Could somebody explain to me how lagrange multipliers works in finding extrema of constrained functions? also, what is calculus of variations and lagrangian mechanics, and can somebody explain to me what the lagrangian function is and the euler-lagrange equation. And, i read something about noethers theorem which seems very interesting. could somebody explain this theorem to me?(adsbygoogle = window.adsbygoogle || []).push({});

I came up with this proof of the law of conservation of energy. It makes sense to me. Is it right?

suppose f[x] is the force on an object with mass=1 in a force field that depends on position alone(is conservative).

f[x]=d2x/dt2

the definition of energy:

E=.5(dx/dt)^2-S[f]dx (here S is my symbol of integration)

because energy is constant the derivative of energy is zero.

dE/dt=0=.5d[(dx/dt)^2]/dt-d[S[f]dx]/dt

by the chain rule, .5d[(dx/dt)^2]/dt=(d2x/dt2)(dx/dt)

0=(d2x/dt2)(dx/dt)-d[S[f]dx]/dt

the next step is making the substitution d[S[f]dx]/dt=(d[S[f]dx]/dx)(dx/dt), which is possible because the dx above the dt and the dx under the integral cancel out leaving a dt below the integral.

0=(d2x/dt2)(dx/dt)-(d[S[f]dx]/dx)(dx/dt)

By the fundamental theorem of calculus, the differentation undoes the integration of f, giving

0=(d2x/dt2)(dx/dt)-f(dx/dt)

dividing through by dx/dt

0=(d2x/dt2)-f

f=(d2x/dt2)

which is the original equality. QED

Makes sense to me. the trickiest part for me was realizing that i could algebraicly manipulate the liebniz fraction d[S[f]dx]/dt=(d[S[f]dx]/dx)(dx/dt)=f*(dx/dt) to get rid of the integral.

I read another proof that uses lagrangians but i didnt understand it.

Could somebody explain what a lagrangian is to me

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# Euler lagrange equation, mechanics, need help

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