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I know there are many related questions already posted on this topic. And there are several answers too. But I am still confused with obtaining Newton's second law from Lagrange's equation. (I don't have enough reputations to add comments on previously posted questions. That is why I am posting this question)

Let me try to tell what I know Lagrange's equation \begin{equation}\frac{d}{dt} \frac{\partial L}{\partial \dot{x}}-\frac{\partial L}{\partial x}=0\end{equation}

Where Cartesian coordinates are itself generalized coordinates. Let $L=T-V$ and, $T=\frac{1}{2}m \dot{x^2}$

Consider $T$ only, because it is what upset me.

According to my text book \begin{equation} \frac{\partial T}{\partial x}=0 \end{equation}

But according to me,

\begin{equation}\frac{\partial T}{\partial x}=\frac{1}{2}m \frac{\partial}{\partial x}\dot{x}\end{equation} Since \begin{equation} \frac{\partial}{\partial x}\dot{x}= \frac{\partial \dot{x}}{\partial t} \frac{\partial t}{\partial x}=\frac {\ddot{x}}{\dot{x}}\end{equation}

On substitution I get \begin{equation} \frac{\partial T}{\partial x}=m \ddot{x} \end{equation}

That is all...

I know I got mistake somewhere. I spend hours. Still couldn't resolve it. Help me..

Thomas Fritsch
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mig001
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  • Related: Calculus of variations — how does it make sense to vary the position and the velocity independently? and links therein. – Qmechanic Oct 06 '19 at 13:29
  • Remember with variational techniques we are allowed to vary $\dot x$ without varying $t$ or $x$. When we obtain our equations of motion from the Lagrangian then we obtain a specific $x(t)$, but with the Euler-Lagrange equations $x$, $\dot x$, and $t$ are all independent variables. – BioPhysicist Oct 06 '19 at 13:33
  • Thank you @Aaron Stevens – mig001 Oct 06 '19 at 13:39
  • The mass term is $m=\dfrac{\partial }{\partial \dot x} \left( \left( \dfrac{\partial T}{\partial \dot x}\right) \right) $ and the force term $F=-\dfrac{\partial V}{\partial x}$ – Eli Oct 06 '19 at 14:18
  • This is a good question. When you actually compute the variation of the action, indeed velocity and position are related (the former is the time derivatives of the latter). But when you derive Euler-Lagrange equation (which is what you are considering) you treat them independently. This is also the meaning of the notation of the partial derivative ("you differentiate but keeping other variable constant"). – lcv Oct 06 '19 at 14:27

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