Problem 1. Non-dimensionalization and phase-space portraits We consider the EOM mx(t) = ax(t) bx (t) (13.1)...

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Problem 1. Non-dimensionalization and phase-space portraits We consider the EOM mx(t) = ax(t) bx (t) (13.1) where x(t) is the position of a particle of mass m. a) What are the dimensions of a and b? b) We consider the case where a and b take positive values. Determine a length scale L and a time scale T that provide dimensionless coordinates T= =(t - to)/T such that the EOM takes the dimensionless form (T) = {(T) (T) Determine the dimensionless energy , 1 (T) = x(t)/L and sketch the phase-space portrait. c) We consider the case where a takes a negative and b takes a positive value. Determine a length scale L and a time scale T that provide dimensionless coordinates T = (t - to)/T (T) = x(t)/L such that the EOM takes the dimensionless form (T) = (T) (T) Determine the dimensionless energy , and sketch the phase-space portrait. Remark: Beware the different sign of the linear term. How does it come about? Problem 1. Non-dimensionalization and phase-space portraits We consider the EOM mx(t) = ax(t) bx (t) (13.1) where x(t) is the position of a particle of mass m. a) What are the dimensions of a and b? b) We consider the case where a and b take positive values. Determine a length scale L and a time scale T that provide dimensionless coordinates T= =(t - to)/T such that the EOM takes the dimensionless form (T) = {(T) (T) Determine the dimensionless energy , 1 (T) = x(t)/L and sketch the phase-space portrait. c) We consider the case where a takes a negative and b takes a positive value. Determine a length scale L and a time scale T that provide dimensionless coordinates T = (t - to)/T (T) = x(t)/L such that the EOM takes the dimensionless form (T) = (T) (T) Determine the dimensionless energy , and sketch the phase-space portrait. Remark: Beware the different sign of the linear term. How does it come about?