Brian Covello: Mathematics Research Utilizing Differential Equations for a Pendulum
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Transcript of Brian Covello: Mathematics Research Utilizing Differential Equations for a Pendulum
ANALYSIS OF THE TAUTOCHRONE, BRACHISTOCHRONE AND CYCLOIDAL PENDULUM
BRIAN COVELLO, NATHANIEL STAMBAUGH
PREVIEW Background Simple pendulum
• Small angle approximation • Elliptic Integrals • A novel pattern taylor expression
The Brachistochrone The Cycloid The Tautochrone Bridge? The Cycloidal Pendulum
BACKGROUND 1599 Galileo studied cycloids 1659 Hyugens showed cycloid is the solution to the tautochrone problem
• The same time path • Cycloidal pendulum à period not dependent on
amplitude 1697 Bernoulli showed cycloid is the solution to the brachistochrone problem
• The least time path • Coincidence?
THE SIMPLE PENDULUM
SIMPLE PENDULUM – EULER LAGRANGE
SMALL ANGLE APPROXIMATION
à à
What if we attempt to solve the nonlinear second order ODE through a taylor expression?
TAYLOR EXPRESSION
• No obvious pattern • Let’s start collecting terms
anyway…
CONTINUED… N=0,1
• Begin with the first term…
• Collect the linear terms n=1…Imagine the multiplicative possibilities that will generate a first degree linear term
CONTINUED… N=2
CONTINUED… N=3
CONTINUED… N=4
• Fill in a0 as needed to the powers of “y” we are dealing with …
SEPARATE BASED ON PARTITIONS…
GENERALIZED PATTERN
TAYLOR EXPRESSION
• For comparison…elliptic integrals
BRACHISTOCHRONE
BRACHISTOCHRONE – EULER LAGRANGE
BRACHISTOCHRONE – EULER LAGRANGE
TIME FROM TOP TO BOTTOM
TIME FROM SOME INITIAL Y
Same time from some initial y as from the top! • This is known as the tautochrone (same time)
TAUTOCHRONE
EVOLUTE OF A CYCLOID Parameterization for evolute:
CYCLOIDAL PENDULUM