ac Problem 6 (70 points): Assume a particle moves in two-dimension (r,() in a central potential given by V (n) = ar? (a)

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Ac Problem 6 70 Points Assume A Particle Moves In Two Dimension R In A Central Potential Given By V N Ar A 1 (47.99 KiB) Viewed 51 times

ac Problem 6 (70 points): Assume a particle moves in two-dimension (r,() in a central potential given by V (n) = ar? (a) Find the Lagrangian L of the system (10 points). (b) Find the equations of movement and the constant of motion (10 points). (c) Reduce the two-dimensional problem to a one-dimensional problem by finding the effective potential Veff (r) (10 points). (d) Show that using ac E + 0-C ar ao you obtain E=mi2/2 + Veff (r), and also (10 points) dr 2E - Veff (r) L2 m2,2 (e) Use the transformations et des de and y=1/r2 and obtain (10 points) dy 2m Ey 2ma 2 do L2 () complete square, integrate and show what are the conics solution for the differential equation above (20 points) 2 ) m dr dt 2 (2 ) 22