I Confused about applying the Euler–Lagrange equation

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The discussion revolves around deriving the equation of motion from a Lagrangian of the form L = (mv^2)/2 + f(v)v, where f(v) is a velocity-dependent function. The key question is about the correct expression for ∂L/∂v, with initial confusion between two forms. The correct expression is clarified to be mv + f(v) + f'(v)v, emphasizing the need for ordinary derivatives rather than partial derivatives in this context. Additionally, participants suggest rewriting the Lagrangian using g(v) = f(v)v to simplify the analysis. The conversation highlights the importance of proper notation and understanding of derivatives in Lagrangian mechanics.
Malamala
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Hello! I have a Lagrangian of the form:

$$L = \frac{mv^2}{2}+f(v)v$$
where ##f(v)## is a function of the velocity. I would like to derive the equation of motion in general, without writing down an expression for ##f(v)## yet. I have that ##\frac{\partial L}{\partial x} = 0##. However, what is ##\frac{\partial L}{\partial v}##? Is it ##mv+f(v)## or ##mv+f(v)+\frac{\partial f}{v}v##? Thank you!
 
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I guess you meant ##\frac{\partial f}{\partial v}##.

For the purpose of experiment, assume ##f(v)=\alpha v## where ##\alpha## is a constant with appropriate units (or you may assume something else simple if you prefer). You can now calculate your Lagrangian explicitly. Which of your approaches gives the right answer in that case?
 
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Malamala said:
However, what is ##\frac{\partial L}{\partial v}##? Is it ##mv+f(v)## or ##mv+f(v)+\frac{\partial f}{v}v##?
It is essentially the second expression. But the notation is a bit off in the last term where you wrote ##\frac{\partial f}{v}v##. A partial derivative should have the symbol ##\partial## in both the numerator and denominator: ##\frac{\partial f}{\partial v}##. However, note that ##f(v)## is a function of the single variable ##v##. So, a partial derivative is not really appropriate. Instead, the notation should express an ordinary derivative ##\frac{df}{dv}## or ##f'(v)##. Thus, the last term would be ##f'(v)v##.

I assume that you are dealing with a one-dimensional problem with spatial coordinate ##x## and where ##v = \frac{dx}{dt}##.
 
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Malamala said:
Hello! I have a Lagrangian of the form:

$$L = \frac{mv^2}{2}+f(v)v$$
where ##f(v)## is a function of the velocity. I would like to derive the equation of motion in general, without writing down an expression for ##f(v)## yet. I have that ##\frac{\partial L}{\partial x} = 0##. However, what is ##\frac{\partial L}{\partial v}##? Is it ##mv+f(v)## or ##mv+f(v)+\frac{\partial f}{v}v##? Thank you!
Let ##g(v) = f(v)v##. Rewrite your Lagrangian using ##g(v)##. What do you do now that the lone ##v## has gone?
 
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