2 (a) Write down the equation of motion for the point particle of mass m moving in the Kepler potential U(x) = -A/x + B/x where x is the particle displacement in m. (b) Introduce a dissipative term in the equation of motion assuming that the dissipative force acting on a particle is proportional to the partical velocity with the coefficient of proportionality v. Write down the modified equation of motion. (c) Determine the dimensions of parameters A, B, vin SI system of units. (d) Mimimise the number of independent parameters in the modified equation of motion with dissipation by introducing dimensionless varibles 5 and r for x and t. (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 I), and dissipative function F. (f) Plot the potential function for both signs of the free parameter, positive (B = 0.5 and B= 1) and negative (B = -0.5 and B= -1). Find extrema of the potential function and their positions on the 5-axis. In which cases are the extrema stable and unstable? (8) 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 and determine what is the range of values of the free parameter S for which the particle motion is oscillatory or non-oscillatory? What is the critical value of B that determines a transition from the subcritical case with the oscillatory motion to the super-critical case with the non-oscillatory motion? (h) Plot (qualitatively or quantitatively using a technology) the phase portrait of the system for the subcritical (B Ber) values of the free parameter and indicate the type of the equilibrium states in both these cases