Contravariant components and spherical component of acceleration

AI Thread Summary
The discussion focuses on deriving the components of acceleration in spherical coordinates and the confusion surrounding contravariant vectors. The original poster successfully derives acceleration components but encounters discrepancies when applying the contravariant vector definition. A suggestion is made to utilize covariant unit vectors to resolve the inconsistencies, leading to unique results. The conversation emphasizes the difference between coordinate basis vectors in Cartesian and spherical systems, noting that spherical coordinate vectors are not unit vectors. The importance of correctly applying vector definitions in different coordinate systems is highlighted.
world line
Messages
8
Reaction score
0
Hello
i know how to derive the components of acceleration in other coordinates like spherical
start here :
http://up.iranblog.com/images/0mbwuclckbu51bxt8jfa.jpg
and at last we have :
http://up.iranblog.com/images/geotowiaxdya2s6ewxk.jpg

also , i know that acceleration is a contravariant vector :
http://up.iranblog.com/images/fl7eosq4cieeoc1kroy.gif
but when i use that definition i don't derive the above result for acceleration !?
and reach to a different results
so what is my mistake ?
 
Physics news on Phys.org
world line said:
so what is my mistake ?

It is very difficult to find your mistake without seeing what you have done.
 
for example :
http://up.iranblog.com/images/90cm6lacnzsoxcju1v9.png
that it is different with the corresponding spherical component

if we use the covaraint unit vector the problem will be solved !
and both solution will have unique result
is it correct ?
 
world line said:
http://up.iranblog.com/images/fl7eosq4cieeoc1kroy.gif

This expression relates the components of a vector with respect to one coordinate basis to the components of the same vector with respect to a different coordinate basis. In general, coordinate basis vectors are not unit vectors. Standard Cartesian coordinate vectors are unit vectors, but spherical coordinate vectors are unit vectors.

world line said:
if we use the covaraint unit vector the problem will be solved !
and both solution will have unique result
is it correct ?

http://up.iranblog.com/images/geotowiaxdya2s6ewxk.jpg

is with respect to contravariant unit vectors.
 
Thank you
 

Thread 'Is calling fictitious forces "not real" just about terminology?'

Hi there, im studying nanoscience at the university in Basel. Today I looked at the topic of intertial and non-inertial reference frames and the existence of fictitious forces. I understand that you call forces real in physics if they appear in interplay. Meaning that a force is real when there is the "actio" partner to the "reactio" partner. If this condition is not satisfied the force is not real. I also understand that if you specifically look at non-inertial reference frames you can...
View full post »

Thread 'Derivation of Hamilton's Principle: Questions'

I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
View full post »

Similar threads

Radial and tangential components of acceleration
Replies
7
Views
3K
I Seeking an intuition linking definitions of contravariant and covariant
Replies
1
Views
2K
Transformation matrix for components of acceleration
Replies
1
Views
4K
Component forces of a pendulum
Replies
2
Views
1K
I Contravariant Vector Transformation in Spherical Polar Coordinates
Replies
13
Views
2K
A Problems with the interpretation of the Torsion tensor and the Lie Bracket
Replies
16
Views
5K
Newtonian gravity and differential equations
Replies
1
Views
1K
B What is the Absolute Value of the Normal force on a block on this wedge?
Replies
3
Views
1K
Showing that the gradient of a scalar field is a covariant vector
Replies
5
Views
4K
Covariant & Contravariant Components
Replies
31
Views
5K

Hot Threads

Recent Insights

Back
Top