Consider a single particle of mass m in spherical coordinates, with the kinetic energy P² p²+ + 7.2 p² r2 sin² 0 IT= 1 2m (a) Write the hamiltonian H and find Hamilton's equations in the case of a central potential V(r). Discuss how the z axis of spherical coordinates should be chosen to simplify the problem to a 2D situation. Write the corresponding 2D hamiltonian. Recover the standard expression for dt in terms of dr, r, E, V(r), l, m. (b) Consider now V(r) = V₁(r)âà f, where â is a fixed unit vector and is the unit vector along the direction of F. V₁ (r) <0 is a function of the distance r = r. (i) Choose an appropriate direction for the z axis, write the hamiltonian H, and find Hamilton's equations. Show that one component of angular momentum is conserved. (ii) Propose a circular motion solution for the equations of motion, with its axis of rotation along â. Find a condition of the form fr(ro) = fe(o)
Consider a single particle of mass m in spherical coordinates, with the kinetic energy P² p²+ + 7.2 p² r2 sin² 0 IT= 1 2m (a) Write the hamiltonian H and find Hamilton's equations in the case of a central potential V(r). Discuss how the z axis of spherical coordinates should be chosen to simplify the problem to a 2D situation. Write the corresponding 2D hamiltonian. Recover the standard expression for dt in terms of dr, r, E, V(r), l, m. (b) Consider now V(r) = V₁(r)âà f, where â is a fixed unit vector and is the unit vector along the direction of F. V₁ (r) <0 is a function of the distance r = r. (i) Choose an appropriate direction for the z axis, write the hamiltonian H, and find Hamilton's equations. Show that one component of angular momentum is conserved. (ii) Propose a circular motion solution for the equations of motion, with its axis of rotation along â. Find a condition of the form fr(ro) = fe(o)
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I just need help for part a. Question 3. (Hamilton and Lagrange formalism)
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3. Consider a single particle of mass m in spherical coordinates, with the kinetic energy
1
2m
P
p² + +
7-2
P²
r² sin²0
(a) Write the hamiltonian H and find Hamilton's equations in the case of a central potential V(r). Discuss how the z
axis of spherical coordinates should be chosen to simplify the problem to a 2D situation. Write the corresponding 2D
hamiltonian. Recover the standard expression for dt in terms of dr, r, E, V(r), l, m.
(b) Consider now V(r) = V₁(r)â f, where â is a fixed unit vector and is the unit vector along the direction of F. V₁ (r) <0
is a function of the distance r = 1. (i) Choose an appropriate direction for the z axis, write the hamiltonian H, and
find Hamilton's equations. Show that one component of angular momentum is conserved. (ii) Propose a circular motion
solution for the equations of motion, with its axis of rotation along â. Find a condition of the form fr(ro) = fe(00)
IT=
relating the constant values r = ro and 0 = 0o. Find explicit expressions for po and w in terms of ro, 0o, m, V₁ (ro) where
w is the angular speed of the motion. (iii) Consider V₁(r) = -Ae-/, with A, λ> 0 constants, show that in this case
0o can be chosen arbitrarily, and compute ro, Po, in terms of 0o, m, A, A. (iv) Consider V₁ (r) = -Bra, with B, a > 0
constants, show that in this case ro can be chosen arbitrarily, that 00 is independent of ro, and compute 80, Po, w in
terms of ro, m, B, a.
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10/31/2023
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