Advanced Classical Mechanics/Continuum Mechanics

We can extend the formalism of Lagrangian mechanics to cover the motion of continuous materials: membranes, strings and fields.

Deriving the Lagrange Equations of a ContinuumEdit

We saw for a string that the potential and kinetic energy can be written as a integral over the length of the string.

 

where   is called the Lagrangian density.

The ActionEdit

Let's write out the action

 

We would like to find the function   that minimizes the action, so we let   and calculate  

Let integrate the last two items by parts and drop the boundary term because we assume that the value of   vanishes on the boundary, we get

 

Lagrange's EquationsEdit

Because the function   is completely arbitrary then for the value of   to vanish, the integrand itself must vanish and we obtain the Lagrange equation for a field in one spatial dimension:

 

that the actual motion must satify.

Each new dimension that the field lives in adds a new term of the form

 

to the equations of motion.

ExamplesEdit

ElectrostaticsEdit

Let's do an example of an electric field with a bunch of charges around. The energy density of an electric field with charges around is

 

We would like to minimize the integral of   over a volume so it is natural to take  . We have

 

and Lagrange's equations are

 

A StringEdit

As a problem toward the beginning of the term, you found that the potential energy of a string could be given by

 

where   is the string tension.

We can also write the kinetic energy of the string as an integral

 

where  , the mass per unit length of the string. We can combine these two expressions to get the Langrangian density

 

Now let's apply Lagrange's equations to this Lagrangian density,

 

 

 

We can find a solution to this equation of the form,   where  . Let's substitute this into the equation to get

 

This solution is a wave travelling in the positive or negative x-direction. The function f can take any form whatsoever and it always travels at the same speed. This type of wave is called a 'simple wave'.

The Discrete Fourier TransformEdit

When a string vibrates it causes the air around it to vibrate as well; these vibrations travel through the air as waves that we can detect with the acoustic spectrometers attached to the sides of our heads. Our ears are sensitive to the pitch or frequency of the sound that the string produces so it is natural to look at the waves along a string as a sum of waves of a particular frequency, so we have   where φ is the phase of the wave.

Let's substitute this into the equation of motion to get

  so   and  

Now we must apply the boundary conditions of the string. Specifically, we have   so

 

where n is an integer so we have a formula for the frequency of the wave in terms of the form of the oscillation along the string

 

The general solution will allow a sum of different frequencies so

 

Let's suppose that we pluck the string so that a t=0, the string is not straight but it is not moving either, so we have   in the formula above, and we get

 

We would like to find the coefficients   that tell us what pitches the string will emit when it is released.

Let's take the integral of both sides times   to get

 

 

Rearranging we have

 

which defines the discrete Fourier transform.

A MembraneEdit

As a problem toward the beginning of the term, you found that the potential energy of a string could be given by

 

where   is the surface tension.

We can also write the kinetic energy of the membrane as an integral

 

where  , the mass per unit area of the membrane. We can combine these two expressions to get the Langrangian density

 

If we apply Lagrange's equations we obtain,

 

 

Different CoordinatesEdit

The value of the action does not depend on the coordinates that you use, but as we shall see the Lagrangian density does. First let's write the action using Cartesian coordinates

 

Sometimes it is useful to use polar coordinates for a circular drum perhaps. Here the action is

 

Notice that an r appears in front of the "Cartesian" Lagrangian density. This is just so the units work out, so we find that

 

In both coordinate systems the action has the same units but the Lagrangian density does not. We also need to rewrite the spatial derivatives in polar coordinates to get

 

If we apply the Lagrange equations we find that the Lagrange density depends explicitly on the coordinate r. We obtain,

 

and divide both sides by r to get

 

or using the definition of the Laplacian in cylindrical coordinates we get

 

Solutions to the Two-Dimensional Wave EquationEdit

When analyzing a drum or something with boundary conditions, it is natural to look for solutions to these equations with the form   in Cartesian coordinates,   in cylindrical coordinates, or   Generally you can pick the function g to vary harmonically in time and the solutions to the spatial dependence are known as harmonics. In Cartesian coordinates we get sines and cosines (the harmonic functions). In cylidrical coordinates, we get sines and cosines in the  -direction and Bessel functions in the radial direction. If your drum is the surface of a sphere, the solutions are called spherical harmonics: sines and cosines in the  direction and polynomials of   in the other direction.

A Slung SlinkyEdit

As a final example we will study a slinky hanging from a horizontal surface and the waves that can propagate along it. Let's first draw a picture to show what we are talking about

 

Let's use the variable   to denote the angle that each pendulum makes with the vertical. Also let's restrict ourselves to the small-angle approximation. We can write the kinetic energy of each mass as

 

There are two sources of potential energy: gravity and the springs. The expression for gravity is straightforward we have

 

The potential energy of the springs is a bit more complicated. First, we have to come up with a convention so that we don't double count the springs. Let's identify a spring with the mass to its left so we have

 

Now let's take the small-angle approximation,

 

We can simplify this further. The first term is a constant; let's drop it. The second term sums to zero when one combines the contribution from every spring. The final term is

 

where we have replaced   with  .

The Lagrange density is given by

 

The Klein-Gordon EquationEdit

Let's apply Lagrange's equations to the Lagrange density to get the following equation of motion

 

where   and  . This equation of motion is very similar to that of the stretched string but there is an extra term. Let's divide both sides by   to get

 

where   is the natural frequency of the hanging masses. This is known as the Klein-Gordon equation, and it describes the evolution of massive scalar field. If we let the spring constant   go to zero, then   goes to zero as well and we have the simple equation for a pendulum.

 

To get an appreciation for the solutions, let's substitute a harmoncally varying wave

 

to get

  and  

If the frequency of the wave   is greater than the natural frequency of the pendulum, the wave varies sinusoidally in space as well as time (it propagates along the slinky). On the other hand, low-frequency waves are damped, so the slung slinky acts as a high-pass filter.

The form of the equation above encourages us to use solutions of the form

 

with the dispersion relation

 

Phase VelocityEdit

Again we see that when   things get a bit funny. We can define the phase velocity of the wave

 

This measures the speed at which the crest of a wave of a particular frequency passes by.

Group VelocityEdit

Let's say that we want to send a signal down the slinky. How long fast will the signal travel? Specifically, we would like to displace the end of the slinky back and forth and watch the compression travel along the spring.

We could naturally think of a signal as the sum of two waves of a single frequency.

 

Using some trignometric identites we see that this is just an amplitude-modulated wave

 

The amplitude modulation is a wave that travels on top of the carrier wave at a speed of   for narrow bandwidth.

The group velocity measures the speed of a "wave packet" or how long a signal takes to travel along the slinky

 

If we think about the wave described above in more detail, we have a rapid back-and-forth oscillation that travels at the phase velocity and on top of it a variation in the amplitude of the rapid oscillation. If we ignored the frequency of the pendulums (taking   or  ), these two variations would travel at the speed   and one would simply see a pattern travelling at a constant speed along the slinky.

Hanging the slinky introduces dispersion, this means that the pattern generally changes as it travels along the slinky.

The Sine-Gordon or Pendulum EquationEdit

The slung Slinky is related to another important equation of physics. Let's take the exact expression for the gravitational potential energy

 

The new Lagrange density is given by

 

and the equation of motion is

 

This is known as the sine-Gordon equation (those physicists are really good punsters). This equation is no longer linear in the variable   so it is much more difficult to solve than the Klein-Gordon equation. Specifically, we expect that the velocity of the wave will depend not only on the frequency but also the amplitude.

We can define our scale of time ( ) and distance ( ) so that the equation takes the simpler form

 

Let's look for solutions that have a fixed pattern travelling at a fixed velocity. This was not possible (except for a monochromatic wave) for the Klein-Gordon equation because of the dispersion. Here we have dispersion and non-linearity. Let's use

 

to get

 

so

 

 

Here the value of velocity depends on the amplitude of the wave,   for an amplitude of 0.01,   for an amplitude of 1 and 1.64 for an amplitude of 2.

SolitonsEdit

The solutions above are strictly periodic, but we can also find waves that travel at velocities less than one that are not periodic (i.e. they are solitary waves or solitons).

The wider waves have amplitudes closer to   and velocities closer to zero. The speed of the solitons are 0.1, 0.5, 0.8 and 0.9 from widest to narrowest and the amplitudes range from 6.17 down to 6.11.

The differential equation actually has a simple solution as well. It is called the "kink" solution

 

The speed at which the kink travels depends on the maximum slope of the kink.

 

The steepest slope travels at  ; then we have 0.8 and zero (a frozen kink).