5 ms·

FWIW, they start counting function arguments from 0, so _2 is indeed the velocity. But I do agree with your main point that the order of arguments is irrelevan

FWIW, they start counting function arguments from 0, so _2 is indeed the velocity. But I do agree with your main point that the order of arguments is irrelevant, and it is a mistake to make it a first-class citizen of the notation.

codethief 3y ago
The order isn't really irrelevant, though, is it? If you take the total derivative and represent it as Jacobian matrix, you hopefully won't argue that the order of the matrix entries won't matter. (Especially if you later on employ it in a chain rule.)
- MatteoFrigo 3y ago
  I think that there are two cases of practical importance, which have incompatible requirements. The first case is where you have a N-dimensional vector space where all dimensions have the same units. The standard example would be the Newtonian 3D space. Depending on what you are trying to do, you can view it as a collection of coordinate-free abstract vectors, as a triple (x, y, z) of real numbers, or as an array X[i] of three coordinates in a given basis. In this case I would agree that X[0], X[1], X[2] is better than (x, y, z), the order matters, and you can define the Jacobian is a 2D array that represents a certain abstract derivative in a given coordinate system. I would argue that the formalism of Sussman and Wisdom (which they got from Spivak) is totally adequate to this case, and perhaps even the best possible. The second case is the one of the Lagrangian that parent mentioned, where L is a function of the triple (t, x, v). You could pretend that (t, x, v) form a vector space, but this definition won't get you far. I would regard (t, x, v) = t * (1, 0, 0) + x * (0, 1, 0) + v * (0, 0, 1) as meaningless because it is adding time, space, and velocity. You cannot really do rotations or general linear transformations in this space. You can define a Jacobian matrix if you want, but now all entries in the matrix have different physical units. In this case I would say the fact that v is the third element of the tuple is irrelevant, and that the tuple is better regarded as a map from symbolic names "t", "x", and "v" to real numbers. I would argue that the Spivak formalism is inadequate in this case, and it seems that many physicists on this thread think the same for essentially the same reason. This difference is kind of analogous to double X[3]; vs struct { double t; double x; double v; }; From one point of view they are the same, but in practice they have totally different meanings.
kergonath 3y ago
> FWIW, they start counting function arguments from 0, so _2 is indeed the velocity. Dammit yes, you’re right! Well, it’s not a bit less confusing. The most frustrating is that they have a point: we need to be stricter about disambiguating functions and numbers, and derivation really should be an operator. But you don’t need to go all the way to zero-indexing (which is definitely not a thing in the fields I know) or positional arguments. This is putting abstract notation purity above practical concerns. It’s not surprising they like Scheme.
- tobinfricke 3y ago
  > But you don’t need to go all the way to zero-indexing (which is definitely not a thing in the fields I know) or positional arguments. This is putting abstract notation purity above practical concerns. Yes, it's definitely a perspective influenced strongly by computer science. > It’s not surprising they like Scheme. In fact one of the authors, Gerry Sussman, is one of the original inventors of Scheme. https://en.wikipedia.org/wiki/Gerald_Jay_Sussman https://en.wikipedia.org/wiki/Gerald_Jay_Sussman
amluto 3y ago
By that standard, if f is a function of x and y, then writing f(1,2) is syntactically bad because the argument slots don’t have a meaningful order. One could surely invent a valid mathematical formalism with exclusively named argument slots, but this isn’t how math is generally done. (I admit it might be a lot easier to avoid losing track of which thing is a row and which is a column in a gnarly linear algebra expression if all dimensions were explicitly named, and this would come with a tradeoff of verbosity. Also, the interpretation of a matrix as a linear function from vectors to vectors would need some clarification as to which dimension is input and which is output, so maybe it would look a bit like Einstein notation with superscript dimensions and subscript dimensions?)
deleted 3y ago
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codethief 3y ago

The order isn't really irrelevant, though, is it? If you take the total derivative and represent it as Jacobian matrix, you hopefully won't argue that the order of the matrix entries won't matter. (Especially if you later on employ it in a chain rule.)

MatteoFrigo 3y ago
I think that there are two cases of practical importance, which have incompatible requirements. The first case is where you have a N-dimensional vector space where all dimensions have the same units. The standard example would be the Newtonian 3D space. Depending on what you are trying to do, you can view it as a collection of coordinate-free abstract vectors, as a triple (x, y, z) of real numbers, or as an array X[i] of three coordinates in a given basis. In this case I would agree that X[0], X[1], X[2] is better than (x, y, z), the order matters, and you can define the Jacobian is a 2D array that represents a certain abstract derivative in a given coordinate system. I would argue that the formalism of Sussman and Wisdom (which they got from Spivak) is totally adequate to this case, and perhaps even the best possible. The second case is the one of the Lagrangian that parent mentioned, where L is a function of the triple (t, x, v). You could pretend that (t, x, v) form a vector space, but this definition won't get you far. I would regard (t, x, v) = t * (1, 0, 0) + x * (0, 1, 0) + v * (0, 0, 1) as meaningless because it is adding time, space, and velocity. You cannot really do rotations or general linear transformations in this space. You can define a Jacobian matrix if you want, but now all entries in the matrix have different physical units. In this case I would say the fact that v is the third element of the tuple is irrelevant, and that the tuple is better regarded as a map from symbolic names "t", "x", and "v" to real numbers. I would argue that the Spivak formalism is inadequate in this case, and it seems that many physicists on this thread think the same for essentially the same reason. This difference is kind of analogous to double X[3]; vs struct { double t; double x; double v; }; From one point of view they are the same, but in practice they have totally different meanings.

kergonath 3y ago

> FWIW, they start counting function arguments from 0, so _2 is indeed the velocity. Dammit yes, you’re right! Well, it’s not a bit less confusing. The most frustrating is that they have a point: we need to be stricter about disambiguating functions and numbers, and derivation really should be an operator. But you don’t need to go all the way to zero-indexing (which is definitely not a thing in the fields I know) or positional arguments. This is putting abstract notation purity above practical concerns. It’s not surprising they like Scheme.

tobinfricke 3y ago
> But you don’t need to go all the way to zero-indexing (which is definitely not a thing in the fields I know) or positional arguments. This is putting abstract notation purity above practical concerns. Yes, it's definitely a perspective influenced strongly by computer science. > It’s not surprising they like Scheme. In fact one of the authors, Gerry Sussman, is one of the original inventors of Scheme. https://en.wikipedia.org/wiki/Gerald_Jay_Sussman https://en.wikipedia.org/wiki/Gerald_Jay_Sussman

amluto 3y ago

By that standard, if f is a function of x and y, then writing f(1,2) is syntactically bad because the argument slots don’t have a meaningful order. One could surely invent a valid mathematical formalism with exclusively named argument slots, but this isn’t how math is generally done. (I admit it might be a lot easier to avoid losing track of which thing is a row and which is a column in a gnarly linear algebra expression if all dimensions were explicitly named, and this would come with a tradeoff of verbosity. Also, the interpretation of a matrix as a linear function from vectors to vectors would need some clarification as to which dimension is input and which is output, so maybe it would look a bit like Einstein notation with superscript dimensions and subscript dimensions?)

deleted 3y ago

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