Write down the equation of motion for the point particle of

Question

Question 1:

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

Neglect the dissipation.

(ii) Introduce a dissipative term in the equation of motion assuming that the viscous force acting on a particle is proportional to the partical velocity. Write down the modified equation of motion.

(iii) Mimimise the number of independent parameters in the modified equation of motion by introducing dimensionless varibles for x and t.

(iv) Present the equation derived in the form of an energy balance equation and find the expressions for the effective kinetic and potential energies (T and V), and dissipative function F.

(v) Plot the potential function for different signs of the free parameter. Find extrema of the potential function and their position on the exis ξ. In which cases are the extrema stable and unstable?

(vi) From the dimensionless equation of motion modified by viscosity (see item (iii)) derive a linearised equation of motion around the stable equilibrium state. Solve it and determine what is the range of values of the free parameter for which the particle moves with oscillations or without oscillations?

(vii) Plot the phase portrait of the system for subcritical and supercritical values of the free parameter and indicate the type of the equilibrium states in both these cases.

Question 2:

Read independently of Study Book and answer to to following questions:

(i) What is the fundamental result of the Calculus of Variations?

(ii) What is the equation which minimises the functional F[y] = _x1∫^x2[2[(y')⁴ - 3y⁵]dx

(iii) Find the first integral of the equation which minimises the functional in the previous question.

Question 3:

Find the quadratic polynomial in x that minimises the functional F[y] = ₀∫³[y']² + y²] dx and which satisfies the conditions y(0) = 0 and y(3) = 38. Find the minimum value of F.