# General solution of harmonic oscillations

• I
• skyesthelimit
In summary, the harmonic oscillator with a restoring force of F=-mω2x has a solution for the x-component at x=exp(±iωt). However, the solution can also be generalized as x=Ccosωt+Dsin(ωt) by using two linearly independent solutions. This is because the general solution of the equation of motion allows for all possible linear combinations of solutions. The solution can also be made unique by assuming appropriate initial conditions, which is typical for mechanics problems. Another way to understand the solution is by expressing one set of linearly independent functions in terms of the other using Euler's formula.
skyesthelimit
For a harmonic oscillator with a restoring force with F= -mω2x, I get that the solution for the x-component happens at x=exp(±iωt). But why is it that you can generalise the solution to x= Ccosωt+Dsin(ωt)? Where does the sine term come from because when I use Euler's formula, the only real part seems to be the cosine term?

Last edited:
Well, you need two linearly independent solutions of the equation of motion
$$m \ddot{x}=F \; \Rightarrow \; \ddot{x}=-\omega^2 x$$
(BTW you forgot the ##x## in your force formula).

You can use any two linearly independent solutions. Obviously your two functions will do. The general solution of the equation of motion is given by all possible linear combinations, i.e.,
$$x(t)=A \exp(\mathrm{i} \omega t) + B \exp(-\mathrm{i} \omega t).$$
Obviously also ##\cos(\omega t)## and ##\sin(\omega t)## solve the equations, and since they are linearly independent the complete solution is also given as superpositions of them, i.e., also
$$x(t)=A' \cos(\omega t) + B' \sin(\omega t)$$
give the complete set of solutions.

You can make the solution unique by assuming the appropriate initial conditions, which is of course the natural thing for a typical mechanics problem: You need to give the position and velocity of the particle at some initial time, say ##t=0##, i.e., you give the values ##x(0)=x_0## and ##\dot{x}(0)=v_0##. Now you can use both superpositions to evaluate the now unique solution ##x(t)## for these initial conditions. You'll see that of course you get the same result.

Another way to see this is to express the one set of linearly independent functions in terms of the other. For that just remember how cos and sin are expressed in terms of exponential functions or use Euler's formula to derive them.

oh right! thanks I just edited the equation! this was super helpful - i didn't quite think about it in terms of linearly independent solutions!

## 1. What is a general solution of harmonic oscillations?

A general solution of harmonic oscillations is a mathematical expression that describes the motion of a system undergoing simple harmonic motion. It takes into account the initial conditions and parameters of the system, such as the mass, spring constant, and amplitude of oscillation.

## 2. How is a general solution of harmonic oscillations derived?

A general solution of harmonic oscillations can be derived using the differential equation for simple harmonic motion, which is a second-order linear differential equation. The solution involves finding the roots of the characteristic equation and using them to form the general solution.

## 3. What are the key components of a general solution of harmonic oscillations?

The key components of a general solution of harmonic oscillations include the position function, velocity function, and acceleration function. These functions are determined by the initial conditions and parameters of the system and can be used to describe the motion of the system at any given time.

## 4. How does the general solution of harmonic oscillations relate to real-world systems?

The general solution of harmonic oscillations can be applied to a variety of real-world systems that exhibit simple harmonic motion, such as a mass on a spring, a pendulum, or an electrical circuit. It provides a mathematical model for understanding and predicting the behavior of these systems.

## 5. Are there any limitations to the general solution of harmonic oscillations?

While the general solution of harmonic oscillations is a useful tool for understanding simple harmonic motion, it does have some limitations. It assumes ideal conditions, such as no friction or external forces, and may not accurately describe the behavior of systems with more complex dynamics.

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