Physics

Physics · mechanics

Problem

(b) Equation 4 is a version of the damped harmonic oscillator, and can be solved by guessing a solution

\eta=\alpha e^{\lambda t}

. Plugging in this guess, what must

\lambda

be? Given constants: 4. A complex dance In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number

i

, strangely defined to obey

i^{2}=-1

. In other words, rather than using real numbers, it is more convenient for us to work in terms of complex numbers. Exponentials are usually associated with rapid growth or decay. However, with the inclusion of complex numbers, imaginary "growth" and "decay" can be translated into oscillations by the Euler identity:

e^{i \theta}=\cos \theta+i \sin \theta \tag{1}

Context question: (a) The usual form of Newton's second law

(\vec{F}=m \vec{a})

breaks down when we go into a rotating frame, where both the centrifugal and Coriolis forces become important to account for. Newton's second law then takes the form

\vec{F}=m(\vec{a}+2 \vec{v} \times \vec{\Omega}+\vec{\Omega} \times(\vec{\Omega} \times \vec{r})) \tag{2}

For a particle free of forces confined to the

x-y

plane in a frame which rotates about the

z

axis with angular frequency

\Omega

, this becomes the complicated-looking system of differential equations,

\begin{aligned} & 0=\ddot{x}+2 \Omega \dot{y}-\Omega^{2} x \\ & 0=\ddot{x}-2 \Omega \dot{x}-\Omega^{2} y \end{aligned} \tag{3}

where dots represent time derivatives. Defining

\eta=x+i y

, show that Equations 3 are equivalent to the following single (complex) equation:

0=\ddot{\eta}-2 i \Omega \dot{\eta}-\Omega^{2} \eta \tag{4}

Context answer: 证明题

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