While doing some calculations I have stumbled into some step, that I am unable to justify.
The question is regarding harmonic oscillator and is as follows:
Given simple harmonic oscillator we have equation:
$$\frac{d^2x}{dt^2}+\frac{k}{m}x=0$$
Which yields to general solution:
$$x(t)=c_1\cos(\omega t)+c_2\sin(\omega t)\space\space\space\space \text{where}\space\omega^2=\frac{k}{m}$$
Setting our constants in our solution to be:
$$c_1=A\cos(\phi),\space\space\space c_2=-A\sin(\phi)$$
we can write our solution as:
$$x(t)=A\cos(\omega t+\phi)$$
The thing I have concern with is, since c1, c2 are arbitrary constants, how we can impose a condition for them by expressing both as another constant times a function that each of the constants are dependent on.
In short: How we can justify that setting our constant by conditions described aboved yields to arbitrary constants.
Best Answer
Following comment made by Lutz Lehmann.
$$\text{Since}\space c_1, \space c_2\space \text{are arbitrary constants, they should span an}\space R^2\space \text{space}$$ I.e. they can be written as (c1, c2).
Expressing our constants in polar coordinates, we have: $$(c_1,c_2)=(A\cos(\phi),A\sin(\phi))$$ Which satisfies that constants can be arbitrary chosen.