Exploring the Physical Significance of Force as a 1-Form in Line Integrals

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In summary: I think this is a good place to stop summarizing and let people read the article)Differential Geometry and Linear and Abstract Algebra are the right places to put this information. Differential Geometry would be the best place to go because it covers vectors and 1-forms. Linear Algebra would be a good place to go because it covers maps and vector spaces.
  • #1
bronxman
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Hello again,

Yesterday, through your help, I took my understanding to a new level.

I can now phrase my single question and I hope you will allow me to make it a separate thread...

  • I understand that the integrand in a line integral is a natural 1-form.
  • I understand that to obtain WORK = F.dx that Force is thus a 1-form.
  • (In fact, I understand that velocity and acceleration are vectors but force and moments are 1-forms.)

BUT WHY IS FORCE A 1-FORM, PHYSICALLY? I understand the outcome of the MATH, but what is the physical meaning of Force (acceleration weighted with mass) being a 1-form?
 
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  • #2
Is it, perhaps, just this?

Gradient is, mathematically, a 1-form. And in claiming force to be a one-form, we are assuming it is derivable from a potential function?

If so, then can I assume that forces derived from, say, friction, are not 1-forms?

Is that it? Force being a one-form presupposed potential energy?
 
  • #3
bronxman said:
BUT WHY IS FORCE A 1-FORM, PHYSICALLY? I understand the outcome of the MATH, but what is the physical meaning of Force (acceleration weighted with mass) being a 1-form?

There is no physical meaning. Depending on what you want to do, forces can be represented mathematically as either vectors or 1-forms.

If you want to find the acceleration of an object then using a vector is best. If you want to find the work done in moving along a given path then using a 1-form is best.
 
  • #4
This would be better placed in "Differential Geometry" than "Linear and Abstract Algebra".
 
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  • #6
I think its natural. Because when we apply a vector(displacement vector) to the force, we want to get a scalar(work). But something that gets a vector and gives a scalar is a 1-form.
 
  • #7
Shyan said:
I think its natural. Because when we apply a vector(displacement vector) to the force, we want to get a scalar(work). But something that gets a vector and gives a scalar is a 1-form.

Then we might as well see the displacement vector as the 1-form and the force as the vector, and mathematically that works out fine, but it's (rightfully) not how we do things.
The intuition behind 1-forms is that "anything you want to integrate is a form". Since you want to integrate force, it should be a form.
 
  • #8
A 1-form, formally, is just a linear map, an element of a dual space . Given an f.d vector space V, any linear map defined on v in V into the base field is a 1-form. So, what arguments does a fdx take in order to spit out a scalar? And, yes, we integrate k-forms over k-manifolds.
 
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  • #9
bronxman said:
Hello again,

Yesterday, through your help, I took my understanding to a new level.

I can now phrase my single question and I hope you will allow me to make it a separate thread...

  • I understand that the integrand in a line integral is a natural 1-form.
  • I understand that to obtain WORK = F.dx that Force is thus a 1-form.
  • (In fact, I understand that velocity and acceleration are vectors but force and moments are 1-forms.)

BUT WHY IS FORCE A 1-FORM, PHYSICALLY? I understand the outcome of the MATH, but what is the physical meaning of Force (acceleration weighted with mass) being a 1-form?

A force, given as a function, is actually a 0-form. fdx is a 1-form, as the wedge of the 0-form f and the 1-form dx.

EDIT: still, as atyy said , if V is a f.d vector space with a distinguished non-degenerate form, then V

and V* are naturally isomorphic, i.e., the isomorphism does not depend on a choice of basis ( you
can make this more precise using language from category theory ) . Still, I don't think the dual of
a 0-form is a 0-vector, but I am not sure.
 
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  • #10
WWGD said:
A force, given as a function, is actually a 0-form. fdx is a 1-form, as the wedge of the 0-form f and the 1-form dx.

OH! I SEE what I was now misreading. Yes. In Frankel's Geometry of Physics, I see how the integrand in a line integral is a one form. And yes, that means the F function AND the dx TOGETHER. Now I see that with the PULL BACK! It seems that when one pulls back the integral of the 1-form F, to a map on the real line, one gets the one-form fdx
 
  • #11
Last edited by a moderator:

1. What is a 1-form in physics?

A 1-form is a mathematical concept used in physics to describe a physical quantity that has both magnitude and direction. It is represented as an arrow and is used to quantify forces, velocities, and other physical quantities.

2. Why is force considered a 1-form?

Force is considered a 1-form because it has both magnitude and direction, making it a vector quantity. In physics, 1-forms are often used to describe vector quantities, making force a natural fit for this mathematical concept.

3. How is force represented as a 1-form?

Force is represented as a 1-form by using a mathematical object called a differential form. This form includes both the magnitude and direction of the force, making it a suitable representation for a 1-form.

4. What is the significance of force being a 1-form?

The significance of force being a 1-form lies in its ability to be described using mathematical concepts and equations. This allows scientists to make precise calculations and predictions about the behavior of objects in the physical world.

5. How does understanding force as a 1-form benefit scientific research?

Understanding force as a 1-form benefits scientific research by providing a mathematical framework for analyzing and predicting the behavior of physical systems. This allows for more accurate and efficient experimentation and allows scientists to make more precise conclusions about the world around us.

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