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5. (Lagrange equations in spherical coordinates for the central force problem; 15 points) A particle of mass $m$ moves in three dimensions under the influence of a central potential $V(r)$. Using spherical coordinates as generalized coordinates answer the following: a) Write explicitly the Lagrangian in spherical coordinates. b) Find the explicit Lagrange equations in spherical coordinates. Recall that the spherical coordinates $(r, \theta, \phi)$ are related to the Cartesian coordinates with the following transformation equations: $x = r \sin\theta \cos\phi$ $y = r \sin\theta \sin\phi$ $z = r \cos\theta$ (1)

          5. (Lagrange equations in spherical coordinates for the central force problem; 15 points)
A particle of mass $m$ moves in three dimensions under the influence of a central potential $V(r)$.
Using spherical coordinates as generalized coordinates answer the following:
a) Write explicitly the Lagrangian in spherical coordinates.
b) Find the explicit Lagrange equations in spherical coordinates.
Recall that the spherical coordinates $(r, \theta, \phi)$ are related to the Cartesian coordinates with the
following transformation equations:
$x = r \sin\theta \cos\phi$
$y = r \sin\theta \sin\phi$
$z = r \cos\theta$
(1)

5. (Lagrange equations in spherical coordinates for the central force problem; 15 points)
A particle of mass m moves in three dimensions under the influence of a central potential V(r).
Using spherical coordinates as generalized coordinates answer the following:
a) Write explicitly the Lagrangian in spherical coordinates.
b) Find the explicit Lagrange equations in spherical coordinates.
Recall that the spherical coordinates (r, θ, ϕ) are related to the Cartesian coordinates with the
following transformation equations:
x = r sinθcosϕ
y = r sinθsinϕ
z = r cosθ
(1)

Added by Eileen M.

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