Central Force Problem
Central Force Problem
Central Force Problem
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Kinetic Energy<br />
mi<br />
T = ∑<br />
2<br />
i<br />
2<br />
i<br />
• Using the chain rule<br />
r<br />
• This wouldn’t work if r = r( q ,..., q , t)<br />
because<br />
dr<br />
i<br />
dt<br />
r<br />
= r(<br />
q ,..., q )<br />
i i 1 n<br />
dr<br />
∂<br />
= ∑ ∂ q<br />
j<br />
dt<br />
i<br />
∂<br />
= ∑ ∂ q<br />
i i 1 n<br />
ri<br />
q <br />
j<br />
j<br />
ri<br />
q <br />
j<br />
j<br />
∂ri<br />
+ ∂ t<br />
Time-independent<br />
m m m<br />
r = ∂r ⋅ ∂r qq = qq <br />
∂r ⋅<br />
∂r<br />
∑ ∑ ∑ ∑ ∑<br />
j<br />
i 2<br />
i i i i i i<br />
i j k j k<br />
i 2 i 2 jk , ∂qj ∂qk jk , i 2 ∂qj ∂qk<br />
2 nd order homogeneous No q