Question

Why does \[x(t) \frac{d}{dx}f(x(t)^2)=2f(x(t)^2)\]?

Answer 1

http://mathworld.wolfram.com/EulersHomogeneousFunctionTheorem.html
Is, nominally, the explanation, but I'm not getting it.

Answer 2

XIV??

Answer 3

can't view ... unless I sign up, try uploading snapshot.

Answer 4

http://gyazo.com/344cc91e03e14eed7e06669d9a266ee8

Answer 5

Oh- I see it now!

Answer 6

Euler assumes that \[T(v)=kv^2\], and so \[T'(v)=2kv\]!

Answer 7

My god I've been stupid!

Answer 8

Thanks!

Answer 9

I guess by square of velocity as square ... they mean.
\[ v^2 = v_x^2 + v_y^2 +v_z^2\]

Answer 10

*as scalar

since Lagrangian has been differentiated as \( \dot q \) , the potential part is zero ,, the product with \( \dot q \) will give the above expression ... as you realized.

Answer 11

*wrt \( \dot q \)

Answer 12

looks like i should download this book too ... I have Goldstein ... but haven't gone through it.

Answer 13

I mean pirate