Why use normal coordinates and what is the benefit?

In summary, the conversation is discussing a method for finding eigen vectors and eigen values for a system of two masses connected by springs. The original method using a matrix did not work, but using a coordinate transform to normal coordinates yielded a successful solution.
  • #1
noamriemer
50
0
Hello!
I have a question regarding a method I saw every now and then:
Say I have a system containing of two masses, attached to one another by a spring. Each is attached to a wall by another spring.
Now I wish to know the eigen vectors and eigen values( [itex]\omega[/itex]) of movement.

I get:
[itex]\ddot x_1= \frac {-2k} {m} x_1+\frac {k} {m} x_2[/itex]
[itex]\ddot x_2= \frac {-2k} {m} x_2+\frac {k} {m} x_1[/itex]

That is because the sysem is completely symmetric. Now, I get
[itex]{\omega_1}^{2}= \frac {3k} {m}[/itex] and [itex] {\omega_2}^{2}= \frac {k} {m} [/itex]
But if I try to find the eigen vectors using a matrix,

|5 -1 |
|-1 5 |
As you can see, the only solution is the trivial one.

So, what I saw was done in this case, was defining new coordinates:
[itex]y_1=x_1+x_2[/itex]
[itex]y_2=x_1-x_2[/itex]
And now it works, and everything is fine.
What I don't understand is why.

How is it that when I move to y-s it is ok?
Shouldn't I get the same result?
Thank you!
 
Physics news on Phys.org
  • #2
noamriemer said:
Hello!
I have a question regarding a method I saw every now and then:
Say I have a system containing of two masses, attached to one another by a spring. Each is attached to a wall by another spring.
Now I wish to know the eigen vectors and eigen values( [itex]\omega[/itex]) of movement.

I get:
[itex]\ddot x_1= \frac {-2k} {m} x_1+\frac {k} {m} x_2[/itex]
[itex]\ddot x_2= \frac {-2k} {m} x_2+\frac {k} {m} x_1[/itex]

That is because the sysem is completely symmetric. Now, I get
[itex]{\omega_1}^{2}= \frac {3k} {m}[/itex] and [itex] {\omega_2}^{2}= \frac {k} {m} [/itex]
But if I try to find the eigen vectors using a matrix,

|5 -1 |
|-1 5 |
As you can see, the only solution is the trivial one.

So, what I saw was done in this case, was defining new coordinates:
[itex]y_1=x_1+x_2[/itex]
[itex]y_2=x_1-x_2[/itex]
And now it works, and everything is fine.
What I don't understand is why.

How is it that when I move to y-s it is ok?
Shouldn't I get the same result?
Thank you!

I'm not quite sure how you got your eigenvectors here. It's been a while but my recollection is that your matrix is something like [1 -1; 1 1]. Or maybe even that is wrong. With the original equation, you have two equations of second order so that's four unknowns for your coefficients that arise because each oscillator will be influenced by both eigenfrequencies. But the thing to note about the coordinate transform is that you have now gone to what are called normal coordinates. These normal coordinates are uncoupled and independent which allows you to write the equations of motion to be dependent upon a single eigenfrequency. So the uncoupled solutions makes this the more desirable method of analysis.
 

1. Why is it necessary to change coordinates?

The coordinates of a location can change due to various reasons such as plate tectonics, continental drift, or human activities. It is necessary to change coordinates to accurately represent the location and ensure consistency in mapping and navigation systems.

2. How often do coordinates need to be changed?

The frequency of changing coordinates depends on the rate of change in the location. For example, if the location is affected by plate tectonics, the coordinates may need to be updated every few years. However, if the location is stable, the coordinates may not need to be changed for decades.

3. Can changing coordinates affect the accuracy of maps?

Yes, changing coordinates can affect the accuracy of maps. However, it is necessary to update coordinates to maintain accuracy and avoid confusion in navigation systems. The changes in coordinates are carefully measured and recorded to ensure minimal impact on the accuracy of maps.

4. Who is responsible for changing coordinates?

Changing coordinates is a collaborative effort between scientists, cartographers, and mapping agencies. Scientists collect data and analyze changes in the location, while cartographers and mapping agencies use this data to update and adjust coordinates on maps and navigation systems.

5. How does changing coordinates impact GPS systems?

GPS systems rely on accurate coordinates to determine the location of a device or individual. Changing coordinates can affect the accuracy of GPS systems, but these systems are equipped with algorithms to adjust and adapt to changes in coordinates. GPS systems also regularly receive updates from mapping agencies to ensure accurate coordinates are being used.

Similar threads

Replies
3
Views
621
  • Introductory Physics Homework Help
Replies
4
Views
684
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
1K
Replies
2
Views
596
  • Linear and Abstract Algebra
Replies
17
Views
1K
  • Classical Physics
Replies
2
Views
1K
Replies
3
Views
959
  • Electromagnetism
Replies
2
Views
803
Replies
27
Views
2K
Back
Top