Advanced Classical Mechanics/Small Oscillations and Perturbed Motion

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In Linear Motion, we argued that all sufficiently small oscillations are harmonic. In this section we will exploit this result in several ways to understand

  1. The motion of systems with many degrees of freedom near equilibrium,
  2. The motion of systems perturbed from known solutions, and
  3. The motion of systems with Lagrangians perturbed from systems with known solutions.

All three of these points are applications of perturbation theory, and they all start with the harmonic oscillator.

Normal Modes[edit]

The modes of oscillation of systems near equilibrium are called the normal modes of the system. Understanding the frequencies of the normal modes of the system is crucial to design a system that can move (even if it isn't meant to). Let's look at a system with many degrees of freedom; we have

Let be an equilibrium position and expand about this point so .

We can expand the potential energy to give

The first term is a constant with respect to and constant terms do not affect the motion. The second term is zero, because is a point of equilibrium so we are left with

where

and

yielding the equations of motion

This is a linear differential equation with constant coefficients. We can try the solution

so we have

This is a matrix equation such that

with

and

This equation only has a solution is . This gives a th-degree polynomial to solve for . We will get solutions for that we can substitute into the matrix equation and solve for .

Is this guaranteed to work? Yes, it turns out. Look at the equation in terms of matrices we have

The matrix is symmetric and real. The matrix should be positive definite (because a negative kinetic energy doesn't make sense). Technical issue: If has a null space, the degrees of freedom corresponding to the null space are massless and cannot be excited unless they are in the null space of . Either way, you can drop the null space from both sides of the equation.

Assuming that is invertable we have

and we have a standard eigenvalue equation. In most examples, the kinetic energy matrix will be diagonal, so it is straightforward to construct the quotient matrix and diagonize it.

Perturbations about Steady Motion[edit]

Let's say I have some solution to the equations of motion and I would like to look at small deviations from the solution. Let's satisfy

and let's look at

where is small. Let's expand the entire Lagrangian to find the equations of motion for the deviations . We have

               

Now let's apply Lagrange's equations for the deviations

to give

       

The two terms without actually cancel each other out, leaving the following equations of motion.

       

In steady motion, the partial derivatives are taken to be constant in time yielding the even simpler result

Again we have a linear differential equation with constant coefficients, and all of the results from the previous section carry over.

Perturbed Lagrangians[edit]

What about finding solutions to Lagrangians that are almost like ones that we have already solved? Let's say we have

where is considered to be small compared to Let's say I have some solution to the equations of motion for and I would like to look at small deviations from the solution induced by the change in the Lagrangian. Let's say satisfy

and let's look at

where is small. Let's expand the entire Lagrangian to find the equations of motion for the deviations . We have

               

               

Now let's apply Lagrange's equations for the deviations

to give

       

The two lowest orders terms without actually cancel each other out, leaving the following equations of motion.

       

Let's specialize and assume that the unperturbed motion is steady so the partial derivatives of the unperturbed Lagrangian are constant in time, to obtain

which is the equation of a coupled set of driven harmonic oscillators.