In mathematical classical mechanics, the momentum is a cotangent
vector, while the velocity is a tangent vector. I don't get this!
calculus. In this usage, these phrases describe how a vector (a.k.a
a rank-1 tensor) transforms under a change of coordintes: a tangent
vector (a.k.a a "contravariant vector") is a vector which transforms
the same way a coordinate position $x^i$ does, while a cotangent vector >(a.k.a a "covariant vector") is a vector which transforms the same way
a partial derivative operator $\partial / \partial x^i$ does.
moderator jt wrote or quoted:
calculus. In this usage, these phrases describe how a vector (a.k.a
a rank-1 tensor) transforms under a change of coordintes: a tangent
vector (a.k.a a "contravariant vector") is a vector which transforms
the same way a coordinate position $x^i$ does, while a cotangent vector
(a.k.a a "covariant vector") is a vector which transforms the same way
a partial derivative operator $\partial / \partial x^i$ does.
Yeah, that explanation is on the right track, but I got to add
a couple of things.
Explaining objects by their transformation behavior is
classic physicist stuff. A mathematician, on the other hand,
defines what an object /is/ first, and then the transformation
behavior follows from that definition.
You got to give it to the physicists---they often spot weird
structures in the world before mathematicians do. They measure
coordinates and see transformation behaviors, so it makes sense
they use those terms. Mathematicians then come along later, trying
to define mathematical objects that fit those transformation
behaviors. But in some areas of quantum field theory, they still
haven't nailed down a mathematical description. Using mathematical
objects in physics is super elegant, but if mathematicians can't
find those objects, physicists just keep doing their thing anyway!
A differentiable manifold looks locally like R^n, and a tangent
vector at a point x on the manifold is an equivalence class v of
curves (in R^3, these are all worldlines passing through a point
at the same speed). So, the tangent vector v transforms like
a velocity at a location, not like the location x itself. (When
one rotates the world around the location x, x is not changed,
but tangent vectors at x change their direction.)
A /cotangent vector/ at x is a linear function that assigns a
real number to a tangent vector v at the same point x. The total
differential of a function f at x is actually a covector that
linearly approximates f at that point by telling us how much the
function value changes with the change represented by vector v.
When one defines the "canonical" (or "generalized") momentum as
the derivative of a Lagrange function, it points toward being a
covector. But I was confused because I saw a partial derivative
instead of a total differential. But possibly this is just a
coordinate representation of a total differential. So, broadly,
it's plausible that momentum is a covector, but I struggle
with the technical details and physical interpretation. What
physical sense does it make for momentum to take a velocity
and return a number? (Maybe that number is energy or action).
(In the world of Falk/Ruppel ["Energie und Entropie", Springer,
Berlin] it's just the other way round. There, they write
"dE = v dp". So, here, the speed v is something that maps
changes of momentum dp to changes of the energy dE. This
immediately makes sense because when the speed is higher
a force field is traveled through more quickly, so the same
difference in energy results in a reduced transfer of momentum.
So, transferring the same momentum takes more energy when the
speed is higher. Which, after all, explains while the energy
grows quadratic with the speed and the momentum only linearly.)
moderator jt wrote or quoted:
calculus. In this usage, these phrases describe how a vector (a.k.a
a rank-1 tensor) transforms under a change of coordintes: a tangent
vector (a.k.a a "contravariant vector") is a vector which transforms
the same way a coordinate position $x^i$ does, while a cotangent vector
(a.k.a a "covariant vector") is a vector which transforms the same way
a partial derivative operator $\partial / \partial x^i$ does.
Yeah, that explanation is on the right track, but I got to add
a couple of things.
Explaining objects by their transformation behavior is
classic physicist stuff. A mathematician, on the other hand,
defines what an object /is/ first, and then the transformation
behavior follows from that definition.
You got to give it to the physicists---they often spot weird
structures in the world before mathematicians do. They measure
coordinates and see transformation behaviors, so it makes sense
they use those terms. Mathematicians then come along later, trying
to define mathematical objects that fit those transformation
behaviors. But in some areas of quantum field theory, they still
haven't nailed down a mathematical description. Using mathematical
objects in physics is super elegant, but if mathematicians can't
find those objects, physicists just keep doing their thing anyway!
A differentiable manifold looks locally like R^n, and a tangent
vector at a point x on the manifold is an equivalence class v of
curves (in R^3, these are all worldlines passing through a point
at the same speed). So, the tangent vector v transforms like
a velocity at a location, not like the location x itself. (When
one rotates the world around the location x, x is not changed,
but tangent vectors at x change their direction.)
A /cotangent vector/ at x is a linear function that assigns a
real number to a tangent vector v at the same point x. The total
differential of a function f at x is actually a covector that
linearly approximates f at that point by telling us how much the
function value changes with the change represented by vector v.
When one defines the "canonical" (or "generalized") momentum as
the derivative of a Lagrange function, it points toward being a
covector.
I think Stefan is using "tangent vector" and "cotangent vector"
in the sense of differential geometry and tensor calculus. In
this usage, these phrases describe how a vector (a.k.a a rank-1
tensor) transforms under a change of coordintes: a tangent vector
(a.k.a a "contravariant vector") is a vector which transforms the
same way a coordinate position $x^i$ does, while a cotangent vector
(a.k.a a "covariant vector") is a vector which transforms the same
way a partial derivative operator $\partial / \partial x^i$ does.
On 2024-08-07 11:37:02 +0000, the moderator said:
I think Stefan is using "tangent vector" and "cotangent vector"
in the sense of differential geometry and tensor calculus. In
this usage, these phrases describe how a vector (a.k.a a rank-1
tensor) transforms under a change of coordintes: a tangent vector
(a.k.a a "contravariant vector") is a vector which transforms the
same way a coordinate position $x^i$ does, while a cotangent vector
(a.k.a a "covariant vector") is a vector which transforms the same
way a partial derivative operator $\partial / \partial x^i$ does.
Thank you. That makes sense.
--=20
Mikko
Explaining objects by their transformation behavior is
classic physicist stuff. A mathematician, on the other hand,
defines what an object /is/ first, and then the transformation
behavior follows from that definition.
How can I see that (given that q' is a tangential vector)
p is a cotangential vector?
Sysop: | Keyop |
---|---|
Location: | Huddersfield, West Yorkshire, UK |
Users: | 344 |
Nodes: | 16 (2 / 14) |
Uptime: | 30:02:15 |
Calls: | 7,518 |
Calls today: | 15 |
Files: | 12,713 |
Messages: | 5,642,183 |
Posted today: | 2 |