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Set 03 — Fundamental Theorems of Physics

Noether's Theorem Every differentiable symmetry of the action of a physical system corresponds to a continuous conservation law. Invariance under translation in time yields conservation of energy, invariance under spatial translation yields conservation of momentum, and invariance under rotation yields conservation of angular momentum.

The theorem unifies what might otherwise appear as separate rules, showing that the deep symmetries of nature are precisely what make its quantities conserved. See the Noether's theorem entry for a fuller treatment.

Heisenberg Uncertainty Principle The product of the standard deviations of a particle's position and momentum can never fall below half the reduced Planck constant: Δx·Δp ≥ ℏ/2. This is not a limitation of measurement but a structural feature of quantum states.

It follows from the non-commutativity of the position and momentum operators, and implies that a wave function cannot be simultaneously localized in both position and momentum space. Learn more at the uncertainty principle page.

Second Law of Thermodynamics The total entropy of an isolated system never decreases over time. In every real process the entropy of the universe increases, which gives macroscopic phenomena their statistical arrow of time and sets the direction of heat flow.

Gauss's Law The total electric flux through any closed surface equals the enclosed charge divided by the permittivity of free space, Φ_E = Q_enc/ε₀. It is one of Maxwell's equations and follows directly from the inverse-square form of the electrostatic force.

In differential form ∇·E = ρ/ε₀, it relates local charge density to the divergence of the electric field. Read more at Gauss's law article.

Schrödinger Equation The time-dependent equation iℏ ∂ψ/∂t = Ĥψ governs the evolution of a quantum state, while stationary states satisfy the eigenvalue equation Ĥψ = Eψ. Its probabilistic interpretation underpins all of non-relativistic quantum mechanics.

The Hamiltonian operator Ĥ encodes the total energy, and its spectrum determines the possible measurement outcomes. Explore further at Schrödinger equation overview.

Navier–Stokes Equations For a viscous incompressible fluid the momentum balance reads ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + f. Whether smooth solutions exist for all initial data in three dimensions remains one of the seven open millennium problems.

The nonlinear advection term v·∇v makes turbulence a fundamental unsolved challenge in classical physics. Details at Navier–Stokes equations page.

Central Limit Theorem The standardized sum of a large number of independent random variables converges in distribution to a normal, regardless of the original common distribution. This is why the bell curve appears across fields as varied as biology, finance, and measurement.

The Lindeberg and Lyapunov conditions provide precise criteria for the convergence. More at central limit theorem article.

Fermat's Last Theorem No three positive integers a, b, c satisfy aⁿ + bⁿ = cⁿ for any integer n > 2, a result proven by Andrew Wiles in 1995 using modular forms and elliptic curves.

Maxwell's Equations Four relations unify electricity and magnetism: ∇·E = ρ/ε₀, ∇·B = 0, ∇×E = −∂B/∂t, and ∇×B = μ₀J + μ₀ε₀∂E/∂t. Together they predict electromagnetic waves propagating at the speed of light.

The displacement current term μ₀ε₀∂E/∂t was Maxwell's crucial addition that closed the theory and revealed light as an electromagnetic phenomenon. Learn more at Maxwell's equations overview.

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