Lectures on integral invariants and Hamiltonian equations

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Discussion Overview

The discussion revolves around the teaching and understanding of advanced concepts in mathematical physics, particularly related to integral invariants and Hamiltonian equations. Participants share their experiences and thoughts on the accessibility of such topics within classical mechanics courses, as well as the prerequisites needed to fully grasp the material presented in a linked paper.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants express uncertainty about where advanced topics in mathematical physics, such as those in the linked notes, would fit into classical mechanics education.
  • There is a suggestion that many students may not encounter classical mechanics formulated in the language of differential forms during their studies.
  • Some participants note that exposure to these ideas may depend on the mathematical inclination of professors or the inclusion of mathematical physics courses.
  • One participant mentions a personal goal to reach a level of understanding sufficient to appreciate the linked paper by the end of the 2027 academic year.
  • Another participant challenges the feasibility of this goal, citing the complexity of the material and the typical educational timeline for mastering the necessary prerequisites.
  • There is a discussion about the age and educational background of the original poster, who is a high school sophomore, and the implications this has for their goal of understanding the paper.

Areas of Agreement / Disagreement

Participants generally agree that the material is advanced and may not be readily accessible to those at earlier stages of their physics education. However, there is disagreement regarding the timeline and feasibility of achieving the necessary understanding, with some expressing skepticism about the original poster's goal.

Contextual Notes

Participants highlight the potential challenges in mastering the prerequisites for understanding the linked paper, including the need for a solid foundation in both mathematics and physics, which may take several years to develop.

wrobel
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I have no idea where to place this post. Just to share my teaching experience . This is a part of my lectures notes. The aim of the text is to present a successive exposition of several pieces of mathematical physics that have common origin. Perhaps it would be of some use for those who teach classical mechanics.
https://arxiv.org/pdf/2507.02878v4
 
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wrobel said:
I have no idea where to place this post. Just to share my teaching experience . This is a part of my lectures notes. The aim of the text is to present a successive exposition of several pieces of mathematical physics that have common origin. Perhaps it would be of some use for those who teach classical mechanics.
https://arxiv.org/pdf/2507.02878v4
oh, awesome! Where in a classical mechanics course would you learn this, out of curiosity? Maybe one day I will have the physics skills to properly appreciate the notes... but when would this be is what I am wondering.
 
TensorCalculus said:
oh, awesome! Where in a classical mechanics course would you learn this, out of curiosity? Maybe one day I will have the physics skills to properly appreciate the notes... but when would this be is what I am wondering.
I would say many (most?) people could go through graduate physics studies and not see classical mechanics formulated in the language of differential forms (as used in these notes). But you would still get a taste for the same ideas in an upper-division undergraduate course and graduate course. You'll have to come across professors more mathematically inclined, or take a mathematical physics class of some kind, if you want to see something similar to wrobel's notes.
 
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Haborix said:
I would say many (most?) people could go through graduate physics studies and not see classical mechanics formulated in the language of differential forms (as used in these notes). But you would still get a taste for the same ideas in an upper-division undergraduate course and graduate course. You'll have to come across professors more mathematically inclined, or take a mathematical physics class of some kind, if you want to see something similar to wrobel's notes.
ah, so graduate physics + beyond. Not something I would come across soon. But the notes look very interesting - from a quick glance I agree that they are a very mathematical approach to mechanics (not that I can claim to understand much)
 
TensorCalculus said:
ah, so graduate physics + beyond. Not something I would come across soon.

Yes, you will have a lot of water under your bridge before you can fully appreciate the article.


1753299424070.webp
 
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Thanks for sharing! This looks like a thoughtful way to connect different topics in mathematical physics. I'm sure it’ll be helpful for others teaching or studying classical mechanics.
 
gleem said:
Yes, you will have a lot of water under your bridge before you can fully appreciate the article.


View attachment 363633
Indeed.
My goal: get there before the end of the 2027 academic year
 
TensorCalculus said:
My goal: get there before the end of the 2027 academic year

To get where?
 
gleem said:
To get where?
To a point at which I have the sufficient prerequisite maths and physics knowledge/experience to be able to fully appreciate the paper linked to in post #1
 
  • #10
TensorCalculus said:
My goal: get there before the end of the 2027 academic year
Will not happen, But, if by some chance you think you do, please let us know.
 
  • #11
gleem said:
Will not happen

Why? I specialize in differential geometric methods in physics, and these notes look like something I attended around 3rd or 4th year of studying. If you study theoretical physics you have to learn fast.
 
  • #12
weirdoguy said:
Why? I
The OP is a sophomore in HS and has a goal of understanding the paper in two years.
 
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  • #13
gleem said:
The OP is a sophomore in HS and has a goal of understanding the paper in two years.
Yeah exactly - I'm a 14 year old kid (will not happen is the answer I was expecting, but it never hurts to hope)...
My physics might be good, but it's not that good.... only as good as a maybe 1st-2nd year uni student... being optimistic
 
  • #14
gleem said:
The OP is a sophomore in HS

Oh lol, I thought OP was another user, sorry. So yeah, @TensorCalculus give yourself 8-10 years.
 
  • #15
weirdoguy said:
Oh lol, I thought OP was another user, sorry. So yeah, @TensorCalculus give yourself 8-10 years.
Awww :(
Oh well, worth hoping.

EDIT: Don't worry, I'm already slowly working my way towards that point - forgot about the book in my hand 😂
 

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  • #16
TensorCalculus said:
Oh well, worth hoping.
No hoping, only doing.
 
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  • #17
gleem said:
No hoping, only doing.
Good correction.
 
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