Point transformation for a constrained particle

Lambda96
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Homework Statement
Lagrangian mechanics for a mass particle
Relevant Equations
none
Hi,

unfortunately, I'm not that fit concerning the Lagrangian formalism, so I'm not sure if I solved the problem 1a correctly.

Bildschirmfoto 2022-11-24 um 12.51.39.png


I have now proceeded as follows

the Lagrangian is

$$L=T-U$$

Since there are no constraining or other forces acting on the point mass, I assume that the potential energy is 0 and thus the system has only kinetic energy, i.e.

$$T=\frac{1}{2}m*(\dot{x}^2+\dot{y}^2)$$

I would now represent the entire equation in the x coordinate only, so.

$$x=x$$
$$y=f(x)$$

Insert into the Lagrangian $$T=\frac{1}{2}m*(\dot{x}^2+\dot{f}^2(x))$$

Thus, I would be done with task 1a, or did I do something wrong?
 
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That is correct, except that if y = f(x) then \dot y = f&#039;(x)\dot x, so <br /> T = \frac12 m (1 + f&#039;(x)^2)\dot x^2.
 
Thank you pasmith for your help 👍 , I had completely forgotten the chain rule.

Concerning task b, the Euler-Lagrange equation is as follows

$$\frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}=0$$

For $$\frac{\partial L}{\partial x}$$

I am not quite sure since f(x) depends on x, would the following then apply

$$\frac{\partial L}{\partial x}=m f''(x)$$

and

$$\frac{\partial L}{\partial \dot{x}}=m\dot{x}+m f'(x)^2\dot{x}$$

$$\frac{d}{dt}m\dot{x}+m f'(x)^2\dot{x}=m\ddot{x}+m f''(x)^2\ddot{x}\dot{x}$$

Then the Euler-Lagrange equation is

$$m f''(x)-m\ddot{x}-m f''(x)^2\ddot{x}\dot{x}=0$$
 
Your lagrangian is L = \frac12 m g(x)\dot x^2 where g(x) = 1 + f&#039;(x)^2. Hence <br /> \begin{split}<br /> \frac{d}{dt}\left( \frac{\partial L}{\partial \dot x}\right) - \frac{\partial L}{\partial x} &amp;= \frac{d}{dt}(mg(x)\dot x)<br /> - \tfrac12 m g&#039;(x) \dot x^2 \\<br /> &amp;= mg(x) \ddot x + mg&#039;(x) \dot x^2 - \tfrac12 m g&#039;(x) \dot x^2 \\<br /> &amp;= mg(x) \ddot x + \tfrac12 m g&#039;(x) \dot x^2. \end{split} So what is g&#039;(x) using the chain rule?
 
Lambda96 said:
$$\frac{\partial L}{\partial \dot{x}}=m\dot{x}+m f'(x)^2\dot{x}$$

$$\frac{d}{dt}m\dot{x}+m f'(x)^2\dot{x}=m\ddot{x}+m f''(x)^2\ddot{x}\dot{x}$$
Don't forget the product rule.
 
Lambda96 said:
$$\frac{\partial L}{\partial \dot{x}}=m\dot{x}+m f'(x)^2\dot{x}$$

$$\frac{d}{dt}m\dot{x}+m f'(x)^2\dot{x}=m\ddot{x}+m f''(x)^2\ddot{x}\dot{x}$$
This is not good!

I suggest you take ##f(x) = \sin x## or something like that and check your differentiation against a definite function.
 
I don't want to be rude, but you are missing trivial points about elementary calculus, as chain rule and product rule.

May i ask, have you take those class? Don't try to skip steps in your learning process.
 
Thanks to all for the help, especially pasmith 👍👍

I have now formed the derivative of ##g(x)=1+f'(x)^2## as follows.

The outer derivative is ##2f'(x)##and the inner derivative would be ##f''(x)##

Thus, the derivative of ##g'(x)## would be ##g'(x)=2f'(x)f''(x)##

The Euler-Lagrange equation is ##m\ddot{x}+mf'(x)^2\ddot{x}+mf'(x)f''(x)\dot{x}^2##Concerning task part c and d

##\textbf{task c}##
##|l(x)|=\int_{x_0}^{x} \sqrt{1+f'(x)^2}dx##

##\textbf{task d}##
I would now simply parameterize the curve ##l(x)=\left(\begin{array}{c} x \\ f(x) \end{array}\right)## of ##[x_0,x]## with ##l_1=x## and ##l_2=f(x)##

Then the Lagrange function would look like ##L=\frac{1}{2}m(\dot{l_1}^2+\dot{l_2}^2)##
 
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