Question

To understand how the two standard ways to write the general solution to a harmonic oscillator are related.

Answer 1

Answer:

a)   C = A cos φ , S = A sin φ,  b)   φ = tan⁻¹ (S / C) ,  A² = C² + S²

Explanation:

a) The two forms given are equivalent, let's start by developing the double angle

            Cos (a + b) = cos a cos b - sin a sin b

Call us

           a  = wt     and      b = φ

           x = A (cos wt cosφ- sin wt sin φ

the second equation is

           x = C cos wt + S  sin wt

Let's match

           C cos wt + S sin wt = A cos φ coswt - a sin φ sin wt

The coefficients of the cosine and breasts must be equal,

         C = A cos φ

          S = A sin φ

b) Divide the last two expressions

          tan φ = S / C

          φ = tan⁻¹ (S / C)

Let's square the two equations

         C² = A² cos² φ

         S² = A² sin² φ

Let's add

        C² + S² = A² (cos² φ + sin² φ)

The part in brackets vouchers1

        A² = C² + S²