is much greater than a,and the eigenvalues are all imaginary,with no real components.This result indicates that L4 and L5 are stable. It is interesting to note that evaluation of only the bottom left 2 x 2 portion of the matrix, involving the effective potential,yields two positive eigenvalues.This would indicate an unstable node However,because the system is rotating and not stationary,the coriolis force affects objects near L4 and L5,pulling them toward the point if they are perturbed.Therefore,L4 and L5 are actually stable equilibrium points. III.NUMERICAL SIMULATION TESTING STABILITY In order to verify our analytical solution we created a 3D N-body code to assess the stability regions of our solution.Our code is written using a combination of Python and C/C++.All of the plotting and setting up of initial conditions was done in Python and we outsource the computational work to C++.To conserve energy and accuracy in our code we invoke a fourth order method for updating the position and velocity.The fourth order method is formally known as the fourth order Hermite which is an implicit method that uses a predictor-corrector scheme.Please reference our midterm report for the exact computational formulation we adopt. We tested two different models of the Earth-Sun system to assess the stability of each of the Lagrange points.The first model we tested was one where the initial positions of the bodies placed at the Lagrange points were not perturbed.See Figure 1.Using this model we can compute the stability of each of the points by analyzing the drift timescale or e-folding time.Small departures from equilibrium will grow exponentially with an e-folding time of roughly 1/eigenvalue for the associated Lagrange point. By fitting a function of the form A*exp(B*t)to the distance of a body from its initial point as a function of time we can obtain the drift timescale as 1/B.Below is a table containing the drift times for the 3 unstable Lagrange points.The optimal function was found using the Levenberg-Marquardt minimization algorithm which uses a damped least squares method to solve nonlinear least squares problems.Because the mass of the Sun is much greater than the mass of the Earth, β is much greater than α, and the eigenvalues are all imaginary, with no real components. This result indicates that L4 and L5 are stable. It is interesting to note that evaluation of only the bottom left 2 x 2 portion of the matrix, involving the effective potential, yields two positive eigenvalues. This would indicate an unstable node. However, because the system is rotating and not stationary, the coriolis force affects objects near L4 and L5, pulling them toward the point if they are perturbed. Therefore, L4 and L5 are actually stable equilibrium points. III. NUMERICAL SIMULATION TESTING STABILITY In order to verify our analytical solution we created a 3D N­body code to assess the stability regions of our solution. Our code is written using a combination of Python and C/C++. All of the plotting and setting up of initial conditions was done in Python and we outsource the computational work to C++. To conserve energy and accuracy in our code we invoke a fourth order method for updating the position and velocity. The fourth order method is formally known as the fourth order Hermite which is an implicit method that uses a predictor­corrector scheme. Please reference our midterm report for the exact computational formulation we adopt. We tested two different models of the Earth­Sun system to assess the stability of each of the Lagrange points. The first model we tested was one where the initial positions of the bodies placed at the Lagrange points were not perturbed. See Figure 1. Using this model we can compute the stability of each of the points by analyzing the drift timescale or e­folding time. Small departures from equilibrium will grow exponentially with an e­folding time of roughly 1/eigenvalue for the associated Lagrange point. By fitting a function of the form A*exp(B*t) to the distance of a body from its initial point as a function of time we can obtain the drift timescale as 1/B. Below is a table containing the drift times for the 3 unstable Lagrange points. The optimal function was found using the Levenberg­Marquardt minimization algorithm which uses a damped least squares method to solve nonlinear least squares problems