Small oscillations: How to find normal modes?

In summary, the most correct way to find normal modes is to find the eigenvectors of the matrix ##A_{ab} = V_{ab}/m + \omega_a^2 \delta_{ab}##, normalize them, and use them to form a transformation matrix C. This will result in a set of coordinates that uncouple the equations of motion and can be used to express any oscillation.
  • #1
bznm
184
0
Hi,
I'm studying Small Oscillations and I'm having a problem with normal modes.
In some texts, there is written that normal modes are the eigenvectors of the matrix $V- \omega^2 V$ where V is the matrix of potential energy and T is the matrix of kinetic energy.
Some of them normalize the eigenvector, other don't do it.

In other texts, there is written that normal modes are the coordinates that uncouple the equation of motion and that I can find them as ζ=$B^-1$ η where ζ is the column vector of these normal modes, η is the column vector of initial coordinates and $B^-1$ is the modal matrix (but... for the modal matrix, do I have to normalize eigenvectors?)

Which is the most correct way to find normal modes?
Thank you
 
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  • #2
The normalization is just for convenience. Let's say we have the Hamiltoninan

$$
H = \sum_a \frac{p_a^2}{2m} + \sum_a \frac{1}{2}m \omega_a^2 r_a^2 + \sum_a \sum_b {}^{'} \frac{1}{2} V_{ab} r_a r_b
$$
Any set of eigenvectors of the matrix ##A_{ab} = V_{ab}/m + \omega_a^2 \delta_{ab}## can be used to define new, "normal" basis.

The elements of the transformation matrix C will depend on the choice of the eigenvectors, but otherwise everything is the same for any set. The choice in which all the eigenvectors have unit length is particularly useful since then the transformation matrix C is orthogonal and its inverse, which is also needed, can be obtained easily by simple transposition

$$
C^{-1} = C^T.
$$
 
  • #3
Jano L. said:
The choice in which all the eigenvectors have unit length is particularly useful since then the transformation matrix C is orthogonal and its inverse, which is also needed, can be obtained easily by simple transposition

$$
C^{-1} = C^T.
$$

You are right! I haven't thought about it! :biggrin:

So if I have to find normal modes, I have to find eigenvectors, then I normalize them (using $$Tu \cdot u=1$$) and find modal matrix C. Then, I find $$C^T$$ and post-moltiplicate it for the vector column of the coordinates that at the beginning of the exercise I have choosen to describe the situation: I'll obtain normal modes. Is it correct?
 
  • #4
I think the normalization we speak about does not involve any T. I will try to explain how I would do the transformation.

Let ##\mathbf v_n## be the eigenvector of the matrix A with eigenvalues ##\lambda_n##. Since the matrix is symmetric, these eigenvectors can be chosen to be orthogonal. Then they can be used to form a new basis (basis of normal modes), so any original oscillation can be expressed as linear combination of them.

The vector in the old coordinates can be expressed in the new coordinates ##R_n## in this way:

$$
\mathbf r = \sum_n R_n \mathbf v_n.
$$

so in coordinates we have
$$
r_a = C_{an} R_n,
$$

where ##C_{an} ## is the ##a-##th component of the ##n##-the vector ##v_{n,a}##.
This is a special case of the coordinate transformation

$$
r_a = \frac{\partial r_a }{\partial R_n} R_n
$$

The matrix C of transformation is directly given by the old coordinates of the eigenvectors. So far the normalization was not important.

However, we still have to transform the momenta in the kinetic energy. We want them transformed in such a way so that the equations of motion will hold in the new variables ##R_n, P_n##, and this implies that the momenta have to transform according to

$$
p_a = \frac{\partial R_n }{\partial r_a} P_n
$$
which is
$$
p_a = (C^{-1})_{na} P_n.
$$

If we did not normalize the vectors ##\mathbf v##, we would have to keep in consideration this inverse matrix ##C^{-1}##. But if we normalize them so that
$$
\sum_a v_{an} v_{an} = \delta_{nm},
$$
the matrix C will be orthogonal (columns will be orthogonal unit vectors) and the inverse is calculated easily just by transposition. Then the transformation of momenta is written

$$
p_a = C_{an} P_n,
$$

which is the same as for the coordinates.
 
  • #5
for your question. The most correct way to find normal modes depends on the specific system and problem you are studying. Both methods you mentioned can be valid approaches to finding normal modes, but they may differ in their application and usefulness for different systems.

The first method, using the eigenvectors of the matrix V-ω²V, is based on the idea that the normal modes are the natural frequencies of the system. This method is commonly used in linear systems, where the potential and kinetic energies can be expressed as matrices. In this case, the eigenvectors of the matrix represent the directions of motion that are decoupled and oscillate at a single frequency. Whether or not the eigenvectors need to be normalized depends on the specific problem and what information you are looking for.

The second method, using the modal matrix, is based on the idea that the normal modes are the coordinates that uncouple the equations of motion. This method is often used in non-linear systems, where the equations of motion cannot be easily solved analytically. The modal matrix represents the transformation between the original coordinates and the normal modes. Whether or not the eigenvectors need to be normalized in this method also depends on the specific problem and what information you are trying to obtain.

In summary, both methods can be valid ways to find normal modes, but it is important to understand the underlying assumptions and limitations of each approach. I would recommend consulting with your instructor or doing further research on the specific system you are studying to determine the most appropriate method for your problem.
 

Related to Small oscillations: How to find normal modes?

1. What are small oscillations?

Small oscillations refer to the motion of a system around its equilibrium position, where the amplitude of the oscillations is small in comparison to the equilibrium position. This type of motion is often seen in systems that are stable and can be described using linear equations.

2. What are normal modes?

Normal modes are the different ways in which a system can oscillate due to small perturbations from its equilibrium position. Each normal mode has a specific frequency and shape of oscillation, and the combination of all normal modes determines the overall motion of the system.

3. How do you find the normal modes of a system?

To find the normal modes of a system, you need to solve the equations of motion for the system using the appropriate mathematical techniques, such as eigenvalue problems or Fourier transforms. The resulting solutions will give you the frequencies and shapes of the normal modes.

4. Why is it important to find the normal modes of a system?

Finding the normal modes of a system is important because it allows us to understand the behavior of the system and predict its response to different stimuli. It also helps in designing and optimizing systems for specific purposes, as well as in identifying any potential instabilities.

5. Can small oscillations occur in any type of system?

Small oscillations can occur in any system that is stable and can be described using linear equations. This includes mechanical systems, electrical circuits, and even biological systems. However, the conditions for small oscillations may vary depending on the specific system and its parameters.

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