Easy way to obtain spherical coordinate system speed vector

 

Professional translation is mistranslation, not many. I'm sorry.

" I think. Therefore I am. "

Renee Descartes, a French physicist, mathematician and philosopher, mathematically

mathematically defined the space of the world in her book Geometry.

This is called Cartesian coordinate system or orthogonal coordinate system.

Cartesian coordinate systems allow us to accurately represent the position of an object in numbers. 

Divide the space into the X-axis, the Y-axis, and the z-axis, and then mark the distance for each axis.

However, there are other ways to indicate the position of the object.

Alternatively, you can display the position vector 

as the sum of units vectors of size 1 and each axis direction.

The direct election from origin to object is defined as r.

The angle from the reference plane to the straight line is defined as θ.

Define the angle of the plane perpendicular to the reference plane as Φ.

You can then mark the position of the object as the coordinates of r, θ, Φ.

This is called a spherical coordinate system.

The spherical coordinate system uses the units vector

r cap, θ cap, Φ cap, and Φ cap, 

which indicate the direction in which r, θ, and Φ change in size,

to mark the position and speed vector of the object.

An object in the space has moved from S1 to S2.

There are several ways to move from S1 to S2, 

but eventually the displacement is the same.

But in a very, very, very short moment, very small changes in position can vary from moment to moment.

This very small change is marked by a micro which means very small, with a micro-displacement symbol.

The ds vector is the smallest displacement that moves when an object moves from its current position to another position. 

In other words, it is a small displacement that should be assumed that you cannot move with a smaller displacement than this.

When an object like that moves this displacement to micro-time called dt,

ds/dt = v

It's a differential method that saves the instantaneous speed.

The microdisplacement ds may vary from moment to moment.

But every moment of micro-displacement ds pile up to create a path.

All sum of microdisplacement creates final displacement △S.

We will focus on momentary microdisplacement ds.

Because the space of the orthogonal coordinate system is defined only by the x-axis, y-axis, and z-axis.

The object here should be described as movement 

in the direction of unit vector i cap, j cap, and k cap.

So the microdisplacement ds is the result of a combination of very small changes in the i cap direction at some point, very small changes in the j cap direction, and very small changes in the kcap direction.

Marked as vector sum: ds = dxi(cap) + dyj(cap)+ dzk(cap)

The spherical coordinate system shall describe the micro-displacement ds as changes to the r-axis, θ-axis, and Φ-axis.

It defines the movement of an object as a unit vector of r cap, θ cap, Φ cap.

So I'm sure there are a lot of friends who want to do this.

It may seem reasonable at first glance,

If you think about it a little bit, you find something strange.

Where the object is located at any moment,

the r cap, Φ cap, and Φ cap each indicate the direction in which the coordinates (r,θ,Φ) of the object change.

Then, mark the microdisplacement ds as in the space the sum of the small changes in the direction of r cap, c cap, and Φ cap.

The coordinates (r, θ, Φ) of the spherical coordinate system

change slightly by micro-displacement ds.

The amount is called dr, dθ, and dΦ.

Then A is the changed length in the direction of the r cap, so it becomes dr.

However, dθ is the amount of variation in the angle, so it cannot be the length of B in the coordinate space.

Therefore, the size of B should be rdθ 

using the proportional relationship between arc and center angle.

C cannot be dΦ for the same reason as B.

Also, Φ cap moves in the vertical direction of θ cap.

The size of C is the location of the object

and the vertical distance of the axis of rotation

rsinθ multiplied by the center angle dΦ.

In other words, the micro-displacement ds is the equivalent of a spherical coordinate system.

It is a vector with a change of dr in the direction of r cap, 

a change of rdθ in the direction of θ cap, 

and a change of rsinθdΦ in the direction of Φ cap.

The speed can then be obtained by dividing the microdisplacement by the micro-time it took to move.

This is the velocity vector expressed in the spherical coordinate system.

The component in the square is the velocity in the direction of the rvector in the spherical coordinate system.

The component in a circle is called angular velocity, especially angular velocity.

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