Physics and simulation
Mechanics, gravity, heat, waves, fluids and electromagnetism written as differential equations — and the numerical methods that turn them into games, films, forecasts and engineering software.
10 topics
Physics states its laws as rates of change: ODEs for particles and rigid bodies, PDEs for fields. A simulator picks a discretization in space (grid, particles, mesh) and an integrator in time (Euler, Verlet, Runge–Kutta, implicit), and then fights stability, accuracy and cost. The same mathematics drives a game engine at 60 frames per second and a climate model on a supercomputer.
Topics
Classical mechanics
Newton's second law turns forces into a second-order ODE for the motion. Gravity, springs, friction and collisions are all modelled this way; energy and momentum are the conserved quantities that check a simulation.
Physics engines
Software that advances a world of bodies in small time steps: accumulate forces, integrate velocities and positions (semi-implicit Euler or Verlet), detect collisions, resolve contacts. A numerical ODE solver tuned for speed and stability rather than accuracy.
N-body gravitational simulation
Integrate the mutual gravity of bodies — planets, stars, dark-matter particles. Symplectic integrators (leapfrog) keep orbits stable for billions of steps; tree codes (Barnes–Hut) and FMM cut the cost from to or .
Heat equation and diffusion
: temperature (or concentration, or probability) flows from high to low and smooths out. Solved by finite differences or Fourier series; Gaussian blur of an image is exactly running this equation.
Wave equation
: disturbances travel at speed . Sound, vibrating strings, seismic waves and light obey it; games use discretized versions for water ripples and room acoustics.
Electromagnetism (Maxwell's equations)
Four equations in divergence and curl govern all of electricity, magnetism and light. Simulated (FDTD, FEM) to design antennas, chips, MRI coils and wireless links.
Fluid dynamics and CFD
The Navier–Stokes equations for a velocity field and pressure . Engineering CFD solves them on meshes with finite volumes; film and games use "stable fluids" (Stam, 1999): advect, add forces, and project to zero divergence by solving a Poisson equation.
Finite element method
Solve a PDE on a complicated shape by splitting it into small elements, writing the weak form (integration by parts) and approximating the solution by piecewise polynomials. The standard tool of structural, thermal and electromagnetic engineering.
Weather and climate modelling
Integrate the equations of the atmosphere and oceans on a global grid. Chaos limits deterministic forecasts to about two weeks, so services run ensembles; data assimilation fits the initial state to observations using adjoint (reverse-mode) gradients. Machine-learned forecasters now compete with physics models.
Population and epidemic models
Computational biology with ODEs: logistic growth, predator–prey cycles, and SIR epidemics whose basic reproduction number decides whether an outbreak grows. These models informed policy during COVID-19.
The mathematics this domain runs on
v⃗ Linear algebra (bridge) ★★★★★
- Vectors★★★★★→Physics engines
f′ Derivatives ★★★★★
≈ Numerical methods ★★★★★
∫ Integrals ★★★★★
∇ Multivariable calculus ★★★★★
∮ Vector calculus ★★★★★
- Vector fields★★★★★→Physics engines, Electromagnetism (Maxwell's equations), Fluid dynamics and CFD
- Divergence★★★★★→Electromagnetism (Maxwell's equations), Fluid dynamics and CFD
- Curl★★★★★→Electromagnetism (Maxwell's equations), Fluid dynamics and CFD
- Laplacian★★★★★→Heat equation and diffusion, Wave equation, Fluid dynamics and CFD
- Divergence theorem (Gauss)★★★★★→Electromagnetism (Maxwell's equations), Fluid dynamics and CFD
- Stokes' theorem★★★★★→Electromagnetism (Maxwell's equations)
- Scalar fields★★★★★→Weather and climate modelling
- Line integrals★★★★★→Classical mechanics
- +1
ẏ Differential equations ★★★★★
- Ordinary differential equations★★★★★→Classical mechanics, Physics engines, Population and epidemic models
- Second-order linear equations (oscillations)★★★★★→Classical mechanics, Physics engines
- Systems of ODEs★★★★★→N-body gravitational simulation, Population and epidemic models
- Euler's method★★★★★→Physics engines
- Partial differential equations★★★★★→Heat equation and diffusion, Wave equation, Fluid dynamics and CFD, Finite element method
- Separable equations★★★★★→Population and epidemic models
- Runge–Kutta methods★★★★★→N-body gravitational simulation, Weather and climate modelling
- Stiffness and implicit methods★★★★★→Physics engines
- +1
φ Dynamical systems and chaos ★★★★★
- Dynamical systems★★★★★→Weather and climate modelling
- Chaos and sensitivity to initial conditions★★★★★→N-body gravitational simulation, Weather and climate modelling
- Attractors★★★★★→Weather and climate modelling
- Phase space★★★★★→N-body gravitational simulation
- Bifurcations★★★★★→Population and epidemic models