Central force and orthogonal transformation

[AmateurNS]

How can I get the central force law by using orthogonal transformation of position vector, x=Ar where A is an orthogonal matrix and r is a position vector?

Thanks!

 

[AmateurNS]

Actually, how can I prove that central force field has a spherical symmetry by using orthogonal transformation of position vector?

 

[charng]

Central force refers to a type of force that acts on an object towards or away from a fixed point, known as the center of force. This force is dependent only on the distance between the object and the center of force, and not on the direction of motion of the object. Orthogonal transformation, on the other hand, refers to a mathematical operation that preserves the length and angle of a vector.

To obtain the central force law using orthogonal transformation, we can start by expressing the position vector, r, in terms of the orthogonal matrix, A. This can be done by using the formula x=Ar, where x is the transformed position vector.

Next, we can apply the transformation to the central force law, which is given by F= -k/r^2, where k is a constant and r is the distance between the object and the center of force. This results in F= -k/(Ar)^2.

Simplifying this equation, we get F= -k/A^2r^2. Since A is an orthogonal matrix, A^2 is equal to the identity matrix, I. Therefore, the equation can be written as F= -k/Ir^2.

We know that the identity matrix does not change the value of a vector, so we can remove it from the equation. This leaves us with the central force law in its original form, F= -k/r^2.

In summary, by using orthogonal transformation of the position vector, we were able to obtain the central force law in terms of the distance between the object and the center of force. This demonstrates the power and usefulness of mathematical transformations in simplifying and understanding complex physical laws.