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hw1_scribe - CDS140a - Introduction to Dynamics Homework 1...

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Homework 1 Exercises 1, 2, and 5 Fernando Ferrari de Goes October 7, 2009 1. Consider the following planar system for ( x,v ) R 2 : b ˙ x = v ˙ v = x 3 (1) (a) Find the equilibrium points for the system. The equilibrium points of this system are obtained by setting the right hand side to zero. Thus, the equilibria occurs in the xv -plane when ˙ x = v = 0 and when x satisFes ˙ v = x 3 = 0 x = 0 . (0 , 0) is the equilibrium point of the system. (b) Find a conserved energy for the system. This system corresponds to a conservative mechanical system and can be rewritten as: b ˙ x = v ˙ v = 1 m ( −∇ V ( x )) . Replacing the Equation 1 in this form, we can compute the potential energy V ( x ): 1 m ( −∇ V ( x )) = x 3 V ( x ) = m x 4 4 . ±inally, we Fnd the conserved energy: E ( x,v ) = m v 2 2 + m x 4 4 . 1
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This note was uploaded on 01/04/2012 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.

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hw1_scribe - CDS140a - Introduction to Dynamics Homework 1...

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