{"id":266,"date":"2016-09-06T20:44:50","date_gmt":"2016-09-06T20:44:50","guid":{"rendered":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/?p=266"},"modified":"2016-09-13T02:13:36","modified_gmt":"2016-09-13T02:13:36","slug":"week-2","status":"publish","type":"post","link":"https:\/\/jhubisz.expressions.syr.edu\/phy360\/2016\/09\/06\/week-2\/","title":{"rendered":"Week 2"},"content":{"rendered":"

Complex Exponentials and Simple Harmonic Motion<\/strong><\/p>\n

The Superposition Principle<\/strong><\/p>\n

In lecture on Tuesday, we covered the use of complex exponentials to represent simple harmonic motion. \u00a0The reason we can use this notation is due to the following three facts:<\/p>\n

    \n
  1. The projection<\/strong> of 2 dimensional rotation at constant angular velocity onto one axis is identical to simple harmonic motion<\/li>\n
  2. the complex plane carries the same amount of information as the 2D plane<\/li>\n
  3. Euler’s formula provides a very concise packaging of this 2d motion into a single formula (just one expression rather than separate ones for the x and y component, for example)<\/li>\n<\/ol>\n

    This notation using the complex plane will provide useful for the following scenarios (as examples):<\/p>\n

      \n
    1. Superposition of multiple simple harmonic oscillators (avoiding trig)<\/li>\n
    2. Studying the behavior of 1D systems that include damping\/dissipation and\/or driving forces (rather inconvenient using trig) (Note that this is also the case in engineering when you have RLC circuits).<\/li>\n
    3. it is the fundamental tool in quantum mechanics, where the complex phases then have real physical meaning, so you better get used to it!! \u00a0\ud83d\ude09<\/li>\n<\/ol>\n

      Here is a link to the mathematica file we are using this week:<\/p>\n

      https:\/\/jhubisz.expressions.syr.edu\/phy360\/wp-content\/uploads\/sites\/5\/2016\/09\/Week-2.nb<\/a><\/p>\n

      As we discussed briefly in class on Tuesday, in response to a question from Emily Syracuse, there is a very interesting and reason for why<\/strong><\/em> we can use this complex notation.\u00a0Recall that in introducing complex notation we bring in extra information, taking 1D motion and turning it into 2D\u00a0circular<\/strong> motion in the complex plane (the real and imaginary axes are the 2 dimensions). \u00a0In the end, we can always throw this extra information (the imaginary part) away to get at the physics (the projection of the circular motion onto one axis).<\/p>\n

      The reason for why this works is due to the fact that the equation of motion for a simple harmonic oscillator is linear.<\/strong><\/p>\n

      Recall the simple harmonic oscillator equation is<\/p>\n\\ddot{x}(t) = - \\omega^2 x(t)\n

      There are no terms like x^2(t) or \\dot{x}^2(t), or other higher powers of x(t) and its derivatives.<\/p>\n

      The SHO equation is linear because all potential energies near equilibrium are quadratic in displacement, meaning the force<\/p>\n\\overrightarrow{F} = - \\overrightarrow{\\nabla} U\n

      is linear.<\/strong><\/p>\n

      Equations like this – linear equations – obey a very important principle that is called the Superposition Principle.<\/strong><\/em><\/p>\n

      The\u00a0Superposition Principle<\/strong><\/em> states that if we have two\u00a0solutions to an equation, the sum of the solutions is also a solution (or the difference, or any arbitrary linear combination of the two solutions). \u00a0You may be familiar with this principle from electromagnetism, and there is it also a result of the equations (Maxwell’s equations, in this case) being linear.<\/strong><\/p>\n

      It is worth revisiting the “big picture” at this point:<\/p>\n