U(r) = K [(\frac{σ}{r} )^{12} − (\frac{σ}{r} )^{6}]

f(x) = f(x_{0}) + (x − x_{0})\frac{df}{dx}|_{x_{0}} + \frac{1}{2}(x − x_{0})^{2} \frac{d^{2}f}{dx^{2}} |_{x_{0}} + \frac{1}{6}(x − x_{0})^{3} \frac{d^{3}f}{dx^{3}} |_{x_{0}} + ...

f(x) = f(x_{0}) + ∑_{n=1}^{∞} \frac{1}{n!}(x − x_{0})^{n} \frac{d^{n}f}{dx^{n}} |_{x_{0}}

f(x) = f(x_{0}) + 0 + \frac{1}{2}(x − x_{0})^{2} \frac{d^{2}f}{dx^{2}} |_{x_{0}} + \frac{1}{6}(x − x_{0})^{3} \frac{d^{3}f}{dx^{3}} |_{x_{0}} + ...

f(x) ≃ f(x_{0}) + \frac{1}{2}(x − x_{0})^{2} \frac{d^{2}f}{dx^{2}} |_{x_{0}} + ...

f(x) ≃ f(x_{0}) + [\frac{1}{2} \frac{d^{2}f}{dx^{2}} |_{x_{0}}] (x − x_{0})^{2}

U(x) ≃ U(x_{0}) + [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})^{2}

Force F (x) = − \frac{∂U}{∂x}

(U(x) − U(x_{0})) ≃ [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})^{2}

(\frac{U(x) − U(x_{0})}{x − x_{0}}) ≃ [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})

F (x) ≃ − [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})

F (x) ≃ −K(x − x_{0})

F (x) = −Kx

m a(x) = −Kx

m \frac{d^{2}x}{dt^{2}} = −Kx

\frac{d^{2}x}{dt^{2}} = − (\frac{K}{m} ) x

ω ≡ \sqrt{\frac{K}{m}}

\frac{d^{2}x}{dt^{2}} = − ω^{2}x

x(t) = A cos(ωt + ϕ)

x(t) = A sin(ωt) + B cos(ωt)

h(x) = H [1 − cos(\frac{2πx}{L})]

m \frac{d^{2}x}{dt^{2}} + Kx = 0

∫ (m \frac{d^{2}x}{dt^{2}})dx + ∫ (Kx)dx = ∫ (0)dx

∫ (m \frac{d^{2}x}{dt^{2}})dx + \frac{1}{2}Kx^{2} = C

mv^{2}

\frac{d}{dx}(mv^{2}) = 2mv \frac{dv}{dx}

2mv \frac{dv}{dx} = 2m \frac{dx}{dt} \frac{dv}{dx}

2mv \frac{dv}{dx} = 2m \frac{dx}{dt} \frac{dv}{dx} = 2m \frac{dv}{dt} = 2m \frac{d^{2}x}{dt^{2}}

\frac{1}{2}mv^{2} + \frac{1}{2}Kx^{2} = C

∫ (m \frac{d^{2}x}{dt^{2}})dt + ∫ (Kx)dt = ∫ (0)dt

∫ (m \frac{d^{2}x}{dt^{2}})dt + ∫ (Force)dt = ∫ (0)dt

m v + ∫ (Force)dt = C

m(v_{f} − v_{i}) + ∫ (Force)dt = C