3. Consider a single particle of mass m in spherical coordinates, with the kinetic energy lees so that the vertical separation between them is constant. Some possibly useful formulas: F = ma, FkHkFN, F = P₁ T = p = mv, r2 sin² 0 (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. 1 2m (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 = 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(80) P + 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 o can be chosen arbitrarily, and compute ro, Po, w 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 fo, Po, in terms of ro, m, B, a. 0≤Fs ≤s FN デューデ F21 = - Gm₁m2 -2. Q Search 21 F21 = Q1Q21-3, L = F xp, Ñ = ŕ x F = 1 (0)
3. Consider a single particle of mass m in spherical coordinates, with the kinetic energy lees so that the vertical separation between them is constant. Some possibly useful formulas: F = ma, FkHkFN, F = P₁ T = p = mv, r2 sin² 0 (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. 1 2m (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 = 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(80) P + 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 o can be chosen arbitrarily, and compute ro, Po, w 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 fo, Po, in terms of ro, m, B, a. 0≤Fs ≤s FN デューデ F21 = - Gm₁m2 -2. Q Search 21 F21 = Q1Q21-3, L = F xp, Ñ = ŕ x F = 1 (0)
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Part b
Transcribed Image Text:3. Consider a single particle of mass m in spherical coordinates, with the kinetic energy
lees so that the vertical separation between them is constant.
Some possibly useful formulas:
F = ma,
FkHkFN,
F = P₁
p = mv,
T =
r2 sin² 0
(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.
1
2m
(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 = 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)
0 ≤ F≤s FN
デューデ
F21 = -Gm₁m21-2³
P +
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
o can be chosen arbitrarily, and compute ro, Po, w 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 fo, Po, in
terms of ro, m, B, a.
Q Search
21
F21 = Q1Q2-3,
L = r xp,
Ñ = 7 × F = Ì
(0)
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