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Goldstein(3rd) 1.15

Generalized potential, U as follows.

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

L is angular momentum and [tex]\sigma[/tex] is a fixed vector.

(b) show thate the component of the forces in the two coordinate systems(cartesin, spherical polar) are related to each other as

[tex]Q_{j}=F_{i}\cdot \frac{\partial r_{i}} {\partial q_{j}} \cdots (a) [/tex]

So I did,

[tex] Q_{j}= - \frac{\partial U}{\partial q_{j}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot q_{j}})[/tex]

[tex]

=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial \dot x_{k}}{\partial \dot q_{j}} \frac{\partial U}{\partial \dot x_{k}})

[/tex]

[tex]

=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial \dot x_{k}})

[/tex]

[tex]

=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}+ \frac{\partial x_{k}}{\partial q_{j}} \frac{d}{dt}(\frac{\partial U}{\partial \dot x_{k}})

[/tex]

[tex]=\frac{\partial x_{k}}{\partial q_{j}}(

- \frac{\partial U}{\partial x_{k}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot x_{k}}))+ \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}

[/tex]

[tex]=\frac{\partial x_{k}}{\partial q_{j}} (

F_{k})+ \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}

[/tex]

If the last term in the last line vanishes, [tex] Q_{j} [/tex] and [tex] F_{k} [/tex] satisfies the relation (a), but it DOESN't vanish. What's my problem??

I've evaluated the last term in this condition, but It doesn't....

Generalized potential, U as follows.

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

L is angular momentum and [tex]\sigma[/tex] is a fixed vector.

(b) show thate the component of the forces in the two coordinate systems(cartesin, spherical polar) are related to each other as

[tex]Q_{j}=F_{i}\cdot \frac{\partial r_{i}} {\partial q_{j}} \cdots (a) [/tex]

So I did,

[tex] Q_{j}= - \frac{\partial U}{\partial q_{j}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot q_{j}})[/tex]

[tex]

=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial \dot x_{k}}{\partial \dot q_{j}} \frac{\partial U}{\partial \dot x_{k}})

[/tex]

[tex]

=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial \dot x_{k}})

[/tex]

[tex]

=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}+ \frac{\partial x_{k}}{\partial q_{j}} \frac{d}{dt}(\frac{\partial U}{\partial \dot x_{k}})

[/tex]

[tex]=\frac{\partial x_{k}}{\partial q_{j}}(

- \frac{\partial U}{\partial x_{k}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot x_{k}}))+ \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}

[/tex]

[tex]=\frac{\partial x_{k}}{\partial q_{j}} (

F_{k})+ \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}

[/tex]

If the last term in the last line vanishes, [tex] Q_{j} [/tex] and [tex] F_{k} [/tex] satisfies the relation (a), but it DOESN't vanish. What's my problem??

I've evaluated the last term in this condition, but It doesn't....

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