Solving Line & Velocity Elements in Spherical Coordinates

In summary, to find the line element in spherical coordinates and the velocity element, start from Cartesian coordinates and calculate the differentials of x, y, and z. Then substitute the differentials in the equation ds^2 = dx^2+dy^2+dz^2 and simplify to get the desired result. To find the velocity vector, use the product rule and calculate the derivatives of the position vector using the unit vectors in spherical coordinates. These unit vectors can be derived using the expressions for x, y, and z in spherical coordinates.
  • #1
Hypnotoad
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I'm trying to find the line element in spherical coordinates as well as a velocity element. I know that they are (ds)^2=(dr)^2+r^2(sin(theta))^2(dtheta)^2+r^2(dphi)^2 and sqrt[(dr/dt)^2+r^2(sin theta)^2(dtheta/dt)^2+r^2(dphi/dt)^2].

I know that this should be a quick and easy problem, but I simply can not figure it out. I would really appreciate some help on this one.
 
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  • #2
Start from Cartesian coordinates in which [itex]ds^2 = dx^2+dy^2+dz^2[/itex] then calculate the differentials dx, dy and dz using:

[tex]x = r \sin \theta \cos \phi[/tex]
[tex]y = r \sin \theta \sin \phi[/tex]
[tex]z = r \cos \theta[/tex]

Substitute for dx, dy and dz in [itex]ds^2 = dx^2+dy^2+dz^2[/itex] and after a bit of algebra you should get the desired result.
 
  • #3
To find the velocity vector, write the position vector as [itex] r(t) \hat r [/itex].
Then [itex]v(t)=\frac{d}{dt} \left[ r(t) \hat r \right] [/itex].
Use the product rule.
You'll have to compute [itex]\frac{d}{dt} \hat r [/itex],
where [tex]\hat r= \sin\theta\cos\phi \hat\imath + \sin\theta\sin\phi \hat\jmath + \cos\theta \hat k[/tex].

To simplify what you get, you might find it useful to know that
[tex]\hat \theta= \cos\theta\cos\phi \hat\imath + \cos\theta\sin\phi \hat\jmath - \sin\theta \hat k[/tex]
and [tex]\hat \phi= -\sin\phi \hat\imath + \cos\phi \hat\jmath [/tex]

You can derive these expressions for the spherical-polar unit vectors if you calculate the vectorial element
[tex]d \vec s = (dx)\hat \imath + (dy)\hat \jmath + (dz)\hat k [/tex]
using Tide's expressions for x, y, and z. [The strategy is to group the terms in [itex] dr[/itex], [itex]d\theta[/itex], and [itex]d\phi[/itex].]
 

1. What is the purpose of solving line and velocity elements in spherical coordinates?

The purpose of solving line and velocity elements in spherical coordinates is to accurately describe the motion and properties of objects in three-dimensional space. It is particularly useful in fields such as physics and engineering, where spherical coordinates are often used to model and analyze systems.

2. What are the three coordinates used in spherical coordinates?

The three coordinates used in spherical coordinates are radius (r), inclination (θ), and azimuth (φ). The radius represents the distance from the origin to the point, while inclination and azimuth describe the angles at which the point is located relative to the origin.

3. How are line elements calculated in spherical coordinates?

In spherical coordinates, the line element is calculated using the Pythagorean theorem, which states that the square of the hypotenuse (line element) is equal to the sum of the squares of the other two sides (radius and change in inclination or azimuth).

4. Can velocity elements be solved in spherical coordinates?

Yes, velocity elements can be solved in spherical coordinates. The velocity elements are calculated using derivatives of the spherical coordinates with respect to time. This allows for the precise determination of the speed and direction of an object's motion in three-dimensional space.

5. What are some real-world applications of solving line and velocity elements in spherical coordinates?

Spherical coordinates are used in a variety of real-world applications, including navigation and astronomy, where they are used to describe the position and movement of celestial bodies. They are also used in physics to model the motion of particles and in engineering to analyze and design complex systems such as aircraft and satellites.

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