I Calculus Question within Lagrangian mechanics

AI Thread Summary
The discussion revolves around calculating derivatives in the context of a Lagrangian function in mechanics. The user is attempting to differentiate the expression for the partial derivative of the Lagrangian with respect to \(\dot{x}\) but is confused about applying the product and chain rules correctly. Clarifications are provided on using the chain rule for derivatives, emphasizing the need to differentiate with respect to time while considering the relationships between variables. The user acknowledges the confusion regarding notation, particularly the use of primes versus dots for derivatives. Overall, the conversation highlights the complexities of applying calculus within Lagrangian mechanics.
Hennessy
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TL;DR Summary
Product rules with hidden chain rules
Hi all currently got a lagrangian function which i've found to be :
\begin{equation}\mathcal{L}=\frac{1}{2}m(\dot{x}^2+\dot{y}^2+4x^2\dot{x}^2+4y^2\dot{y}^2+8xy\dot{x}\dot{y})- mg(x^2+y^2)
\end{equation}
Let us first calculate
$$(\frac{\partial L}{\partial \dot{x}})$$ which leads us to $$m\dot{x}+4x^2m\dot{x}+8xy\dot{y}$$ now we also have to differentiate this again with respect to t.
$$\frac{d}{dt}(m\dot{x}+4x^2m\dot{x}+8xy\dot{y}) $$ Now this is where i'm stuck. I'm stuck because of the 2 product rule in the middle of this term and then the triple product rule on the right hand and then within them i know there are chain rules as $$x,y,\dot{x},\dot{y}$$ are all f(t). Im basically asking how to use the product rule again. using $$x,\dot{x}$$ as my uv then the product rule is $$x'\dot{x}+x\dot{x}'$$ but when i calculate the primes i get confused. so for example $$x'\dot{x}+x\dot{x}'$$ does this mean differentiate the entire function wrt x and then multiply it just by $\dot{x}$ or does it mean multiply it by the entire thing? Advice would be appreciated , i know this is more a calculus question but just trying to figure it out apologies if this is in the wrong place. Put it here as it requires knowledge of lagrangian mechanics for $$x,y,\dot{x},\dot{y}$$ as being time derivatives is all. Thank you!
 
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Use the chain rule for derivatives. If you have a function ##F(t,f_1(t),f_2(t),\ldots)## then
$$
\frac{dF}{dt} =
\frac{\partial F}{\partial t} + \frac{\partial F}{\partial f_1} \dot f_1 + \frac{\partial F}{\partial f_2} \dot f_2 + \ldots
$$

You may want to redo this derivative:
Hennessy said:
Let us first calculate
$$(\frac{\partial L}{\partial \dot{x}})$$ which leads us to $$m\dot{x}+4x^2m\dot{x}+8xy\dot{y}$$

It is also unclear what you mean by ##x’##. We already use dots to denote time derivatives.
 
Orodruin said:
Use the chain rule for derivatives. If you have a function ##F(t,f_1(t),f_2(t),\ldots)## then
$$
\frac{dF}{dt} =
\frac{\partial F}{\partial t} + \frac{\partial F}{\partial f_1} \dot f_1 + \frac{\partial F}{\partial f_2} \dot f_2 + \ldots
$$

You may want to redo this derivative:


It is also unclear what you mean by ##x’##. We already use dots to denote time derivatives.
Hi there, apologies for the confusion. I rewrote the product rule from uv' +vu' and I wrote it in terms of my two functions x and $\dot{x}$ i also understand that i originally wrote my product rule as u'v+v'u , but this shouldn't of changed the result if im not mistaken?
 
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