1. Consider the 2D motion of a particle of mass in a central force field with potential V(r). a) Find the r, o polar-coordinate expression of the Lagrangian for this system and write down the corresponding Euler-Lagrange e.o.m.s. b) Note that the angular variable o is cyclic. What is the physical interpretation of the correspond- ing integral of motion? (For the definitions of the italicized terms see this link.) c) Solve for in terms of this integral of motion and substitute the result into the Euler-Lagrange equation for r. Show that the result can be arranged to look like a purely 1D e.o.m. of the form dVet (r) dr (1) Identify in the process the explicit expression for Ver(r), which will depend among other things on the integral of motion.
1. Consider the 2D motion of a particle of mass in a central force field with potential V(r). a) Find the r, o polar-coordinate expression of the Lagrangian for this system and write down the corresponding Euler-Lagrange e.o.m.s. b) Note that the angular variable o is cyclic. What is the physical interpretation of the correspond- ing integral of motion? (For the definitions of the italicized terms see this link.) c) Solve for in terms of this integral of motion and substitute the result into the Euler-Lagrange equation for r. Show that the result can be arranged to look like a purely 1D e.o.m. of the form dVet (r) dr (1) Identify in the process the explicit expression for Ver(r), which will depend among other things on the integral of motion.
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Transcribed Image Text:1. Consider the 2D motion of a particle of mass u in a central force field with potential V(r).
a) Find the r, o polar-coordinate expression of the Lagrangian for this system and write down the
corresponding Euler-Lagrange e.o.m.s.
b) Note that the angular variable o is cyclic. What is the physical interpretation of the correspond-
ing integral of motion? (For the definitions of the italicized terms see this link.)
c) Solve for o in terms of this integral of motion and substitute the result into the Euler-Lagrange
equation for r. Show that the result can be arranged to look like a purely 1D e.o.m. of the form
dVef(r)
(1)
dr
Identify in the process the explicit expression for Vef(r), which will depend among other things on
the integral of motion.
d) Take now
k
V (r) =
with k > 0 to be an attractive electrostatic/gravitational-type potential. Sketch the profile of the
corresponding effective potential function Vef(r). Find the equilibrium solution for the correspond-
ing e.o.m. (1). What is its physical interpretation? Compute the angular frequency of the small
oscillations around the equilibrium configuration.
e) Same questions if we take instead
1
V (r) = ;kr²
(3)
with k > 0 to be an isotropic harmonic potential.
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