Editing Simple harmonic motion (section)

==Dynamics==
In [[Newtonian mechanics]], for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear [[ordinary differential equation]] with '''constant''' coefficients, can be obtained by means of [[Newton's second law]] and [[Hooke's law]] for a [[mass]] on a [[spring (device)|spring]].

<math display="block"> F_\mathrm{net} = m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx,</math>
where {{mvar|m}} is the [[Mass#Inertial mass|inertial mass]] of the oscillating body, {{mvar|x}} is its [[displacement (vector)|displacement]] from the [[mechanical equilibrium|equilibrium]] (or mean) position, and {{math|''k''}} is a constant (the [[Hooke's law#Formal definition|spring constant]] for a mass on a spring).

Therefore,
<math display="block"> \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -\frac{k}{m}x</math>

Solving the [[differential equation]] above produces a solution that is a [[sine wave|sinusoidal function]]:
<math> x(t) = c_1\cos\left(\omega t\right) + c_2\sin\left(\omega t\right),</math> where <math display="inline"> {\omega} = \sqrt{{k}/{m}}.</math>
The meaning of the constants <math> c_1</math> and <math> c_2</math> can be easily found: setting <math> t=0</math> on the equation above we see that <math> x(0) = c_1</math>, so that <math> c_1</math> is the initial position of the particle, <math> c_1=x_0</math>; taking the derivative of that equation and evaluating at zero we get that <math> \dot{x}(0) = \omega c_2</math>, so that <math> c_2</math> is the initial speed of the particle divided by the angular frequency, <math> c_2 = \frac{v_0}{\omega}</math>. Thus we can write:
<math display="block"> x(t) = x_0 \cos\left(\sqrt{\frac{k}{m}} t\right) + \frac{v_0}{\sqrt{\frac{k}{m}}}\sin\left(\sqrt{\frac{k}{m}} t\right).</math>

This equation can also be written in the form:
<math display="block"> x(t) = A\cos\left(\omega t - \varphi\right),</math>
where
* <math> A = \sqrt{{c_1}^2 + {c_2}^2} </math>
* <math>\tan \varphi = \frac{c_2}{c_1}, </math>
* <math>\sin \varphi = \frac{c_2}{A}, \; \cos \varphi = \frac{c_1}{A} </math>
or equivalently
* <math> A = |c_1 + c_2i|, </math>
* <math>\varphi = \arg(c_1 + c_2i) </math>
In the solution, {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are two constants determined by the initial conditions (specifically, the initial position at time {{math|1=''t'' = 0}} is {{math|''c''<sub>1</sub>}}, while the initial velocity is {{math|''c''<sub>2</sub>''ω''}}), and the origin is set to be the equilibrium position.{{Cref2|A}} Each of these constants carries a physical meaning of the motion: {{math|''A''}} is the [[amplitude]] (maximum displacement from the equilibrium position), {{math|1=''ω'' = 2''πf''}} is the [[angular frequency]], and {{math|''φ''}} is the initial [[phase (waves)|phase]].{{Cref2|B}}

Using the techniques of [[calculus]], the [[velocity]] and [[acceleration]] as a function of time can be found:
<math display="block"> v(t) = \frac{\mathrm{d} x}{\mathrm{d} t} = - A\omega \sin(\omega t-\varphi),</math>
*Speed: <math> {\omega} \sqrt {A^2 - x^2} </math>
*Maximum speed: {{math|1=''v'' = ''ωA''}} (at equilibrium point)
<math display="block"> a(t) = \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = - A \omega^2 \cos( \omega t-\varphi).</math>
*Maximum acceleration: {{math|''Aω''<sup>2</sup>}} (at extreme points)

By definition, if a mass {{math|''m''}} is under SHM its acceleration is directly proportional to displacement.
<math display="block"> a(x) = -\omega^2 x.</math>
where
<math display="block"> \omega^2=\frac{k}{m}</math>

Since {{math|1=''ω'' = 2''πf''}},
<math display="block">f = \frac{1}{2\pi}\sqrt{\frac{k}{m}},</math>
and, since {{math|1=''T'' = {{sfrac|1|''f''}}}} where {{math|''T''}} is the time period,
<math display="block">T = 2\pi \sqrt{\frac{m}{k}}.</math>

These equations demonstrate that the simple harmonic motion is [[wikt:isochronous|isochronous]] (the period and frequency are independent of the amplitude and the initial phase of the motion).