Derivation of the equation for simple harmonic motion

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There must be an easier way to derive this.

The desired closed-form solution is:   for some constants  ,  .

A differential equation for a 1-dimensional harmonic oscillator is:

 

Since force is mass times acceleration,

 

To solve this second-order differential equation, we introduce a new variable v, reducing it to a system of linear first-order differential equations. Also writing   for convenience:

 
 

Rewrite this system using matrices:

 

The coefficient matrix has characteristic polynomial

 

so its complex eigenvalues are  . A complex eigenbasis is then:

 ,

which leads to the general solution of our system,

 

Here,   and   are the (complex) coordinates of   (that is, the initial state vector of the system) with respect to the complex eigenbasis  .

Applying Euler's formula:

 

Using the trigonometric identities   and  , this simplifies to:

 

Now, let us return to the coordinates   and  . We know that

 

Inverting the matrix, we have

  where   and   are both real.

Substituting back into the equation for  , we have

 

Another trig identity gives:

 , where  .

If we then let

  and
 ,

then we finally end up at the desired solution

 .

Gradient of an ellipsoid

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The basic ellipsoid equation is

 

So the gradient is

 

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