Contravariant components and spherical component of acceleration

In summary, the conversation discusses deriving the components of acceleration in different coordinate systems, specifically spherical coordinates. It is important to use the correct type of unit vectors, either contravariant or covariant, in order to obtain the correct results.
  • #1
world line
8
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 ?
 
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  • #2
world line said:
so what is my mistake ?

It is very difficult to find your mistake without seeing what you have done.
 
  • #3
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 ?
 
  • #4
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.
 
  • #5
Thank you
 

1. What are contravariant components of acceleration?

Contravariant components of acceleration refer to the acceleration of an object in different directions, relative to a set of coordinate axes. These components are independent of the coordinate system chosen, and are typically represented as vectors in three-dimensional space.

2. How are contravariant components of acceleration calculated?

The contravariant components of acceleration can be calculated using the formula ai = ai/gi, where ai represents the contravariant component, ai represents the acceleration in the chosen coordinate direction, and gi represents the corresponding unit vector.

3. What is the significance of spherical component of acceleration?

The spherical component of acceleration is the component of an object's acceleration that is directed towards or away from the center of a spherical coordinate system. It is particularly useful in analyzing motion in circular or spherical paths, such as planetary orbits.

4. How is the spherical component of acceleration calculated?

The spherical component of acceleration can be calculated using the formula ar = a cos(θ), where ar represents the radial component of acceleration, a represents the magnitude of the total acceleration, and θ represents the angle between the acceleration vector and the radial direction.

5. Can contravariant components and spherical component of acceleration be used in any coordinate system?

Yes, contravariant components and spherical component of acceleration can be used in any coordinate system. However, they are most commonly used in rectangular and spherical coordinate systems due to their simplicity and applicability in many physical situations.

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