5
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Pauli matrix relation of dot and cross products with complex numbers
This is a confusing form of the Clifford-algebraic identity $ab = a\cdot b + a\wedge b$, where $a$ and $b$ are vectors. It's confusing because it mixes Clifford-algebraic and 3D-vector-algebraic ...
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3
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Why the normal vector addition does not seem to work in centripetal acceleration?
Its important to note that the velocity is infact not constant,as it is changing its direction.
But,I'm going to show you why the speed is constant if we have the fact that the angle between the force ...
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How can I formalize better this proof that angular momentum is conserved for a small impulse?
Summary of the changes in $\delta L$ for a decomposed $\delta p$
To be a bit more complete, here is a mathematical overview how $\delta p$ concretely changes the $L$.
Parallel to $\mathbf{L}$: $\...
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Why the normal vector addition does not seem to work in centripetal acceleration?
The concern raised in this question, as I understand it, is that we start with a tangential velocity:
But then, the acceleration, being perpendicular to the velocity, causes the increment $\mathbf{a} ...
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Accepted
Derivative of the product of a scalar function and a vector valued function
The author's point here is that one can treat a vector function like a scalar function when it comes to differentiation. This may be obvious but the book is meant for students with a limited ...
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I have confusion between the concept of distance and displacement
We define distance to be a "directionless" quantity and displacement to be a vector quantity because its useful to define them this way. Consider a round trip on a line between two points. ...
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1
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Accepted
How can I formalize better this proof that angular momentum is conserved for a small impulse?
In general, $\delta \mathbf{L} = \delta \mathbf{r} \times \mathbf{p} + \mathbf{r} \times \delta \mathbf{p}$, which in this case becomes $\delta \mathbf{L} = \mathbf{r} \times \delta \mathbf{p} = \...
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Connection between contra-/covariant vectors in SR and complex numbers?
I think the thing you're looking for here is the Wick rotation. You start with a real vector space with coordinates $(t,x) \in \mathbb{R}^2$ and replace the real $t$-coordinate with a purely complex ...
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How can I formalize better this proof that angular momentum is conserved for a small impulse?
This is a comment that got too long. OP's question is very good. The quoted paragraph of infinitesimal estimates on p. 188 in Ref. 1 seems like an anticlimax after introducing a rigorous Def. 4.6 of ...
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How can I formalize better this proof that angular momentum is conserved for a small impulse?
I'll attempt to sum up here my understanding in a "community wiki" fashion. Anyone who reads this and thinks it can be improved please edit / comment.
The definition of Lagrange stability (...
1
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How can I formalize better this proof that angular momentum is conserved for a small impulse?
Overview
If total momentum is conserved between two bodies due to Newton's 3rd law, then angular momentum will also be conserved as the impulse will act and react on the same line of action.
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