- #1

Soren4

- 128

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[itex] (\ddot r - r\dot{\varphi}^2)\hat{\mathbf r} + (2\dot r \dot\varphi+r\ddot{\varphi}) \hat{\boldsymbol{\varphi}} [/itex] (1)

I do not understand what is the correct explanation for the presence of terms that seem the ones found in non inertial frames, like [itex]2\dot r \dot\varphi[/itex], mathematically equal to the Coriolis term present in the expression of acceleration for non inertial frames of reference, which I write down here. Does this mean that the polar coordinate system is non inertial?

[itex] \vec{a}= \vec{a}'+2\vec{\omega}\times\vec{v}'+\dot{\vec{\omega}}\times\vec{r}'+\vec{\omega}\times(\vec{\omega}\times\vec{r}')[/itex] (2)

I have the idea that polar coordinates are just a particular case of a

**non-inertial rotating frame**. The "special" thing about it is that the point is constantly on the [itex]x[/itex] axis (that is the axis oriented as the unit vector [itex]\hat{\mathbf r}[/itex] ), which is rotating as far as the particle move away from the radial direction. Is this the correct way to see it?

I found on Wikpedia (https://en.wikipedia.org/wiki/Polar_coordinate_system#Centrifugal_and_Coriolis_terms) this answer to my question.

The term [itex]r\dot\varphi^2[/itex] is sometimes referred to as the centrifugal term, and the term [itex]2\dot r \dot\varphi[/itex] as the Coriolis term. Although these equations bear some resemblance in form to the centrifugal and Coriolis effects found in rotating reference frames, nonethelessthese are not the same things. In particular, the angular rate appearing in the polar coordinate expressions is that of the particle under observation, [itex]\dot{\varphi}[/itex], while that in classical Newtonian mechanics is the angular rate [itex]Ω[/itex] of a rotating frame of reference. The physical centrifugal and Coriolis forces appear only in non-inertial frames of reference. In contrast, these terms, that appear when acceleration is expressed in polar coordinates, area mathematical consequence of differentiation; these terms appear wherever polar coordinates are used. In particular, these terms appeareven when polar coordinates are used in inertial frames of reference, where the physical centrifugal and Coriolis forces never appear.

I highlighted the things that confuses me the most. In particular here it is claimed that such coordinate system can be "used in inertial frame of reference", which is obviously against my idea of polar coordinates as non inertial frame. Now my question is: if the polar coordinate system is inertial, then how to interpret the terms that appear in the expression for acceleration (1)?

Here it says that these terms are not to be interpreted as caused by fictitious forces, but that they just come from differentiation. That is true, but isn't it the same for the (real ?) non inertial frames? In order to derive the expression for acceleration in non inertial frames (2) a differentiation (which takes in account the variation of unit vectors) is done, nothing more than that.

Did I misunderstand Wikipedia or am I wrong to consider polar coordinates a non inertial frame of reference? If this is the case, how can I interpret in the proper way those terms in (1)?