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Oerg

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## Homework Statement

A point particle moves in space under the influence of a force derivable from a

generalized potential of the form

U(r,r&) = V(r) + σ ⋅L

where r is the radius vector from a fixed point, L is the angular momentum about

that point and σ is a fixed vector in space.

Deduce the generalized force Q = (Qr, Qθ, Qφ ) in spherical polar coordinates.

Hence derive Lagrange’s equations of motion.

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I actually have the solution to this question, but I do not really understand part of the solution. This is the part that I do not understand from the solution:

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Let the polar axis of the polar spherical coordinates (r, θ, ϕ) be in the direction

of σ. Note that [tex] L =(0, - mrv_\phi,mrv_\theta) [/tex], where m is the mass of the particle.

[tex]U(r, v) = V (r) + \sigma \cdot \vec{L} [/tex]

[tex]= V (r) + \sigma ( L_r \cos \theta - L_\theta \sin \theta ) [/tex]

[tex]= V (r) + \sigma mv_\phi r \sin \theta[/tex]

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Firstly, [tex] L_r [/tex] is zero? I have difficulty visualizing [tex] L_r [/tex].

Why is there a negative in front of [tex] mrv_\phi [/tex]?

Which is the polar axis for spherical coordinates?

It would be good if someone could provide a link or explain how to visualize [tex] L_r , L_\theta[/tex] and [tex] L_\phi[/tex].

Any help would be appreciated, thanks.

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