Green's function - Fig 1 Displacement of a harmonic...

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Fig. 1: Displacement of a harmonic oscillator, , with a complicated driving force . Green's function From Physics Notes A Green's function is a solution to an inhomogenous differential equation with a delta function source. Its main purpose is to provide an easy method to solve a differential equation with arbitrary "driving" terms. This is applicable to physical phenomena such as the motion of a mechanical oscillator subjected to an arbitrary time-dependent driving force, or the sound wave radiated from a loudspeaker of arbitrary shape. As an introduction to the Green's function method, we will study its usage in the simple context of a driven harmonic oscillator. This is a physical system consisting of a damped harmonic oscillator subjected to an arbitrary driving force . Its equation of motion is where is the mass of the particle is the damping constant is the natural frequency of the oscillator is the time-dependent driving force. We are interested in solving for in the presence of an arbitrarily complicated driving force , such as the one as shown in Fig. 1. Contents 1 The Green's function 2 Finding the Green's function 3 Basic features 4 Causality 5 Further reading 6 Exercises The Green's function Let's first consider a more "basic" differential equation: This is called the Green's function equation , and is a function of two variables, and , called the Green's function . Note that the differential operator on the left-hand side involves only derivatives in . Hence, we can regard as an independent parameter. Referring back to the equation of motion for a driven harmonic oscillator, we see that the Green's function describes the motion of a damped harmonic oscillator which is subjected to a "pulse" of force, where is a delta function centered at (as noted above, plays the role of a parameter).
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Why do we care about the Green's function? Because as soon as we can find the Green's function, we can automatically produce a solution to the driven harmonic oscillator equation for any given driving force . That solution is To show mathematically that this is indeed a solution, plug this into the equation of motion: (Note that we can move the differential operator inside the integral over because are independent variables.) Thus, this equation for works as a solution to the oscillator equation with driving force . Let's contemplate the physical meaning of this. As we know, a non-zero driving force causes the oscillator to move. However, the value of at time does not depend only on the instantaneous force being applied at time , because even after the driving force is turned off, the oscillator would continue oscillating (see, for example, Fig. 1). Instead, should depend on the values of over a continuous range of times . (Specifically, it should depend on the force at all previous
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