Mathematical Physics

Newton formulated classical mechanics in terms of forces and accelerations. Lagrangian mechanics reformulates the same physics in terms of energy: the Lagrangian L equals kinetic energy T minus potential energy V. Nature chooses the path that minimizes the action — the time integral of the Lagrangian. This principle of least action yields the Euler-Lagrange equations that reproduce Newton's laws. Hamiltonian mechanics goes further, describing systems in terms of position and momentum in phase space with Hamilton's equations. Noether's theorem is a crown jewel: every continuous symmetry of the action implies a conserved quantity. Time-translation symmetry gives conservation of energy. Spatial-translation symmetry gives conservation of momentum. Rotational symmetry gives conservation of angular momentum.

Quantum mechanics describes nature at the smallest scales. Its central equation is the Schrödinger equation: i hbar times the time derivative of the wavefunction psi equals the Hamiltonian operator H applied to psi. The wavefunction encodes the complete state of a quantum system, and the square of its magnitude gives the probability density of finding a particle at a given position. The Heisenberg uncertainty principle arises mathematically from the non-commutativity of operators: the commutator of position x and momentum p equals i hbar. This means you cannot simultaneously know both position and momentum with arbitrary precision — measuring one disturbs the other. The Copenhagen interpretation says the wavefunction "collapses" upon measurement, but other interpretations (Many-Worlds, de Broglie-Bohm pilot wave theory) interpret the same mathematics differently.

The Standard Model of particle physics describes three of the four fundamental forces (electromagnetism, weak nuclear, strong nuclear) as gauge theories. A gauge theory has a symmetry group — a set of transformations that leave the physics unchanged. Electromagnetism has U(1) symmetry (a circle). The weak force has SU(2) (3x3 special unitary matrices). The strong force has SU(3) (8 gluons). The full Standard Model gauge group is SU(3) × SU(2) × U(1). Each symmetry generates a force-carrying boson: U(1) gives the photon, SU(2) gives W and Z bosons, SU(3) gives gluons.

Noether's theorem connects symmetries to conservation laws: time-translation symmetry → conservation of energy. Spatial translation → conservation of momentum. Rotation → conservation of angular momentum. Gauge symmetry → conservation of electric charge. This deep principle unifies the structure of physical law with the structure of abstract algebra.

Einstein's general theory of relativity (1915) is pure geometry: gravity is not a force but the curvature of spacetime caused by mass and energy. The Einstein field equations relate the Einstein tensor (describing spacetime curvature) to the stress-energy tensor (describing the distribution of matter and energy). John Wheeler summarized it: "Spacetime tells matter how to move; matter tells spacetime how to curve." The equations predicted black holes (regions where curvature is so extreme that not even light can escape), gravitational waves (ripples in spacetime detected for the first time in 2015 by LIGO), and the expansion of the universe. The Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is the cosmological solution describing a homogeneous, isotropic expanding universe.

See how a quantum wavefunction evolves over time. Adjust the potential to see how different energy landscapes affect the probability distribution.