Phy 340 class notes for Fall, 1999

Aug 25 Sep 8 Oct 4 Nov 1 Phy 340 page
Aug 30 Sep 13 Oct 11 Nov 8
Sep 20 Oct 18 Nov 15
Sep 27 Oct 25 Nov 22
Nov 29

Phy 340, Week 5

20 Sep What are complex numbers? I want to show that they are no more nor less abstract than are real numbers. If you assume that you know what real numbers are, then you can define complex numbers as ordered pairs of reals -- subject to certain rules for addition and multiplication:

( a, b ) + ( c, d ) = ( a + c, b + d )            ( a, b )( c, d ) = ( ac - bd, ad + bc )

With these rules, the multiplicative identity is ( 1, 0 ). The combination ( 0, 1 ) has the property that its square is ( -1, 0 ). It plays the role of "i." Once you have persuaded yourself that this construction does indeed reproduce the desired properties of complex numbers, you can forget about it and use the normal rules for manipulating objects such as x + iy.

Derive Euler's formula for the complex exponential. Do it by setting up and solving a set of differential equations for its real and imaginary parts. Show the geometric interpretation of the complex numbers, and in particular this complex exponential.

Back to the general solution of the harmonic oscillator equation. Is the sum of two solutions a solution? Is this true for all equations? (Yes and No respectively) Look at the damped case. This is where the power of the complex notation comes into its own.

22 Sep Do some homework problems.

24 Sep Do the details of the damped simple harmonic oscillator. The exponential solution leads to complex or real solutions, depending on the values of the parameters, m, b, and k. Look closely at the case where there are complex, and therefore oscillatory solutions: b2 < 4 k m. It looks like the values for the position will then be complex, but appearances are deceiving. Pick some initial conditions for the problem, say x( 0 ) = x0 and vx( 0 ) = 0. Now just grind out the equations that determine the values of the arbitrary constants in the solution, the ones that I called A1 and A2. It's messy, but I eventually got to the point where I was able to recognize that the solution is in fact real, though I didn't finish all the algebra.

The pendulum: Use the material about velocity and acceleration in polar coordinates, to set up the equations of motion. I have to make a physical assumption about the cord holding the mass to proceed, and I choose to assume that its length is constant. This kills all the dr/dt terms. The resulting equation for q is nota simple harmonic oscillator, but if I assume that the angle is small, it becomes approximately one.

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