
Structure and Interpretation of Classical Mechanics (2015) - Tomte
https://mitpress.mit.edu/sites/default/files/titles/content/sicm_edition_2/toc.html
======
jessriedel
Sussman is one of the very few classical mechanics textbooks that gives a
reasonable definition of the Legendre transform. Most physicists cannot
actually tell you what that transform is, even though it sits at the heart of
both classical and quantum mechanics.

[http://blog.jessriedel.com/2017/06/28/legendre-
transform/](http://blog.jessriedel.com/2017/06/28/legendre-transform/)

~~~
jules
The description in this article is great, but the why is still rather
mysterious. How would somebody come up with that?

If you are familiar with the method of Lagrange multipliers, then what's
happening can be explained as follows. Given the Lagrangian L(x,v) the problem
of classical mechanics is to find a trajectory x(t),v(t) that extremises the
integral of L(x(t),v(t))dt under the constraint x'(t) = v(t). Lagrange
multipliers are a method to deal with constraints in optimisation problems.
Usually it's taught in the finite dimensional case, but it also works in the
infinite dimensional case. We introduce a Lagrange multiplier p(t) and add the
constraint to the objective: integral of L(x(t),v(t)) + p(t)(x'(t) - v(t)) dt.
To solve the problem we minimise this over x,v,p. If we carry out the
minimisation over v first then we're left with two variables x,p. That's the
Hamiltonian formulation of the problem, and it's called the dual problem in
convex optimisation. So the momentum p is the Lagrange multiplier for the
constraint x' = v.

In more detail: we rewrite L(x(t),v(t)) + p(t)(x'(t) - v(t)) = L(x(t),v(t)) -
p(t)v(t) + p(t)x'(t). Now we separate out H(x,p) = min_v L(x,v) - pv, so the
original problem becomes to minimise the integral of H(x,p) + p(t)x'(t). After
applying the Euler-Lagrange equations we obtain Hamilton's equations:

dH/dx = dp/dt

dH/dp = -dx/dt

~~~
jessriedel
Agreed, thanks. That blog post was just trying to explain _what_ it is, not
why it is, mostly as a basis to complain about education rather than teach
physics.

~~~
Misdicorl
The why is actually quite simple- and the same reason the Fourier transform is
sometimes used. Some problems are simply more elegantly expressed in a
particular basis system. Nobody tries to express a ball's motion in flight via
Fourier analysis, but you certainly could.

In the same way, sometimes solutions for the position(s) of a system is the
most natural basis system for describing/investigating a problem (use the
Lagrangian) and sometimes solutions for its momentum are (use the
Hamiltonian).

Of course, they're intrinsically linked since the evolution of one determines
the evolution of the other.

------
montalbano
At first glance, this looks like an excellent textbook.

Personally, my favourite theorem in classical mechanics is the so called
'tennis racket theorem', sometimes known as the 'intermediate axis theorem'.

It explains why objects with roughly three different moments of intertia have
unstable rotation about their intermediate moment.

It can be easily demonstrated with a tennis racket, or even most smartphones
(be careful not to break it though).

[https://en.wikipedia.org/wiki/Tennis_racket_theorem](https://en.wikipedia.org/wiki/Tennis_racket_theorem)

~~~
barking
There's a link on that wikipedia page to this youtube demo which makes it
really obvious

[https://www.youtube.com/watch?v=4dqCQqI-
Gis](https://www.youtube.com/watch?v=4dqCQqI-Gis)

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namin
You can play with the software online here:
[http://io.livecode.ch/learn/namin/scheme-
mechanics](http://io.livecode.ch/learn/namin/scheme-mechanics)

The notation chapter with live snippets is reproduced here:
[http://io.livecode.ch/learn/namin/scheme-
mechanics/chapter9](http://io.livecode.ch/learn/namin/scheme-
mechanics/chapter9)

------
nestorD
Here is a link to the unofficial html edition :
[https://tgvaughan.github.io/sicm/toc.html](https://tgvaughan.github.io/sicm/toc.html)

~~~
kdtop
I came to say how horrible the formatting of the original post is. But this
site fixes that. Much better! Thanks!

------
RoboTeddy
I love this book! Prior to reading it, I had been getting confused when trying
to learn classical mechanics. The book writes out everything explicitly in
code, which let me use the software engineering part of my brain, and made
everything easy to follow. Apparently I had been being held back by
unfamiliarity with math formalisms!

If you're interested in the topic and know how to program, it might be worth a
read in case it turns out you're in the same boat.

------
Illniyar
Apparently written by the same person who wrote Structure and Interpretation
of Computer Programs - Gerald Jay Sussman .

~~~
privong
Who also wrote "Functional Differential Geometry":
[https://mitpress.mit.edu/books/functional-differential-
geome...](https://mitpress.mit.edu/books/functional-differential-geometry)

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dang
For the curious, 2015:
[https://news.ycombinator.com/item?id=9560567](https://news.ycombinator.com/item?id=9560567)

2013:
[https://news.ycombinator.com/item?id=6947257](https://news.ycombinator.com/item?id=6947257)

2010:
[https://news.ycombinator.com/item?id=1581696](https://news.ycombinator.com/item?id=1581696)

------
hyeonwho4
Unfortunately, the code in this book uses a modified version of scheme which
is no longer maintained.

~~~
kkylin
Looks like MIT Scheme (on which the ScmUtils system runs) still gets some
regular maintainence releases, though I don't know how active development is
these days:

[https://www.gnu.org/software/mit-scheme/](https://www.gnu.org/software/mit-
scheme/) [https://ftp.gnu.org/gnu/mit-
scheme/stable.pkg/](https://ftp.gnu.org/gnu/mit-scheme/stable.pkg/)

Installation instructions for the ScmUtils package:

[http://groups.csail.mit.edu/mac/users/gjs/6946/index.html](http://groups.csail.mit.edu/mac/users/gjs/6946/index.html)

As far as I know, Gerry and Jack still teach the course every year (can
someone currently at MIT verify?) and still use the system.

~~~
mspecter
Yep!
[https://groups.csail.mit.edu/mac/users/gjs/6.945/](https://groups.csail.mit.edu/mac/users/gjs/6.945/)

I'm in his group at MIT.

~~~
kkylin
Good to hear! Please say hi to everyone. I used to hang out in the group back
when it was still at Tech^2.

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billfruit
I haven't read beyond the first two chapters, but the clarity and
expressiveness of the notation used is amazing.

I do wish more maths/physics etc would adopt similar, clearer, unambiguous
notations.

------
analognoise
What are the books you need to read, before this book, so it makes sense?

I'm an EE, but I admit I made no progress. Clearly I lack some deeper
understanding - where can I best look to fix that?

~~~
hnarayanan
It's not a book that I would recommend but the lectures from a course:
[https://theoreticalminimum.com/courses/classical-
mechanics/2...](https://theoreticalminimum.com/courses/classical-
mechanics/2011/fall)

This should give you plenty of intuition to then tackle SICM.

------
ArtWomb
Excellent! Good companion to this is Richard Fitzpatrick's Computational
Physics notes

[http://farside.ph.utexas.edu/teaching/329/329.html](http://farside.ph.utexas.edu/teaching/329/329.html)

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AlexCoventry
> _We hope others, especially our competitors, will adopt these methods, which
> enhance understanding while slowing research._

:)

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app4soft
Is there a PDF-version?

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martincmartin
Is there something similar for Quantum Mechanics? Both MIT and Harvard use
Griffith's "Introduction to Quantum Mechanics," but it seems to emphasize
computation and symbol manipulation over physical intuition.

~~~
amelius
There's [1], also by Sussman, which is "just" about differential forms and
related math, and in the summary they refer to QM: "An explanation of the
mathematics needed as a foundation for a deep understanding of general
relativity or quantum field theory."

[1] [https://mitpress.mit.edu/books/functional-differential-
geome...](https://mitpress.mit.edu/books/functional-differential-geometry)

~~~
thatcherc
Having read through that it's not really directly relevant to quantum physics
(at least what I learned in undergrad). It is super applicable to general
relativity and electrodynamics though! It's a much shorter book than SICM but
I still have yet to understand it through to the end. Tricky stuff, but fun to
think about!

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aj7
“If you can’t explain it simply, you don’t understand it well enough,”
followed by “For example, we write Lagrange’s equations in functional notation
as follows: D(∂2L ∘ Γ[q]) − ∂1L ∘ Γ[q] = 0” Whoa Nellie! Gonna get bumpy.

------
rongenre
Interesting -- I'm super rusty on my physics -- is there discussion of the
exercises anywhere? It'd be fun to go through this and feel like I "got" it.

