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okkvlt

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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