Advanced Classical Mechanics/Small Oscillations and Perturbed Motion

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 ModesEdit

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 MotionEdit

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 LagrangiansEdit

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.