Question

(2017 Comprehensive exam, problem 3) A particle of mass $m$ moves in a potential $V(r) = -V_0 e^{-\lambda^2 r^2}$. a) Write down the Lagrangian for the particle, and its equations of motion. b) Obtain an expression for the effective potential, $V_{eff}(r)$, in terms of the particle's angular momentum $l$. c) For sufficiently small $l$, a stable circular orbit exists. In this case, sketch $V_{eff}(r)$, clearly indicating the location of any critical points as well as its asymptotic behavior. d) Assuming that a stable circular orbit exists, obtain an equation that the radius $r_0$ of the orbit must satisfy. e) Find the largest value of the angular momentum, $l_{max}$, for which a circular orbit exists. What is $V_{eff}(r)$ in this case? f) For $l > l_{max}$, sketch $V_{eff}(r)$, clearly indicating the location of any critical points as well as its asymptotic behavior, and explain why this indicates that there is no stable circular orbit.

          (2017 Comprehensive exam, problem 3) A particle of mass $m$ moves in a potential $V(r) = -V_0 e^{-\lambda^2 r^2}$.
a) Write down the Lagrangian for the particle, and its equations of motion.
b) Obtain an expression for the effective potential, $V_{eff}(r)$, in terms of the particle's angular momentum $l$.
c) For sufficiently small $l$, a stable circular orbit exists. In this case, sketch $V_{eff}(r)$, clearly indicating the location of any critical points as well as its asymptotic behavior.
d) Assuming that a stable circular orbit exists, obtain an equation that the radius $r_0$ of the orbit must satisfy.
e) Find the largest value of the angular momentum, $l_{max}$, for which a circular orbit exists. What is $V_{eff}(r)$ in this case?
f) For $l > l_{max}$, sketch $V_{eff}(r)$, clearly indicating the location of any critical points as well as its asymptotic behavior, and explain why this indicates that there is no stable circular orbit.

(2017 Comprehensive exam, problem 3) A particle of mass m moves in a potential V(r) = -V0 e^-λ^2 r^2.
a) Write down the Lagrangian for the particle, and its equations of motion.
b) Obtain an expression for the effective potential, Veff(r), in terms of the particle's angular momentum l.
c) For sufficiently small l, a stable circular orbit exists. In this case, sketch Veff(r), clearly indicating the location of any critical points as well as its asymptotic behavior.
d) Assuming that a stable circular orbit exists, obtain an equation that the radius r0 of the orbit must satisfy.
e) Find the largest value of the angular momentum, lmax, for which a circular orbit exists. What is Veff(r) in this case?
f) For l > lmax, sketch Veff(r), clearly indicating the location of any critical points as well as its asymptotic behavior, and explain why this indicates that there is no stable circular orbit.

Added by Colleen R.

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