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(a) 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. (d) Mimimise the number of independent parameters in the modified equation of motion with dissipation by introducing dimensionless varibles and r for x and 1. (e) Present the equation derived in the form of an energy balance equation and find the expressions for the effective kinetic and potential energies (7 and V), and dissipative function F.

(a) 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. (d) Mimimise the number of independent parameters in the modified equation of motion with dissipation by introducing dimensionless varibles and r for x and 1. (e) Present the equation derived in the form of an energy balance equation and find the expressions for the effective kinetic and potential energies (7 and V), and dissipative function F.

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) 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. (d) Mimimise the number of independent parameters in the modified equation of motion with dissipation by introducing dimensionless varibles and r for x and 1. (e) Present the equation derived in the form of an energy balance equation and find the expressions for the effective kinetic and potential energies (7 and V), and dissipative function F. (f) Plot the potential function for both signs of the free parameter, positive (B= 0.5 and = 1) and negative (B=-0.5 and B=-1). Find extrema of the potential function and their positions on the -axis. In which cases are the extrema stable and unstable?
Transcribed Image Text:(a) 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. (d) Mimimise the number of independent parameters in the modified equation of motion with dissipation by introducing dimensionless varibles and r for x and 1. (e) Present the equation derived in the form of an energy balance equation and find the expressions for the effective kinetic and potential energies (7 and V), and dissipative function F. (f) Plot the potential function for both signs of the free parameter, positive (B= 0.5 and = 1) and negative (B=-0.5 and B=-1). Find extrema of the potential function and their positions on the -axis. In which cases are the extrema stable and unstable?
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