Hamiltonian/Lagrangian mechanics

[Brian-san]

Homework Statement
Consider a 3D space measured with spherical coordinates (r, θ,Φ). A particle moves under the influence of the potential
$$V(\vec{r})=\frac{1}{2}k(\vec{r}\cdot\vec{r}), k>0$$
(a) Find the generalized momenta p_r, p_θ, p_Φ.
(b) What is the Lagrangian in these coordinates?
(c) Find the Lagrangian equations of motion.
(d) Find the Hamiltonian in these coordinates.
(e) Find the Hamiltonian equations of motion.
(f) What are the conserved quantities?
(g) Are the solutions the ones you expected from elementary methods?

The attempt at a solution
*I used the prime (') notation to represent time derivatives (d/dt)

Lagrangian: L=0.5m(r'²+r²θ'²+r²sin²(θ)Φ'²)-0.5kr²

Generalized Momenta:
p_r=dL/dr'=mr'
p_θ=dL/dθ'=mr²θ'
p_Φ=dL/dΦ'=mr²sin²(θ)Φ'

Lagrangian Equations of Motion:
mr''=mr(θ'²+sin²(θ)Φ'²)-kr
(mr²θ')'=0.5mr²sin(2θ)Φ'²
mr²sin²θΦ'=l (constant)

Hamiltonian: H=(1/2m)(p_r²+p_θ²/r²+p_Φ²/r²sin²(θ))+0.5kr²

Hamiltonian Equations of Motion:
p'_r=-dH/dr=(-1/2mr³)(p_θ²+p_Φ²/sin²(θ))-kr
p'_θ=-dH/dθ=(-1/2mr²)(p_Φ²cos(θ)/sin³(θ))
p'_Φ=-dH/dΦ=0

r'=p_r/m
θ'=p_θ/mr²
Φ'=p_Φ/mr²sin²(θ)=l/mr²sin²(θ)

Conserved Quantities: Total Energy (E), Momentum in Φ (l)

Does all of this look right? As for the final part, I'm not exactly sure what the result would be from elementary methods. I've gotten so used to the advanced/abstract stuff.

 

[Jhero]

Thank you for your post. Your solution looks correct to me. As for the final part, it is difficult to say whether the solutions would be the same using elementary methods without knowing what specific methods you are referring to. However, since the Lagrangian and Hamiltonian formalisms are generally more advanced and abstract approaches, it is likely that the solutions obtained through these methods would differ from those obtained through elementary methods. This is because the Lagrangian and Hamiltonian formalisms take into account the full dynamics of the system, while elementary methods may only consider certain simplified cases. Overall, your solution appears to be a comprehensive and accurate approach. Good job!