Linear Harmonic Oscillator

The equation of motion of a harmonic oscillator subjected to a linear restoring force in one dimension is,



Where q is the displacement of the object from its equilibrium position, q = 0. Note that q denotes the co-ordinate along the direction of motion and we are writing k for C₂ using standard terminology. The constant k is, in general, called force constant.

Let the solution of above equation be

Q = a e t



Substituting these, we get

( ² + ω² )q = 0                    ω² = k/m

Since, q ≠ 0, we find ² + ω² = 0, or = ± i ω

Hence, we get two independent solutions.

q = a₁ e^{i ω t}, and q = a² e^{-i ω t}

The general solution is, q = a₁ e_i ω t + a² e^{-i ω t}

= (a₁ + a₂) cos ω t + i (a₁ – a₂) sin ω t

Of course, q is not a complex number, it is real.

In fact, a₁ and a₂ are complex numbers in such a way that (a₁ + a₂) is real and (a₁ – a₂) is pure imaginary. Let us write the constants a₁ and a₂ as,

a₁ + a₂ = A sin Ø

and, i (a₁ – a₂) = A cos Ø

where A and Ø are two independent real constants. Thus, we get

q = A sin (ω t + Ø)

as the familiar solution for a linear harmonic oscillator.

A is the amplitude, ω is angular frequency, and Ø the phase of the oscillator.

T = 2 π/ω is the time period of oscillations.

The instantaneous velocity of the oscillator is,



Note that the solution of harmonic oscillator can also be written as,

q = A cos (ω t + Ø)

by defining, a₁ + a₂ = A cos Ø, and i (a₁ – a₂) = A sin Ø

Both the solutions are equally are equally valid and which one holds for a given motion is decided by the value of the phase Ø at t = 0. In complex notation, the two solutions can be written as,

q = Ae^{i (ω t + Ø)}

where it is understood that either the real part A cos (ω t + Ø), or the coefficient of imaginary part A sin (ω t + Ø) corresponds to actual physical solution.

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