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Write down the equation of motion for the point particle of mass m moving in the Kepler potential U(x)=-A/x+B/x² where x is the particle displacement in m.

Write down the equation of motion for the point particle of mass m moving in the Kepler potential U(x)=-A/x+B/x² where x is the particle displacement in m.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) (g) Write down the equation of motion for the point particle of mass m moving in the Kepler potential U(x)=-A/x+B/x² where x is the particle displacement in m. From the dimensionless equation of motion modified by dissipation Compare to question (2 b) derive a linearised equation of motion around a stable equilibrium state. Solve it (h) and determine what is the range of values of the free parameter 3 for which the particle motion is oscillatory or non-oscillatory? What is the critical value of that determines a transition from the subcritical case with the oscillatory motion to the super-critical case with the non-oscillatory motion? Plot (qualitatively or quantitatively using a technology) the phase portrait of the system for the subcritical (B<B) and supercritical (B> B) values of the free parameter and indicate the type of the equilibrium states in both these cases.
Transcribed Image Text:(a) (g) Write down the equation of motion for the point particle of mass m moving in the Kepler potential U(x)=-A/x+B/x² where x is the particle displacement in m. From the dimensionless equation of motion modified by dissipation Compare to question (2 b) derive a linearised equation of motion around a stable equilibrium state. Solve it (h) and determine what is the range of values of the free parameter 3 for which the particle motion is oscillatory or non-oscillatory? What is the critical value of that determines a transition from the subcritical case with the oscillatory motion to the super-critical case with the non-oscillatory motion? Plot (qualitatively or quantitatively using a technology) the phase portrait of the system for the subcritical (B<B) and supercritical (B> B) values of the free parameter and indicate the type of the equilibrium states in both these cases.
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