Math 485 3 Feb 11 Page 1 of 4 Feb 3 2011 assume a solution does exist therefore no equilibrium state Math 485 3 Feb 11 Page 2 of 4 Feb 3 2011 Examples in Classical Mechanics Review on second order ODE s Autonomous equation since we can t solve in many cases we turn it into a geometric equation x y are phase variables then trajectory of solution is a curve that is tangent to v vector x y 1 No two solution curves intersect solutions are a collection of nonintersecting curves that ows with the vector eld v vector f g 2 What kind of curves are allowed as a solution How do these solution curves t together in phase space A solution is a curve in the phase space phase portrait graph of solution curves on phase plane once this is done you have solved the problem Math 485 3 Feb 11 Page 3 of 4 Feb 3 2011 Fixed Point Analysis Equilibrium solution x0 y0 Local Phase portrait Local behavior determined primarily by Jacobian Matrix solve the 2 equations Local analysis Math 485 3 Feb 11 Page 4 of 4 Feb 3 2011
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