# Good coordinates and degrees of freedom

• Sahar Ali
In summary, the conversation discusses the concept of good coordinates and degrees of freedom for various systems, including a bead on a loop, a bean on a helix, a particle on a cylinder, a pair of scissors, a rigid rod, a rigid cross, a linear spring, a fixed rigid body, a hydrogen atom, a lithium atom, and a compound pendulum. The number of degrees of freedom and a set of good coordinates are specified for each system. There is also a discussion on the inclusion of spin motion in the counting of degrees of freedom.
Sahar Ali
Moved from a technical forum, so homework template missing
I have to present a topic "Good coordinates and degree of freedom" I know what are good coordinate and degree of freedom. but I will have to explain examples/question given below(from Liboff's text) I know the answer to all of them but I really do not know how to explain these how will I explain these parts to a class while presenting anyone who can give a little explanation?
For each of the following systems, specify the number of degrees of freedom and a set of good coordinates.

(a) A bead constrained to move on a closed circular loop.
(b) A bean constrained to move on the helix of constant pitch and constant radius.
(c) A particle on a right circular cylinder.
(d) A pair of scissors on a plane.
(e) A rigid rod in 3-space.
(f) A rigid cross in 3-space.
(g) A linear spring in 3-space.
(h) Any rigid body with one point fixed.
(i) A Hydrogen atom
(j) A lithium atom
(k) A compound pendulum (two pendulums attached end to end)

(a) The distance along loop from an arbitrary fixed point on the loop. 1 degree of freedom.
(b) The distance along helix from an arbitrary fixed point on the helix. 1 degree of freedom.
(c) Cylindrical coordinates. 2 degrees of freedom.
(d) 3 numbers to locate the center of scissors. One for angle scissors makes with the chosen axis. One for angle scissors is open. 5 degrees of freedom.
(e) 3 numbers to locate the center of a rod in space. Two numbers to orient rod in space, typically q and f.5 degrees of freedom.
(f) 3 numbers to locate a center of the rod in space. Two numbers to orient the rod in space. Two numbers to rotate about both axes in space. 6 degrees of freedom.
(g) Three numbers to locate the center of spring in space, two numbers to orient spring in space and one number for amount spring is stretched. 5 degrees of freedom.
(h) 3 numbers to locate a body in space. 2 numbers to orient body and 2 numbers about each axis of rotation. 7 degrees of freedom
(i) 3 numbers to locate proton in space. 3 numbers to locate the electron in space. 6 degrees of freedom.
(j) 3 numbers to locate the nucleus in space. 3 numbers for each electron in space. 12 degrees of freedom.
(k) 2 degrees of freedom for the first pendulum. 2 degrees of freedom for the second pendulum.

Sahar Ali said:
(d) 3 numbers to locate the center of scissors. One for angle scissors makes with the chosen axis. One for angle scissors is open. 5 degrees of freedom.
The scissors are moving in a plane. Not in three dimensions.

Sahar Ali said:
(f) 3 numbers to locate a center of the rod in space. Two numbers to orient the rod in space. Two numbers to rotate about both axes in space. 6 degrees of freedom
6 is correct, but it is not equal to 3+2+2.

Sahar Ali said:
g) Three numbers to locate the center of spring in space, two numbers to orient spring in space and one number for amount spring is stretched. 5 degrees of freedom.

3+2+1=?

Sahar Ali said:
(h) 3 numbers to locate a body in space. 2 numbers to orient body and 2 numbers about each axis of rotation. 7 degrees of freedom
The body is fixed in one point. Your counting of rotation angles is also wrong.

Sahar Ali said:
(j) 3 numbers to locate the nucleus in space. 3 numbers for each electron in space. 12 degrees of freedom
This depends on how you consider the degrees of freedom of the nucleus. In general, it would be inconsistent to give all electrons full freedom and not the nucleons. You should specify that you disregard degrees of freedom internal to the nucleus.

Sahar Ali said:
(k) 2 degrees of freedom for the first pendulum. 2 degrees of freedom for the second pendulum
Assuming the penduli move completely independent, yes.

Cant, we consider spin motion in all these cases? what if we take both spin and linear motion?

## 1. What are good coordinates?

Good coordinates refer to a set of values that can uniquely describe the position of an object or system in space. These coordinates can be used to determine the location of the object and its orientation.

## 2. How do you determine the degrees of freedom in a system?

Degrees of freedom refer to the number of independent variables or parameters needed to describe the state of a system. In order to determine the degrees of freedom in a system, you must identify all the independent variables that can affect the system's state, and count the number of these variables.

## 3. Why is it important to have good coordinates and degrees of freedom?

Having good coordinates and degrees of freedom is crucial in order to accurately describe the state of a system or object. These values allow scientists to make precise measurements, predict behavior, and understand the dynamics of the system.

## 4. Can coordinates and degrees of freedom change over time?

Yes, coordinates and degrees of freedom can change over time. This is especially true for dynamic systems where the state of the system is constantly changing. In these cases, it is important to continuously monitor and update the coordinates and degrees of freedom to accurately describe the system's state.

## 5. How do you choose the best coordinates and degrees of freedom for a system?

The selection of good coordinates and degrees of freedom depends on the specific system being studied. Generally, the coordinates and degrees of freedom should be chosen based on the system's geometry and dynamics, and should be as few as possible while still accurately describing the system's state. It is also important to consider the limitations of the measuring equipment and the complexity of the system when choosing these values.

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