Derived the least action principle after some additions and correction to the previous notes on Euler-Lagrange. Introduced the
Hamiltonian formalism, phase space and Legendre transform. Then, I went back on the mathematical tools section to work on cotangent
bundles, one form, metrics and symplectic geometry to have all the tools. Corrected ex. 5.
From now on, because of summer vacations and also to minimize stress I will publish only one longer update per months. The project
starts to develop, new collaboration may hopefully start and the website may evolve. Feel free to contact me !
Started the "Exercises and examples" section with 8 exercises on smooth submanifolds: 4 that are purely math to
get familiar with differential geometry, and 4 more oriented on physics. I have finished correcting 3 of them, and will release
them in this section whenever they are all done. Ultimately, these exercises are already a solid ground to cover completely point
mechanics (newtonian and analytical).
In parallel, I have started to read and correct mistakes in the analytical mechanics notes. Next, I plan to finish the exercises and
make progress on Hamilton equations and variational principle.
Released the first subsection of "mathematical tools" dedicated to differential geometry. The goal was to properly define
concepts such as charts, vector fields, submanifolds and flows. I tried to make this section as mathematical as possible, so
it is understandable by physicist and mathematician alike. Also changed some small stylistic details on the website.
Next: Now that charts and vector fields are cleanly introduced, I will go back to analytical mechanics and introduce Hamilton equation.
This will eventually require getting back to the mathematical section to define abstract manifold (at least for the needs of physics).
Reworked the pdf style, table of content display and added a new summary page template.
Wrote a 3 pages summary at the end of the pdf "newtonian mechanics". Introduced the generalized energy
(also known as hamiltonian), and started to derive least action principle.
Next: Finish the derivation of Euler-Lagrange equation from least action principle and interpret it, introduce Legendre transforms and Hamilton equation.
Then start on the conclusion, appendices and exercices of the analytical mechanics subsection.
Completed the derivation of Euler-Lagrange equation from D'Alembert principle using the virtual displacement theory.
Finished to work on the github automatization to generate the chapter from the .tex file.
Started an additionnal "examples and exercices" pdf, planned for every section of the book.
Next: Before moving on to Hamilton equation and introduce Legendre transforms I wanted to already summarize everything I
learnt and derived. But then I realized I needed stronger understanding of differential geometry. I thus started reading
"The geometry of physics: an introduction" of T. Frankel, and will summarize this in the mathematical physics note.
Finished the harmonic oscillator section and added it to the corresponding pdf in classical mechanics on the website. Derived
the wave equation from the chain of springs in the continuum limit (will be available in the continuum mechanics pdf soon). Started
to introduce Euler Lagrange equation. Changed the note global structure: appendices are now only for long derivation in the notes. Started
new "mathematical tools" and "examples" sections.
After a writing down the principle of least action and Hamilton equation, I'm planning to work on phase space, Liouville theorem, chaos theory and
Fourier transforms in mathematical tools. Then write illustrative examples for those concepts.
Defined what kind of physical quantities can be called conserved and introduced the concepts of state space. Derived
the transport equation as a consequence, and studied what it means using these definitions for the momentum, the angular
momentum and the energy to be conserved. Notes are available in the classical mechanics section.
After a short section on harmonic oscillator and periodic motions, I plan to move to analytical mechanics, Euler-Lagrange and
Hamilton equations. In parallel, I plan to start an appendix dedicated on dynamical systems, phase space and Liouville theorem.
Properly formalized the definition of a frame in a physics from a mathematical point of view, using the language of differential geometry and manifolds. Introduced Newton law and work as a way to define kinetic energy. Started to question what "conservation" really means and derived a mathematical definition based on chain rule for function defined over phase space. Distinguished the advection equation from the conservation equation depending on whether the considered physical quantity is intensive or extensive.
Restructured the notes on electromagnetism. The electrostatics paragraph is now created (not online yet). I have started a new section devoted to the derivation and interpretation of the energy of the electromagnetic field and the wave equation. The plan is to temporarily pause this development and first introduce the Euler–Lagrange and Hamilton equations in the Analytical Mechanics section. I will also derive the general properties of wave equations there, before returning to electromagnetism with all the necessary ingredients in place.