Classical mechanics involving Lagrangian and Hamiltonian mechanics

Classical mechanics involving Lagrangian and Hamiltonian mechanics.Problem 1
A particle of mass ? moves while constrained to a helical racetrack of parametric equations
? =
?θ
2π
? = ? sin θ
? = ? cos θ
where ? and ? are suitable constants indicating the radius and step of the helix,
respectively, while ? is vertical and pointing upwards. Indicate with ? the acceleration of
gravity. In addition to gravity, the particle is also affected by a harmonic spring force

? ? the helix, and ? is the drag coef?icient.
(a) Derive the system’s kinetic energy in the coordinate θ
(b) Derive the system’s Lagrangian in the same coordinate for the system’s conservative
forces
(c) Using Lagrange’s method, derive the equation of motion for the system, and prove it
correspond to a damped and forced harmonic oscillator equation of the form
θ¨ + ?θ? + ω
2 0
θ = ?
by determining the parameters ?, ω and as a function of any of .
0
? ?, ?, ?, ?, ? ??? ?
(d) Assume θ(0) = 0 and ?(0) = 0. Determine the law of motion for the system, as well
as minimum ? reached by the particle during motion.
???
To answer questions (e) ~ (h), please neglect friction (i.e, assume ? ≈ 0)
(e) Determine the total energy of the system and derive the system’s Hamiltonian
(f) Derive two canonical equations of motion for the system
(g) Derive the time-dependent Hamilton-Jacobi partial differential equation (PDE) of
motion of the system in the action ?(θ, ?)
(h) Using additive separation of variables in the form ?(θ, ?) = ?(θ) - ??, derive the
Hamilton-Jacobi equation of motion in the unknown function ?(θ) and show it
leads to identical solutions as in (d) for ? ≈ 0
(i) Now redo questions (c) and (f) by also considering friction, i.e, ? ≠ 0
Problem 2
Consider three bodies of masses ? ( ) in a center-of-mass reference frame (i.e., a
?
? = 1,...., 3

?
→ ?
= 0
potentials of the form ? , where is the distance between the and
?, ?+1
= ?(?
?, ?+1
) ?
?, ?+1
???
? + 1?? body,*1 as shown in figure.
(a) How many degrees of freedom does this system have?
(b) Show that, using ? as a redundant set of nine generalized coordinates
→
= {?
→ 1
2, ?
→ 2
3, ?
→ 3
1
}
for the system, the system’s kinetic energy takes the form
?(??
?, ?+1
) = 1 2 µ?, ?+1??? 2, ?+1
by deriving the three terms (reduced mass µ) µ and as a function of and
1,2, µ2,3 µ3,1 ?1, ?2 ?3
(c) Using the method of undetermined Lagrange multipliers, derive a suitable Lagrangian for
the system, ?(?? , where
?, ?+1, ?
→ ?
, ?+1, λ
→
) λ
→
= λ
?
→?
+ λ
?
→?
+ λ
?
→?
1 Note that summation over repeated indexes and the circular permutation of j (i.e., j+1=1 for j=3) are assumed in the
text of this problem.