hw_1 - MEE 201 Advanced Dynamics. 10/16/2007. Homework set...

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MEE 201 Advanced Dynamics. 10/16/2007. Homework set No. 1 Due Tuesday, 10/23/2007 in class. 1. Write Newton’s equations of motion for two particles of masses m 1 and m 2 interacting via Newton’s law of gravitation. How many degrees of freedom does the system have? 2. Consider the following equation of motion for a mass-spring system: m ¨ x = - kx - δx 2 . where m is the mass, k is the spring constant and δ is a positive constant representing the strength of the quadratic nonlinearity. Write this as a dynamical system in the phase space. Find the equilibrium points. Find the kinetic and potential energy. Show by an explicit calculation that the sum of the kinetic and potential energy is conserved during the motion. Can you solve the resulting system explicitly? Using MATLAB, plot the vector field corresponding to the above equation and, on the same plot, the contours of total energy of the system. Note how the vector field is tangential to the contours of the energy. 3. Write Newton’s equations of motion for the mathematical pendulum (par-
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