# 物理代写|理论力学作业代写Theoretical Mechanics代考|PHYS2041

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Linear Harmonic Oscillator

The harmonic oscillator belongs to the most important and to the most intensively discussed model systems of Theoretical Physics. The range of its application range far exceeds thee canopy of Classical Mechanics. We will be dealing again and again with this model in Electrodynamics and in particular in Quantum Theory. The relevance of this model lies above all in the fact that it belongs to the very few mathematically strictly tractable systems by which many of the fundamental principles of Theoretical Physics can be illustrated. One understands by the harmonic oscillator a self-oscillating system that obeys a characteristic equation of motion of the same type as that for the simple pendulum (2.125).

In order to discuss the basic phenomena we first have in mind an elastic spring to which a mass point $m$ is attached. For small deflections the mass point experiences a backwards directed force being proportional to the displacement $|x|$. According to the sketched arrangement in Fig. $2.25$ the gravitational force will be ignored. The movement happens one-dimensionally along the spring axis. Then Hooke’s law holds:
$$F=-k x .$$
$k$ is the spring constant. As equation of motion er have the following linear homogeneous differential equation:
$$m \ddot{x}+k x=0 .$$
In it is the same differential equation as that for the simple pendulum (2.125). From reasons which become clear later the entity
$$\omega_0=\sqrt{\frac{k}{m}}$$
is called eigen frequency of the harmonic oscillator. When a physical system is described by an equation of motion of the type given in (2.157) then we always speak of a linear harmonic oscillator.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Free Damped Linear Oscillator

Each real oscillator eventually comes to stop because of the unavoidable frictional forces. We therefore want to now include them into our considerations where, however, we will restrict ourselves to the simplest case of the Stokes’s friction. Then the extended equation of motion reads:
$$m \ddot{x}=-k x-\alpha \dot{x} .$$
One can realize this situation by a ‘tongue’, dipping into a liquid and being fixed to the mass $m$ (Fig. 2.27). While the frictional term in Eq. (2.167) in general represents a certain approximation, there exists an exact non-mechanical realization of the damped harmonic oscillator by the electrical oscillator circuit. The sum of the partial voltages in the circuit sketched in Fig. $2.28$ must be zero. The electrical current therefore obeys the following differential equation:
$$\ddot{I}+R \dot{I}+\frac{1}{C} I=0 .$$
The ohmic resistance $R$ simulates the frictional term.

After division by $m$ we get from ( $2.167$ ) the following homogeneous differential equation of second order:
$$\ddot{x}+2 \beta \dot{x}+\omega_0^2 x=0 ; \quad \beta=\frac{\alpha}{2 m} .$$
As ansatz an exponential function appears again plausible:
$$x(t)=e^{\lambda t} .$$
It is exactly then a solution if $\lambda$ fulfills the following relation:
$$\lambda^2+2 \beta \lambda+\omega_0^2=0 .$$
Therefrom one finds:
$$\lambda_{1,2}=-\beta \pm \sqrt{\beta^2-\omega_0^2} .$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Linear Harmonic Oscillator

$$F=-k x .$$
$k$ 是弹簧常数。由于运动方程 er 有以下线性旻次微分方程:
$$m \ddot{x}+k x=0 .$$

$$\omega_0=\sqrt{\frac{k}{m}}$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Free Damped Linear Oscillator

$$m \ddot{x}=-k x-\alpha \dot{x} .$$

$$\ddot{I}+R \dot{I}+\frac{1}{C} I=0 .$$

$$\ddot{x}+2 \beta \dot{x}+\omega_0^2 x=0 ; \quad \beta=\frac{\alpha}{2 m} .$$

$$x(t)=e^{\lambda t}$$

$$\lambda^2+2 \beta \lambda+\omega_0^2=0 .$$

$$\lambda_{1,2}=-\beta \pm \sqrt{\beta^2-\omega_0^2}$$

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