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Hamiltonian/Lagrangian mechanics

  1. Sep 16, 2009 #1
    The problem statement, all variables and given/known data
    Consider a 3D space measured with spherical coordinates (r, θ,Φ). A particle moves under the influence of the potential
    [tex]V(\vec{r})=\frac{1}{2}k(\vec{r}\cdot\vec{r}), k>0[/tex]
    (a) Find the generalized momenta pr, pθ, pΦ.
    (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:
    pr=dL/dr'=mr'
    pθ=dL/dθ'=mr²θ'
    pΦ=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)(pr²+pθ²/r²+pΦ²/r²sin²(θ))+0.5kr²

    Hamiltonian Equations of Motion:
    p'r=-dH/dr=(-1/2mr3)(pθ²+pΦ²/sin²(θ))-kr
    p'θ=-dH/dθ=(-1/2mr²)(pΦ²cos(θ)/sin3(θ))
    p'Φ=-dH/dΦ=0

    r'=pr/m
    θ'=pθ/mr²
    Φ'=pΦ/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.
     
  2. jcsd
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