Estimate the. poriod of osalaon (w). as a funchon of amplitude A by., minimizıng the "objechue fvntion (* +x1 dt with X = A coswt a If x = ) A cosut what is the velouty what isthe acceleraten ( What is the intagrand +x] in terms of A, coswt, ? © Parform the ntegral O do determine dw
Estimate the. poriod of osalaon (w). as a funchon of amplitude A by., minimizıng the "objechue fvntion (* +x1 dt with X = A coswt a If x = ) A cosut what is the velouty what isthe acceleraten ( What is the intagrand +x] in terms of A, coswt, ? © Parform the ntegral O do determine dw
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![For the classical anharmonic oscillator, the differential equation is
\[ \frac{d^2x}{dt^2} = \ddot{x} = -x^3 \]
Estimate the period of oscillation (\(\omega\)) as a function of amplitude \(A\) by minimizing the objective function
\[ O = \int_{0}^{\frac{2\pi}{\omega}} [\dot{x} + x^3]^2 \, dt \]
with \( x = A \cos(\omega t) \)
**a) If \( x = A \cos(\omega t) \), what is the velocity \(\left(\frac{dx}{dt}\right)\)? What is the acceleration \(\left(\frac{d^2x}{dt^2}\right)\)?**
**b) What is the integrand \([\dot{x} + x^3]^2\) in terms of \(A, \cos(\omega t)\)?**
**c) Perform the integral**
**d) Determine \(\frac{dO}{d\omega}\)**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2Fddea624e-136b-44ed-9c9c-adb8f841476e%2Ffywmfm8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For the classical anharmonic oscillator, the differential equation is
\[ \frac{d^2x}{dt^2} = \ddot{x} = -x^3 \]
Estimate the period of oscillation (\(\omega\)) as a function of amplitude \(A\) by minimizing the objective function
\[ O = \int_{0}^{\frac{2\pi}{\omega}} [\dot{x} + x^3]^2 \, dt \]
with \( x = A \cos(\omega t) \)
**a) If \( x = A \cos(\omega t) \), what is the velocity \(\left(\frac{dx}{dt}\right)\)? What is the acceleration \(\left(\frac{d^2x}{dt^2}\right)\)?**
**b) What is the integrand \([\dot{x} + x^3]^2\) in terms of \(A, \cos(\omega t)\)?**
**c) Perform the integral**
**d) Determine \(\frac{dO}{d\omega}\)**
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Solution (a)
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