What is angular momentum from a differential geometry…
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What is angular momentum from a differential geometry perspective
I'm I guess what you'd call semi-amateur physics guy. I do have a BSc in Physics but it's many years old and these days I mostly just read random bits from textbooks I own out of interest. I've been interested in the DG formulation of classical mechanics, Sympletic Geometry and Hamiltonian flows, for awhile now. I've been really trying to grok the miracle of vector calculus in the context of the calculus of forms. By which I mean the fact that vectors, 1-forms, and 2-forms all have 3 components in R\^3.
I was reading up on the cross product and as far as I understand it the mental picture is take your vectors, which are members of the Tangent Bundle to the manifold, map those to 1-forms in the Cotangent Bundle, then form the wedge product to get a 2-form, then translate that back to a vector, and that's your "cross product" vector. And that explains why the length of the vector is actually an area, which always seemed so odd to me.
The geometric interpretation I've read is that the vectorial representation of da \^ db corresponds to the flux through the 2d surface element spanned by vec{a} and vec{b}
Which brings me to my question, if angular momentum is classically the cross product of the position vector and the linear momentum vector then what flux does it represent? What vector field are we even talking about?
Also side question. How does it even make sense to take the cross product of a position vector and a momentum vector? It never occurred to me how weird that is. They don't seem like they should be a part of the same vector space. What does it mean to add a position to a momentum?
#science
source
I'm I guess what you'd call semi-amateur physics guy. I do have a BSc in Physics but it's many years old and these days I mostly just read random bits from textbooks I own out of interest. I've been interested in the DG formulation of classical mechanics, Sympletic Geometry and Hamiltonian flows, for awhile now. I've been really trying to grok the miracle of vector calculus in the context of the calculus of forms. By which I mean the fact that vectors, 1-forms, and 2-forms all have 3 components in R\^3.
I was reading up on the cross product and as far as I understand it the mental picture is take your vectors, which are members of the Tangent Bundle to the manifold, map those to 1-forms in the Cotangent Bundle, then form the wedge product to get a 2-form, then translate that back to a vector, and that's your "cross product" vector. And that explains why the length of the vector is actually an area, which always seemed so odd to me.
The geometric interpretation I've read is that the vectorial representation of da \^ db corresponds to the flux through the 2d surface element spanned by vec{a} and vec{b}
Which brings me to my question, if angular momentum is classically the cross product of the position vector and the linear momentum vector then what flux does it represent? What vector field are we even talking about?
Also side question. How does it even make sense to take the cross product of a position vector and a momentum vector? It never occurred to me how weird that is. They don't seem like they should be a part of the same vector space. What does it mean to add a position to a momentum?
#science
source
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