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.