- #1
nomadreid
Gold Member
- 1,674
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To make the confusions both concise and explicit, I will put down some incorrect calculations, and ask for corrections
Take the Lagrangian KE - PE = T - V
Action = S=∫ L dt (with given limits)
Principle of least action: δS= 0: S(t1)-S(t2) =0 if t1-t2 is small (using the (.) as function notation, not multiplication)
I presume δ can be taken as d/dt
Then d(∫ L dt)/dt = L(t1)-L(t2) = T(t1)-T(t2)+V(t2)-V(t1)=0
Next: conservation of energy
KE+PE = k for some k
So d/dt (T+V) = dT/dt + dV/dt = 0
If this is true on all points on the interval (t1,t2), then T(t1)-T(t2)+V(t1)-V(t2) = 0
Putting the two equations together... but obviously I've already gone way too far into the gross errors, and would be grateful for corrections. Thanks.
Take the Lagrangian KE - PE = T - V
Action = S=∫ L dt (with given limits)
Principle of least action: δS= 0: S(t1)-S(t2) =0 if t1-t2 is small (using the (.) as function notation, not multiplication)
I presume δ can be taken as d/dt
Then d(∫ L dt)/dt = L(t1)-L(t2) = T(t1)-T(t2)+V(t2)-V(t1)=0
Next: conservation of energy
KE+PE = k for some k
So d/dt (T+V) = dT/dt + dV/dt = 0
If this is true on all points on the interval (t1,t2), then T(t1)-T(t2)+V(t1)-V(t2) = 0
Putting the two equations together... but obviously I've already gone way too far into the gross errors, and would be grateful for corrections. Thanks.