Inculcation: all of mechanics

Main article: inculcation

I make a very specific use of the words “physics” and “mechanics”, here. Ofcourse I shall explain what I mean, but do note, it is a personal choice.

trying to do physics but failing §

• What is spacetime? What is an inertial frame?

Before doing anything, just watch this:

Mechanics using the Action principle

Continue with GR and action: General Relativity by Prof. Thanu Padmanabhan - YouTube

After that watch these lectures covering Newtonian, Lagrangian, Hamiltonian, Statistical mechanics, special relativity all at once!

• These are nice as an intro “proper” physics, they will look fascinating, but my recommended levels of motivation and precision is absent. These are “Feynman lectures done right”. One may watch his non-linear dynamics and quantum mechanics lectures right after this.
• But nothing is explained rigorously, although hinted at, lots of details are skipped and Balki name drops a LOT of stuff. You may choose to ignore them initially, because each term becomes a rabbit hole for math topics.

what is physics actually then §

From these previous lectures, one must agree that doing physics is a three step process:

• Step 1: Choose your spacetime, you have 3 major types of options:
  • $c^{-1}=0$ Newtonian: the good ol’ “non-relativistic” spacetime
  • $c^{-1}=1,G=0$ Minkowski: the SR spacetime
  • $c^{-1}=1,G=1$ Lorentzian: a whole range of GR spacetimes
• Step 2: Choose what the contents of your spacetime must be:
  • points: point particles, bodies with finite number of degrees of freedom, rays
  • fields: infinite degrees of freedom
  • fluids: “spacetime itself flowing” (they are different from fields, yes)
  • condensed matter
• Step 3: Choose a description for the contents (although not all of the following is possible for all the contents in step 2)
  • $\hslash =0$ “classical mechanics/classical field theory” description - that is, writing equations of motion, $ma=F$ for particles in $c\to \infty$ or Maxwell’s equations for EM fields
  • $\hslash =0$ “classical statistics” description
  • $\hslash =1$ “quantum mechanics/QFT” description
  • $\hslash =1$ “quantum statistical” description

BUT! Does the description of matter really depend very much on the spacetime? Yes sure, the equation of motion will change drastically, but the methods in ODEs do not change at all! We can study about spherical harmonics in waves, electrodynamics and in QM class - their physical interpretation is different but their math interpretation remains the same!

Hence, there is a component to this in the second step of the 3 step process: which is “abstract”. How much of the things we study can we abstract out? Can it be really useful in doing physics?

This makes sense to me because generally quantum mechanics is taught before fluid mechanics, which is true because the former is a linear PDE - which makes it much easier - just introduce eigenvectors and eigenvalues and the method can be explained!

Now for an example:

Around 1850 Maxwell realized that the field strength of the electromagnetic field is modeled by what today we call a closed differential 2-form on spacetime. In the 1930s Dirac observed that more precisely this 2-form is the curvature 2-form of a U(1)-principal bundle with connection, hence that the electromagnetic field is modeled by what today is called a degree 2-cocycle in ordinary differential cohomology . ^[https://ncatlab.org/nlab/show/higher+category+theory+and+physics#GaugeTheory]

This is an example of the table for electromagnetic field (for $\hslash =0$). Gauge fields will be discussed in Doing the proper theory of gauge fields.

Lectures on Geometrical Anatomy of Theoretical Physics by Frederic Schuller

These lectures start from logic!

Although we must do physics as well, so we make this little table for reference with common terminology:

contents	description	$c^{-1}=0$ Newtonian spacetime	$c^{-1}=1,G=0$ Minkowski spacetime (SR)	$c^{-1}=1,G=1$ Lorentzian manifolds (GR)
	$\hslash =0$
points	“actual”	“classical mechanics” Kleppner, Goldstein, David Morin, LandauLifshitz vol 1, Balki’s lectures	“relativistic mechanics” LandauLifshitz vol 2	“general relativity” LandauLifshitz vol 2
	“stat”	“statistical mechanics” Kardar	“relativistic statistical mechanics” Palash Pal
fields	“actual”	“non-relativistic classical field theory”	“classical field theory”	“classical field theory in curved spacetime”
	“stat”		“statistical field theory”
fluids		“fluid mechanics”	“relativistic fluid mechanics”	“fluid mechanics in curved spacetime”
	$\hslash =1$
points	“actual”	“quantum mechanics” Griffiths, Balki’s lectures, Shankar	-	-
	“stat”	“quantum statistical mechanics” QM textbooks ^ should cover this	-	-
fields	“actual”		“QFT”	“QFT in curved spacetime” or “global QFT”
	“stat”
fluids	quantum fluids?	?	?	?
atoms?
condensed matter?
				QG?

In general, there are

• Landau Lifshitz volumes 1-10
• David Tong’s notes for many of the elements in the table
• look below!

So, essentially we did all of physics together. But did we do it properly? Did we went onto understanding the details?

How to do mechanics properly §

understanding the philosophy §

How one should do is by constructing everything linearly:

• first analysis in ${\mathbb{R}}^n$ and/or analysis on smooth manifolds, then
  • inculcation-odes -> Mechanics of points done properly
  • PDEs
  • PDEs -> inculcation-mechanics-fields
  • ? inculcation-mechanics-branes
  • PDEs -> inculcation-mechanics-fluids
• What one must understand is, in the side of ODEs: Lagrangian, Hamiltonian etc really do not care about the spacetime, or what you are trying to describe even: give it a ray of light in Newtonian spcaetime, or a point particle moving around a black hole (Swartzchild spacetime): the description remains the same.
  • Configuration space $\leftrightarrow$ Lagrangian dynamics, any other ODE
  • Phase space $\leftrightarrow$ Hamiltonian dynamics
• I do not mean the equations remains the same btw! I just mean use can use the general prescription (math!) like Lagrangians and Hamiltonians for any spacetime!
• This would therefore mean, the “statistical mechanics” prescription would also be, in this way - because statistical mechanics just starts from the phase space - the methods independent of “spacetimes” per say as phase space only depend on the configuration spaces.

Tip

You might wonder that we do not need this much details, or need to do things “properly”, whatever that means. The physics textbooks do cover most of the content, with a bit of analysis and geometry everything becomes well-understood as well as “proper”. I am assuming things like linear algebra “trivial”, but one has to start from that, thats why inculcation-linear-constructions exists!

doing the spacetimes bit §

This doesn’t need to be done first, or before Mechanics of points done properly, but these lectures are amazing:

The WE-Heraeus International Winter School on Gravity and Light

• THE WE-HERAEUS INTERNATIONAL WINTER SCHOOL ON GRAVITY AND LIGHT (Spanish Translation Available!) (richie291.wixsite.com)
• Maths with Physics: The WE-Heraeus International Winter School on Gravity and Light, Lectures, Tutorials and Solutions

Doing this lectures mean you shall be clear of all spacetimes not just GR.

Mechanics of points done properly §

That is: as we see in “classical mechanics”, but the idea of a configuration space captures rays (as in ray optics) and rigid bodies along with finite number of point particles.

writing the equations §

	Configuration space	Lagrangian	Phase space	Hamiltonian	$V=0$ solutions
1 point	${\mathbb{R}}^3$ (space)				geodesics in ${\mathbb{R}}^3$ with usual metric, that is, straight lines
$n$ points
1 rigid body	$SO(3)⋉{\mathbb{R}}^3$

solving the equations §

• Physics textbooks on “classical mechanics” only worry about local properties of the configuration spaces, local solutions of the equations: so for example it cannot differentiate between a cylinder or a sphere as configuration spaces, because locally they are “same” given how we are describing them (smooth manifolds).
• Both local and global properties of ODEs are studied in inculcation-odes, in for example Perko’s book.

geometry behind the dynamics §

The geometry behind dynamics of a ODE is that of a vector field on a manifold.

vector fields and ODE dictionary §

What we do is, write a differential equation like

$${\dot{x}}_i=f_i(x_1,x_2,\dots ,x_n)$$

for all $1\le i\le n$ so we have $n$ equations for $n$ variables $x_i$ and make it even more compact by

$$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$$

where $\mathbf{f}:\mathcal{U}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ is a vector field on the open domain $\mathcal{U}$.

This gives us a geometric pov on ODEs in ${\mathbb{R}}^n$, and we have a

solving differential equations	analysis and geometry of vector fields
an equation $\dot{x}=\mathbf{f}(x)$	a vector field $\mathbf{f}:\mathcal{U}\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$
solutions of the equation $\begin{array}{rl}\dot{x}(t)& =\mathbf{f}(x(t))\\ x(0)& =x_0\end{array}$	integral curves of the vector field $t\mapsto {\Phi }_t^{\mathbf{f}}(x_0)$
how solutions depend on initial conditions $x_0\mapsto x(t)$	flows of the vector field $x_0\mapsto {\Phi }_t^{\mathbf{f}}(x_0)$
conserved quantities	integrals of the vector field
(linearly) decoupling the differential equation $\dot{z_i}=f_i(z_i)$	(linear) coordinate transformation such that $\mathbf{f}={\sum }_i{f}_i(z_i){\hat{z}}_i$

We may convert ordinary differential equation of any order to first order by taking enough independent variables and defining them to be higher derivatives.

Link to original

The geometry of a Lagrangian is behind the algebra of a chain complex it creates, it seems to me.

The geometry behind hamiltonian dynamics is symplectic geometry.

https://people.math.harvard.edu/~jeffs/SymplecticNotes.pdf

probabilistic description of points §

As done in statistical mechanics.

Todo Main goal of statistical mechanics: derive the empirical laws of thermodynamics from the classical mechanics description.

• Boltzmann, Gibbs invented statistical mechanics to give more meaning to the thermodynamic quantities
• Ergodic theory was invented to ask when does the assumptions made by Boltzmann and others hold.
• First reference should be Balki’s lectures, other physics references include MIT 8.333 Statistical Mechanics I: Statistical Mechanics of Particles, Fall 2013 - YouTube
• Statistical Mechanics From Thermodynamics to the Renormalization Group
• Roderich Tumulka’s notes is best reference I’ve seen
  • actually talks about the main goal
• Information Geometry (ucr.edu)
  • Part 17-21 is where thermodynamics is discussed
• Classical Mechanics versus Thermodynamics (ucr.edu)

Hence what we get is

$$\mathrm{Probabilistic}:\mathrm{CM}(X,\mu )\to \mathrm{StatMech}(Q,\pi )\to \mathrm{Thermo}(Q,S)$$

As of now, I have obtained some justifications for the definitions in statistical mechanics, but it still remains a formal study (and not a replacement of the actual dynamics).

quantization of the mechanics of points §

As done in quantum mechanics. Essentially, functional analysis on (rigged) Hilbert spaces (brings in Representation theory)

Balakrishnan's quantum physics

Frederic Schuller's lectures on quantum mechanics with lecture notes

• Maths with Physics: Frederic Schuller’s Lectures on Quantum Theory with Lecture Notes
• DR. FREDERIC SCHULLER’S COURSE OF QUANTUM THEORY (richie291.wixsite.com)

So

$$\mathrm{Quantizaton}:\mathrm{CM}(X,H)\to \mathrm{QM}(L^2(X),\hat{H})$$

but now we observe the correspondence between them, even though they are very different structure wise:

though the one point problem §

We must notice the analogy

the “classical” problem in Hamiltonian dynamics	the quantum problem in Schrodinger/H picture
(physical) space is ${\mathbb{R}}^3$ and time is $\mathbb{R}$	(physical) space is ${\mathbb{R}}^3$ and time is $\mathbb{R}$
configucation space is set of all positions possible ${\mathbb{R}}^3$	the domain of the wave function is ${\mathbb{R}}^3$ (= configuration space)
the phase space is set of all positions, momentum pairs possible ${\mathbb{R}}^3\times {\mathbb{R}}^3$ whose “dimension” is $6$	the ”quantum phase space” is the set of all wave functions, set of all square integrable functions ${\mathbb{R}}^3\to \text{C}$ $L^2({\mathbb{R}}^3,\text{C})$ whose “dimension” is not finite!
Hamiltonian is a smooth function ${\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{R}$ $H(q,p)=\frac{p^2}{2m}+V(q)$	the “Hamiltonian” is a Hermitian linear functions $L^2({\mathbb{R}}^3)\to L^2({\mathbb{R}}^3)$ $\hat{H}=\frac{{\hat{p}}^2}{2m}+\hat{V}$
Hamilton’s equations	Schrodinger equation
Observables are smooth functions ${\mathbb{R}}^3\to \mathbb{R}$ (just like Hamiltonian)	Observables are Hermitian linear functions $L^2({\mathbb{R}}^3)\to L^2({\mathbb{R}}^3)$ (just like quantum “Hamiltonian”)
The set of all observables is the set of all smooth functions, so ${\mathcal{C}}^{\infty }({\mathbb{R}}^6)$ where we can define the Poisson brackets $\{,\}$	The set of all observables is the set of all Hermitian linear functions $L^2({\mathbb{R}}^3)\to L^2({\mathbb{R}}^3)$, this is a subspace of all linear functions and is closed in the commutator brackets $[,]$
position $q_i$	${\hat{q}}_i$
momentum $p_i$	${\hat{p}}_i=\text{pdvf}{q}_i$
momentum “generates” translations - $\exp (\alpha X_{p_i})$ ($X_{p_i}$ is the Hamiltonian vector field of $p_i$, exponential of vector fields) - translation is a group action by $({\mathbb{R}}^3,+)$ onto the phase space ${\mathbb{R}}^6$ - which is not a symmetry here! because $\{H,p_i\}\ne 0$ if we have a non-constant potential	momentum “generates” translations - $\exp (\alpha {\hat{p}}_i)$ (exponential of linear maps) - translation is a group action by $({\mathbb{R}}^3,+)$ onto the phase space $L^2({\mathbb{R}}^3)$ - which is not a symmetry here! because $\{H,p_i\}\ne 0$ if we have a non-constant potential
$L_i$	${\hat{L}}_i$
angular momentum “generates” rotations - $\exp (\theta X_{L_i})$ ($X_{L_i}$ is the Hamiltonian vector field of $L_i$, exponential of vector fields) - rotation is a group action by $SO(3)$ onto the phase space ${\mathbb{R}}^6$ - which is a symmetry here! because $\{H,L_i\}=0$ if we have a non-constant potential	angular momentum “generates” rotations - $\exp (\theta {\hat{L}}_i)$ (exponential of linear maps) - rotation is a group action by $SO(3)$ onto the quantum phase space $L^2({\mathbb{R}}^3)$ - which is a symmetry here! because $\{H,L_i\}=0$ if we have a non-constant potential
the group action is symplectic, and more specifically Hamiltonian!	the above group actions act by unitary transformations because exponential of Hermitian maps are unitary
$\{f_1,f_2\}=0$ means we are one step closer to Liouville–Arnold integrability	$[{\hat{f}}_1,{\hat{f}}_2]=0$ means we can simultaneously diagonalize them $\to$ one step closer to having an eigenbasis (is this actually correct?)

This analogy can be generalised many folds (geometric quantization of a Hamiltonian group action).

Link to original

Mechanics of fields §

🚧 This section is still under construction.

Why upgrade to Lagrangian fields? §

“Classical field theory” as a course was COOKED up to teach QFT because any quantum theory is defined by a classical theory and then we quantize it

quantization: ClassicalFT → QFT

this is VERY UNPHYSICAL in a way, proper physics says we must have a THEORY OF EVERYTHING and then we may take some limits c → ∞ , ħ, G →0, etc to get “approximate” theories, BUT WE CAN’T HAVE SUCH THINGS, so we just just do mathematically inspired stuff

Doing the proper theory of Lagrangian fields §

so the actual content must be this

• given you know vector bundles on smooth manifolds, we say field configuration bundles = some vector bundle on spacetime
• this is what i call spacetime fields , now write a Lagrangian and study it, the spacetime be a semiriemannian manifold (not flat), then div of stress energy tensor= 0 doesn’t imply global conservation laws, we must have Killing vector fields (I’VE NOT STUDIED THIS WELL YET FORGIVE ME IF IT’S KINDA OFF)
• you might have use the representation Spin(n) → SO(n), which allows you to define spin bundle where Spin(n) bundles

spin geometry:= study of dirac operators on such bundles

Doing the proper theory of gauge fields §

gauge fields DO NOT happen on spacetime! (that is, the field’s domain isn’t the spacetime manifold)

• http://nicf.net/articles/classical-em/
• Gregory L. Naber - Topology, Geometry and Gauge fields - two volumes
• Mikio Nakahara - Geometry, topology, and physics

quantization of field theories §

• https://www.youtube.com/watch?v=fjJsX4ektBA&list=PLbMVogVj5nJRYLTwyuusiiFchFU-WvElW
• https://www.youtube.com/@tobiasjosborne/playlists
• https://www.youtube.com/watch?v=ACZC_XEyg9U
• https://www.youtube.com/playlist?list=PLDlWMHnDwyljrnVxoGoBkHclt3VEkP0Kf
• https://www.youtube.com/playlist?list=PLbMVogVj5nJQ3slQodXQ5cSEtcp4HbNFc
• https://www.youtube.com/watch?v=29v0B2Fol3k&list=PL04QVxpjcnjiByGS5xGhiqC_G0rJVDDem&index=6&t=994

Mechanics of fluids §

The physics of fluid mechanics: Averaging a differential equation system with high number of dynamical variables

Hamiltonian system of a system of particles

$\downarrow$ Statistical mechanics of the system of particles: Boltzmann equation with the probability distribution $f(\mathbf{x},t)$ $\downarrow$ Take the average of quantities at the point $(\mathbf{x},t)$ and get equations for velocity and pressure $\mathbf{v}(\mathbf{x},t),p(\mathbf{x},t)$:

• Continuity of mass
• Continuity of momentum
• Continuity of energy
• Entropy inequality
• …is that it?

My question is can we do this for any Hamiltonian system? Or even any ODE with high enough ($\sim 10^{23}$) dimension? Probably not. But let’s not worry about that!

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There are various physical and computational aspects to it

• Interfacial Phenomena | Mathematics | MIT OpenCourseWare
• Lectures on Finite Element Methods for Fluid Dynamics, a full semester course on CFD using FEM. - YouTube

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Thus there is an analysis side to it.

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But there is a geometric, and topological side to it too,

• The Euler-Arnold equation | What’s new (wordpress.com)
• Vladimir I. Arnold, Boris A. Khesin - Topological Methods in Hydrodynamics
• https://www2.math.upenn.edu/~ghrist/preprints/fluidshandbook.pdf
• Introduction to Topological Fluid Dynamics - Lecture 1 (of 7) (youtube.com) and more: Short Course Video Lectures – Welcome to Renzo Ricca’s website
• Geometric Fluid Dynamics, Fall 2021 (utoronto.ca)
• fea-khesin-alt.qxp (toronto.edu)
• Topological fluid dynamics - Wikipedia - references

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Turbulence is still an unsolved problem.

Mechanics with a lattice somewhere §

Where does the Ising model sit in all these? Does it even need a “spacetime”?