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Method 3: Phase Space

In the previous section, we derived the differential equation that describes the motion of a mass-spring system. This differential equation is a second-order ordinary differential equation (ODE) that can be written as:

In this section, we will explore the third method of solving this differential equation: using the powerful tools of phase space and linear algebra.

Table of Contents​

Phase Space​

We know that the second derivative of the position function is the acceleration of the mass, . Let's define a new function , which represents the velocity of the mass. Then, Equation can be rewritten as:

And since , we have two first-order ODEs:

This is the fun part - now imagine we have a coordinate system with two axes: one for and one for . Each point in this coordinate system represents a state of the system at a given time . Let be the state vector of the system.

Notice that this coordinate system is not a physical space, unlike the usual -plane. When we have a system like this, we call it the phase space of the system.

Then, the system of first-order ODEs can be written as:

This means that at each point in the phase space, the vector field gives the direction of the velocity vector at that point. We can graph this vector field to visualize the behavior of the system:

In the graph above, the horizontal axis represents the position of the mass, and the vertical axis represents the velocity of the mass. By placing a point on the phase space and tracing its path, we can visualize the motion of the mass-spring system:

(The movement is not perfectly accurate since its done numerically, but it gives a good idea of the motion.)

Notice that the path traced by the point is an ellipse, which should make intuitive sense as the system oscillates back and forth. We can analyze some properties of the system by looking at this phase space. For one, we can pick an initial point and trace its path to see how the system evolves over time:

We can notice that:

  • At the horizontal intercepts, the velocity is zero, while the displacements are at their maximum. This corresponds to the mass being at the extreme points of its motion.
  • At the vertical intercepts, the displacement is zero, while the velocity is at its maximum. This corresponds to the mass being at the equilibrium position.

Vectors and Matrices​

To solve the system of first-order ODEs, carefully observe the system of equations:

We can rewrite the right-hand side of the equation as a matrix-vector product, where the vector is just the state vector :

This matrix is known as the state matrix of the system and is denoted by . If we use all the variables in the system, we can write the system of ODEs as:

This should look familiar if you've studied any differential equations. It resembles exponential growth and decay equations, where we have . In that case, the solution is ( is added to account for the initial conditions. It's the same thing as multiplying the constant by , which is another constant). Similarly, we can guess the solution to the system of ODEs as:

Where and are some unknown constants for now. Substitute this into the system of ODEs:

So we have . Once again, this should look familiar if you've studied linear algebra. is an eigenvalue of the matrix , and is the corresponding eigenvector.

Eigenvalues and Eigenvectors​

Now we need to find the eigenvalues of the state matrix . We can use the trick of finding the eigenvalues of a matrix by the following equation:

Where is the trace of the matrix (and the mean of the eigenvalues) and is the determinant of the matrix (and the product of the eigenvalues). For the state matrix , we have:

The eigenvalues are complex numbers, which means the system will oscillate.

Deriving the General Solution​

Now that we have and , as well as the solution , we can derive the general solution to the system of ODEs. First, we need to find values for - they must be eigenvectors of the state matrix .

For , we have:

This gives us the system of equations:

The second equation is just the first equation multiplied by . Thus, for any and , where is a constant, we have an eigenvector. Therefore, the first eigenvector is .

Similarly, for , we have the eigenvector .

Then, the general solution to the system of ODEs is the sum of the two eigenvectors multiplied by their corresponding eigenvalues:

This is the general solution to the system of ODEs that describe the motion of an undamped mass-spring system. However, this solution is in terms of complex numbers, which can be difficult to interpret. So we can separate the real and imaginary parts of the solution to get a more intuitive form.

Using Euler's formula , we can rewrite the solution as:

Let and . Then, the solution is:

Taking just the real part of the solution, we have:

Now we have the general solution to the system of ODEs that describe the motion of an undamped mass-spring system.

Initial Conditions​

Finally, we need to find the constants and using the initial conditions of the system. Let and be the initial position and velocity of the mass, respectively.

Just differentiating we get . Hence, .

For , we can first split the cosine function using the angle addition formula:

Let and , it'll make the calculations easier. Then, we have:

Differentiating with respect to gives us :

At , and . Thus:

Hence:

Next, consider in relation to , , and . Since and , we can construct a right-angled triangle with sides , , and . Furthermore, by the Pythagorean theorem, we have:

Using the triangle, , so:

With all these values, we can now write the general solution in two forms:

Undamped SHO General Solution:

Where:

  • is the initial position of the mass.
  • is the initial velocity of the mass.
  • is the angular frequency of the mass-spring system.
  • is the amplitude of the motion.
  • is the phase shift of the motion.

Similarly, for :

Undamped SHO Velocity:

Where:

  • is the initial position of the mass.
  • is the initial velocity of the mass.
  • is the angular frequency of the mass-spring system.
  • is the amplitude of the velocity.
  • is the phase shift of the velocity.

Summary and Next Steps​

The important takeaway from this section is not just the solution to the system of ODEs. Rather, the concepts of phase space, eigenvalues, and eigenvectors are far more crucial in understanding the behavior of linear systems.