Derivation of Lagrangian

Nov 2013
552
30
New Zealand
I am working through the derivation of the lagrangian but don't get this step:

\(\displaystyle \frac{\partial T}{\partial \dot{q}}=\sum_{i=1}^N m_i \mathbf{\dot{r}}_i \cdot \frac{\partial \mathbf{\dot{r}_i}}{\partial \dot{q}} =\sum_{i=1}^N m_i \mathbf{\dot{r}}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q}\)

T is the kinetic energy initially specified in terms of rectangular cartesian co-ordinate variables r_i and then generalised to general co-ordinates q_i. In this step why is d r_dot / d q_dot = d r / d q?
 
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Pmb

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In the first place you omitted the dot product sign. Put that in and get the expression for T which is T = Sum over all i { (1/2)m_i x v_i^2}

where v_i^2 = v_i dot v_i

Then take the derivative with respect to q_i (which has a dot over it which I don't know how to do)

If this is unclear I'll do it out on paper scan it then post it.
 
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Pmb

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See attachment:
 

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Nov 2013
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In the first place you omitted the dot product sign. Put that in and get the expression for T which is T = Sum over all i { (1/2)m_i x v_i^2}

where v_i^2 = v_i dot v_i

Then take the derivative with respect to q_i (which has a dot over it which I don't know how to do)

If this is unclear I'll do it out on paper scan it then post it.
ok, fixed the latex above. Copied it from the textbook wrong (and didn't have access to codecogs equation editor and my hand typed latex not too great). Didn't realise they were vectors. (Caiken M.G. textbook, pg 42) was a bit unclear on this when they developed the equations. Looked like they were expressing the formulas purely in terms of the vector components.

Read your attached page. I saw your product rule expansion of the kinetic energy formula. Seems you haven't gone as far as showing how the dot's cancel yet? I presume there is more to this than just the product rule.

I did eventually find this video on the internet:

https://www.youtube.com/watch?v=7kzTloEwP1E

Seems to rely on equating two different expressions for r_dot. I can see the result but it seems not to be geometrically intuitive.
 

Pmb

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ok, fixed the latex above. Copied it from the textbook wrong (and didn't have access to codecogs equation editor and my hand typed latex not too great). Didn't realise they were vectors. (Caiken M.G. textbook, pg 42) was a bit unclear on this when they developed the equations. Looked like they were expressing the formulas purely in terms of the vector components.

Read your attached page. I saw your product rule expansion of the kinetic energy formula. Seems you haven't gone as far as showing how the dot's cancel yet? I presume there is more to this than just the product rule.
Sorry about that. It's rather simple but I just came from a long walk and ize poopeden. I'll post it later if/when I get around to it. Goldstein's text shows you how yo do it. Do you have a copy or can you download one?
 

Pmb

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Sorry about that. It's rather simple but I just came from a long walk and ize poopeden. I'll post it later if/when I get around to it. Goldstein's text shows you how yo do it. Do you have a copy or can you download one?
Here it is (Please forgive the lack of the vector sign where it should be). See attachment:
 

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Nov 2013
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I saw someone on the internet complaining about Goldstein's pegagogy. Don't personally have experience of it.

Just pursuing these things for my own edification and Caiken is not terrible. Well, his chapter one is a bit terrible but he makes a lot of sense in chapter two onwards when he start's getting into D'Alembert's principle and have managed to make up for lost explanations using this video series:

[youtube]kfa-1N18_Zc[/youtube]

However, each by themselves is probably inadequate.
 

Pmb

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Nov 2013
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Thanks, glancing thru the contents page and chapter 1 looks like they treat the material very similar to the Norwegian University videos. Will make interesting reading.

Difference from Caikin seems to be that Goldstein seems to treat vectors more seriously than what Caikin does (rather than as a "by the way").
 

Pmb

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Thanks, glancing thru the contents page and chapter 1 looks like they treat the material very similar to the Norwegian University videos. Will make interesting reading.

Difference from Caikin seems to be that Goldstein seems to treat vectors more seriously than what Caikin does (rather than as a "by the way").
May I ask what is the exact title of the book you're using and the entire author's name? I want to take a look at how he explains things.

By the way, I'm one of the people who reviewed the first printing of Goldstein et al book. I found well over a hundred errors in it. Mostly misuse of subscripts etc. In reviewing his text on this point I just this moment found an odd statement in it. He sets up what appears to be an equality but its actually what he's proving. It's a very poor way to do that. I contact one of the co-authors and mention it.