Classical harmonic oscillator

A simple harmonic oscillator  is a system that 
contains a mass attached to the spring connected 
to a rigid support. 

Once the mass is displaced from the equilibrium 
position, the spring exerts an elastic restoring force 
which obeys Hooke's law ( F = - kx) 

1. One-dimensional simple harmonic motion

The second Newton law  and Hooke's law for a classical 
harmonic oscillator is written as:

F = ma = -k x 
or m d2 x/dt2 + kx = 0 

then:
m d2 x/dx2 + kx 
with ω2 = k/m

F is the restoring elastic force exerted by the 
spring of the the spring constant k on the masse m. 
x is the displacement of this mass from the equilibrium 
position 

The solution of this equation is 

x(t)= c1 cos ωt + c2 sin ωt = C cos(ωt - φ)


C is the amplitude tht is the maximum displacement from the 
equilibrium position. ω is the angular frequency, and φ 
is the phase.

with the period  T = 2π/ω = 2π(m/k)1/2 

2. Energy of a simple harmonic oscillator

We have: 

Velocity: 
v(t) = dx/dt = - c1 ω sin ωt + c2ω cos ωt = - C ω sin(ωt - φ) 

Acceleration: 
a(t) = dv/dt = - c1 ω2 cos ωt + c2 ω2 cos ωt 
= - C ω2 cos(ωt - φ)

Kinetic energy 
KE = (1/2) m v2 = (1/2) m (- c1 ω sin ωt + c2ω cos ωt)2 = 
(1/2) m C2ω2sin2(ωt - φ) 

Potential energy 
KE = (1/2) k x2 = (1/2) k (c1 cos ωt + c2 sin ωt)2 = 
(1/2)k C2 cos2(ωt - φ)

With ω2 = k/m. 

Total energy :

TE = KE + PE = (1/2) m C2ω2sin2(ωt - φ) + 
(1/2)k C2 cos2(ωt - φ)  = 

(1/2) C2k sin2(ωt - φ) + 
(1/2)k C2 cos2(ωt - φ) = (1/2)k C2 


Total energy: TE = (1/2)k C2