Solving the Differential Equation of a Simple Harmonic Oscillator

A simple harmonic oscillator, a fundamental concept in physics, describes a system that oscillates with a frequency that is independent of the amplitude of the oscillation. This article will guide you through the steps to solve the differential equation of a simple harmonic oscillator.

The standard form of the differential equation for a simple harmonic oscillator is:

(frac{d^2x}{dt^2} omega^2 x 0)

where (x) is the displacement, (omega) is the angular frequency, and (t) is time.

Steps to Solve the Equation

Characteristic Equation

To solve the equation, we assume a solution of the form:

(x(t) e^{rt})

Substituting this into the differential equation:

(r^2 e^{rt} - omega^2 e^{rt} 0)

Factoring out (e^{rt}) (which is never zero) gives us the characteristic equation:

(r^2 - omega^2 0)

Finding Roots

Solving for (r) gives us:

(r^2 -omega^2) which implies (r pm iomega)

This indicates that the roots are complex and can be expressed as (r iomega) and (r -iomega).

General Solution

The general solution to the differential equation with complex roots is given by:

(x(t) C_1 cos(omega t) C_2 sin(omega t))

where (C_1) and (C_2) are constants determined by initial conditions.

Initial Conditions

To find the specific solution for a given problem, we apply initial conditions such as:

Initial displacement: (x(0) x_0) Initial velocity: (frac{dx}{dt}bigg |_{t0} v_0)

Example

Suppose we have initial conditions (x(0) x_0) and (frac{dx}{dt}bigg |_{t0} v_0).

From (x(0) x_0):

(x(0) C_1 cos(0) C_2 sin(0)) which implies (C_1 x_0)

For the velocity:

(frac{dx}{dt} -C_1 omega sin(omega t) C_2 omega cos(omega t))

Evaluating at (t 0):

(frac{dx}{dt}bigg |_{t0} -x_0 omega sin(0) C_2 omega cos(0))

which implies (C_2 frac{v_0}{omega})

Combining these results, the specific solution is given by:

(x(t) x_0 cos(omega t) frac{v_0}{omega} sin(omega t))

This solution describes the motion of a simple harmonic oscillator, showing how the position varies with time based on the initial conditions.

From a Physical Perspective

Considering a simple oscillator made up of a mass (m) connected to a spring with elastic constant (k), the one-dimensional equation of motion for the mass is:

(m frac{d^2x}{dt^2} kx 0) where (x x(t)) is the elongation from the position of equilibrium (x0).

The equation can be written as:

(frac{d^2x}{dt^2} omega^2 x 0) where (omega sqrt{frac{k}{m}}).

Additionally, the equation can be expressed as:

(frac{d^2x}{dt^2} - omega^2 x 0) or

(frac{d^2}{dt^2} - iomega frac{d}{dt} iomega x 0) which can be split into the system:

(frac{d}{dt} - iomega x v(t))

(frac{d}{dt} v 0).

The second equation, (frac{dv}{dt} -iomega v 0), can be separated as:

(frac{dv}{v} -iomega dt).

Integrating this, we obtain:

(v(t) A e^{-iomega t}) with (A) as an arbitrary constant.

Substituting in the first equation, we obtain:

(frac{dx}{dt} - iomega x A e^{-iomega t}).

The integrating factor is (e^{-iomega t}) and the solution is given by:

(x(t) e^{iomega t} int A e^{-2iomega t} dt B).

After integration, we get:

(x(t) frac{iA}{2omega} e^{-iomega t} B e^{iomega t}).

This can be written as:

(x(t) C_1 e^{iomega t} C_2 e^{-iomega t}) with (C_1) and (C_2) as complex constants.

To have a real solution, the constants must be complex conjugates, so the solution becomes:

(x(t) A cos(omega t) B sin(omega t)).

The constants are usually specified from the initial conditions.

Example Calculation

For example, if (x(0) x_0) and (frac{dx}{dt}bigg |_{t0} 0), we obtain (A x_0) and (B 0) and the solution is:

(x(t) x_0 cos(omega t)).

If (x(0) 0) and (frac{dx}{dt}bigg |_{t0} x_0'), we obtain (A 0) and (B frac{x_0'}{omega}) and the solution is:

(x(t) frac{x_0'}{omega} sin(omega t)).

This solution effectively describes the motion of a simple harmonic oscillator based on the specific initial conditions given.