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The three-body problem is a classic problem in celestial mechanics that involves predicting the motion of three celestial bodies under the influence of gravity. The problem is notoriously difficult because it does not have a closed-form analytical solution and requires numerical methods to solve. To understand the three-body problem, let us consider three point masses A, B, and C that are mutually attracted to each other by gravity. The gravitational force between two masses is given by the form
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Let’s find the transfer function for this translational mechanical system. Here, we have a mass-spring-damper system. It is clear that the input is the force 𝑓(𝑡) and the output is the displacement 𝑥(𝑡). The restoring force of the Hookean spring, along with the viscous damping force, impede the motion brought about by the forcing function 𝑓(𝑡). By drawing a free-body force diagram, we can use Newton’s Second Law to set up a second-order differential equation which formulates a dynamic mode
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