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Why it is said that

$$\frac{dx^2}{dt^2}=\upsilon^2$$

I can only understand the following one:

$$\left (\frac{dx}{dt} \right)^2=\upsilon^2$$

Edit:

Excerpt from Landau's Mechanics:

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Execrpt from Shourt Course of Theoretical Physics by the same authors (Landau and Lifshits):

enter image description here

So it seems to be a universally accepted notation in the old days (both books are from 70s).

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3 Answers

Oh I got it. It means:

$$\upsilon^2 = \left (\frac{dx}{dt} \right)^2=\frac{(dx)^2}{(dt)^2}$$

but they do write it in the form of

$$\frac{dx^2}{dt^2}$$

(for example in the Russian edition of Landau's Mechanics (in the English one they've put it in the right way))

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I guess Russians want to save some ink by avoiding to print two parentheses –  Pygmalion Apr 23 '12 at 14:56
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If this is the intent of the source you are looking at you should be aware that the notation is not standard. –  dmckee Apr 23 '12 at 15:11
    
@dmckee Of course I was joking. –  Pygmalion Apr 23 '12 at 16:50
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I think you maybe realised that it can't be the second derivative $\frac{d^2x}{dt^2}$ because the square is after the x: $\frac{dx^2}{dt^2}$

$\frac{d^2x}{dt^2} \neq \frac{dx^2}{dt^2} = \left ( \frac{dx}{dt} \right ) ^2$

Disregard this if you already knew :)

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Maybe

$$\frac{dx^2}{dt^2} := \frac{d^2x}{dt^2}$$

in Russian notation, but let's just say that left notation is not valid according to standard ISO 80000-2 Mathematical signs and symbols to be used in the natural sciences and technology, item 2-11.14.

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