A foundational chapter introducing classical mechanics from a modern perspective: frames are built using differential geometry, dynamics is formulated as a flow on state space, and conservation laws emerge from symmetry principles.

• Frames and geometric structure of space (Newton → modern maths): Motion requires a precise mathematical structure: space is modeled as a manifold, with tangent spaces encoding velocities. A frame corresponds to choosing a basis of these tangent spaces at each point, together with time and an origin. Coordinate systems induce (holonomic) frames, while inertial frames are those where free motion is uniform and rectilinear.
• Newton’s laws, momentum and work-energy (17th–18th century): Newton’s laws define dynamics through the evolution of momentum, while the concept of work links forces to energy variations. Kinetic energy naturally emerges from the work-energy theorem, clarifying the distinction between instantaneous forces, accumulated work, and state functions such as energy.
• State space and dynamical systems (modern formulation): A mechanical system is described by its state (position and velocity), evolving as a trajectory in state space. Newton’s equations define a vector field whose integral curves are physical motions. Observables are functions on this space, and conservation corresponds to invariance along these trajectories.
• Meaning of conservation laws (Euler → Noether): A quantity is conserved if it remains constant along all solutions of the equations of motion. Momentum, energy, and angular momentum are not always conserved: their conservation depends on specific physical conditions. This formalism clarifies the difference between state functions (observables) and path-dependent quantities like work.
• Momentum conservation and translation symmetry (Newtonian mechanics): Momentum is conserved when the net external force vanishes. Physically, this reflects invariance under spatial translations: the laws of physics are identical everywhere, leading to conservation of momentum.
• Energy conservation and time symmetry (18th–19th century): Kinetic energy alone is not generally conserved, but when forces derive from a time-independent potential, the sum of kinetic and potential energy (mechanical energy) is conserved. This reflects invariance under time translations: physics does not depend on when an experiment is performed.
• Angular momentum and rotational symmetry (Euler, Lagrange): Angular momentum measures rotational motion and evolves according to torque. It is conserved when the total torque vanishes, typically for central forces. This conservation reflects rotational invariance and explains phenomena such as Kepler’s second law for planetary motion.
• Symmetries as the origin of conservation (Noether’s insight): Conservation laws are deeply tied to symmetries: translation symmetry yields momentum conservation, time symmetry yields energy conservation, and rotational symmetry yields angular momentum conservation. This perspective unifies classical mechanics and prepares the transition to Lagrangian and modern physics.

An overview of the history (experiments and models) that led to modern electromagnetism:

• Origin and history of the electric charge (~600 BCE - 1897): A brief historical overview of the electric charge: From early electrostatic observations (amber friction) to Coulomb's quantitative law and finally the identification of the electron through Thomson’s measurement.
• The Millikan experiment and the elementary charge (~1909): A focus on Millikan oil drop experiment: explanation of the setup as well as the theory behind that led to the first evidence of an elementary electric charge. All charges are multiple of this constant. Funnily enough, Millikan experiment has a long controversial history for data falsification.
• Continuum assumption applied to electromagnetism: charge conservation (~1780–1810): Electromagnetism adopts the same continuum assumption used in fluid and solid mechanics. A charge density field is defined by coarse-graining discrete charges over small volumes, such that integrating the density over a region yields the total charge. This introduces the concepts of charge density, current density, and the charge conservation equation.
• The first galvanometer: Orsted experiment and Biot-Savart law (~1819-1821): How Orsted observed magnetic loops acting on a compass when a current goes through a wire, and the formalization by Biot and Savart. This allowed to define the first measure instrument for electric currents, galvanometers.
• Heuristic derivation of Maxwell equations (~1831–1855): An in-depth intuition driven introduction of Maxwell equation. Maxwell-Gauss is a reformulation of Coulomb law with the continuum assumption. Maxwell-Flux is the translation of the conservation of the magnetic flux (Magnetic monopoles have never been observed, so magnetic field lines cannot start or end). Maxwell-Faraday encodes induction (how we nowadays produce electricity), discovered by Faraday, refined by Lenz and reformulated by Maxwell. Finally, Maxwell-Ampère reformulates the Biot-Savart law, with a refinement added by Maxwell linking the magnetic field to the fluctuations of the electric field.

A list of experiments and theories that progressively led physicists to quantum mechanics:

• Early atomic models (~1897–1909): First evidence that matter is composed of elementary constituents. Thomson’s discovery of the electron and Rutherford’s nuclear model established the existence of subatomic structure.
• UV catastrophe & Planck’s quantization (~1900): Crisis in classical electromagnetism caused by blackbody radiation. Planck introduced quantization of energy per radiation mode, introducing the constant that would later be known as Planck’s constant.
• Bohr–Sommerfeld model (~1913–1916): Semiclassical model for hydrogen-like atoms. Quantization of angular momentum explained spectral lines. Sommerfeld’s extension to elliptical orbits introduced early quantum numbers and partially accounted for fine structure.
• Photoelectric effect (1887–1905): Observed experimentally by Hertz and theoretically explained by Einstein. The light-quantization hypothesis accounted for the frequency threshold and kinetic energy dependence.
• Compton effect (~1923): X-ray scattering experiment demonstrating momentum transfer consistent with particle-like photons.
• Wave–particle duality & de Broglie wavelength (~1924–1927): Extension of quantization to matter. Every particle is associated with a wavelength λ = h/p. Confirmed experimentally by electron diffraction (Davisson–Germer experiment) and later single-particle interference experiments.

• Details on Planck quantization of the energy modes of Blackbody radiations. (~1900): A semi-classical derivation inspired by statistical physics where Planck introduced his "bins" quantization idea.