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Direct integration schemes give bad results. You can do much better by using the exact solution in the absence of gravitational interactions between the planets. You can then set up a variation of constants approach where you take the integration constants (which are the orbital parameters) as dynamical variables and write the differential equations (where ...

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The standard way to choose a time-step is to run a test simulation with multiple bodies and plot the total energy of the system versus time. The total energy should remain (roughly) constant. If your step size is too large then you will get energy drift. So simply find the largest time-step that does not produce energy drift. In the case of modelling the ...

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The Fox-Goodwin reference is to a method for numerical integration of ordinary differential equations, which must refer to the algorithm you were to use. Possibly the problem was taken from a paper which referenced this method; that means you can use a scientific citation index to find papers which cite Proc. of Cambridge Phil. Soc. 45 (1949) 373. You can ...

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Find the center of mass (CM) of the two connected particles. Then, determine the distance (r) from the center of mass of the part of the spring where the third particle hit it. Then, if you could allow a collision time and force while in contact, and assume the collision to be frictionless (this is might be hard for point particles and thin spring, cause ...

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Let the discontinuity in thermal conductivity be located half-way between grid points i and i+1. Let the conductivity to the left of the discontinuity be $k_L$ and the conductivity to the right of the discontinuity be $k_R$. Then, for a grid point at i,j (immediately to the left of the discontinuity), the steady state heat balance equation (assuming a ...

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Hint: Use the final equation from my other answer here: $$E(0,t) = E_0\exp(-i\bar{\omega}t)\frac{\sin[(N/2)\Delta\omega t]}{\sin[(1/2)\Delta\omega t]}.$$ This is already a function of time. Now, $I(0,t)\propto |E(0,t)|^{2}$, so just square that function for the intensity (normalize to remove constants if you like). That will ...

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