Conversion of energy expression from Cartesian to spherical coordinates

In summary, the conversation discusses the method for obtaining velocity analogues of terms in 1.5, specifically the second and third terms in parentheses in 1.6 and 1.7. The question asks for a general procedure to convert such equations between arbitrary coordinate frames, and also mentions familiarity with coordinate transformations between Cartesian and spherical coordinates. A helpful resource is provided for further understanding.
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Syrus
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A text I am reading displays the attached image. Can someone explain the general method for obtaining the velocity analogues of those terms (in parentheses) in 1.5? I know the second and third terms in parentheses in 1.6 and 1.7 are the squares of angular velocities, but can a general procedure be expressed to convert such equations between arbitrary coordinate frames?

In other words, how may we derive equation 1.6 (or 1.7) from 1.5 in an entirely mathematical sense?

(I am familiar with the coordinate transformations between Cartesian and spherical coordinates)

*In the attachment, a dot indicates over a symbol indicates the time-derivative of the quantity over which it appears.
 

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Related to Conversion of energy expression from Cartesian to spherical coordinates

1. What is the purpose of converting energy expressions from Cartesian to spherical coordinates?

The purpose of converting energy expressions from Cartesian to spherical coordinates is to simplify complex calculations and equations involving energy in three-dimensional systems. Spherical coordinates provide a more intuitive representation of energy in relation to the radial distance, polar angle, and azimuthal angle.

2. What is the formula for converting energy expressions from Cartesian to spherical coordinates?

The formula for converting energy expressions from Cartesian to spherical coordinates is as follows:E(r, θ, φ) = x² + y² + z² = r² + (θ² + φ²)sin²θ

3. Can energy expressions be converted from spherical to Cartesian coordinates?

Yes, energy expressions can also be converted from spherical to Cartesian coordinates using the following formula:E(x, y, z) = r² + (θ² + φ²)sin²θ = x² + y² + z²

4. How does converting energy expressions from Cartesian to spherical coordinates affect the values of energy?

Converting energy expressions from Cartesian to spherical coordinates does not change the values of energy. The conversion only changes the way energy is expressed and represented in three-dimensional systems.

5. In what situations would it be necessary to convert energy expressions from Cartesian to spherical coordinates?

It is necessary to convert energy expressions from Cartesian to spherical coordinates in situations where energy is being analyzed in three-dimensional systems, such as in classical mechanics, quantum mechanics, and electromagnetism. This conversion is also useful in solving problems involving spherical symmetry, such as those in astrophysics and geophysics.

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