Notation question - what does mean?

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The discussion revolves around the notation in the context of the Lagrangian's rotation, specifically the meaning of "!" after the index j. It is clarified that the expression "xi -> Oijxj" follows the Einstein summation convention, where repeated indices indicate summation. The "!" does not have a mathematical significance in this context and is interpreted as an expression of happiness. Thus, the notation is primarily focused on the summation convention rather than any additional meaning from the exclamation mark. Understanding this notation is essential for grasping concepts in theoretical physics.
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Notation question - what does "!" mean?

Homework Statement


Hi, this is part of a question, in the context of rotation of the Lagrangian:

x_i -> O_ij.x_j, summation over j !

My question is: what does the " !" after the j mean?

Thanks

Homework Equations





The Attempt at a Solution

 
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Hi Pyroadept! :smile:

(try using the X2 tag just above the Reply box :wink:)

xi -> Oijxj is written in the "Einstein summation convention", where any repeated index is summed over.

I don't think the ! means anything, except happiness ! :smile:
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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