Resolved: "Argue for formula about one-dimensional harmonic oscillatorL(x,v)=m2v2−k2x2,k>0"

Dax Gross

Answered question

2022-03-15

Argue for formula about one-dimensional harmonic oscillator

L (x, v) = \frac{m}{2} v^{2} - \frac{k}{2} x^{2}, k > 0

Avery Campbell

Beginner2022-03-16Added 6 answers

Step 1
I think you've miscalculated; you don't need

S (γ_{0}) = 0

because

S (γ) - S (γ_{0}) = \int_{0}^{T} [\frac{12}{m} ({({\dot{γ}}_{0} + \dot{ν})}^{2} - {\dot{γ}}_{0}^{2}) - \frac{12}{k} ({(γ_{0} + ν)}^{2} - γ_{0}^{2})] d x

= \int_{0}^{T} [\frac{12}{m} (2 {\dot{γ}}_{0} \dot{ν} + {\dot{ν}}^{2}) - \frac{12}{k} (2 γ_{0} ν + ν^{2})] d x,

which gives the desired result iff

\int_{0}^{T} [m {\dot{γ}}_{0} \dot{ν} - k γ_{0} ν] d x = 0 .

But that integral is

\int_{0}^{T} m [{\dot{γ}}_{0} \dot{ν} + {\ddot{γ}}_{0} ν] d x = {[m {\dot{γ}}_{0} ν]}_{t_{1}}^{t_{2}},

which vanishes as desired because

ν (t_{1}) = ν (t_{2})

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Recalculate according to your conditions!

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