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Particles initially on a circle of radius $R$ with common tangential speed $v$ move freely on tangent lines. They remain on a circle of radiusrotated through $\theta (t)=\arctan (vt/R)$, while total angular momentum remains $mRv$.

Collision mechanics uses impulse together with conservation of momentum and, for elastic collisions, kinetic energy.

For force minus the derivative of a potential, total kinetic plus potential energy is conserved.

A Lie point symmetry is a continuous transformation of independent and dependent variables that maps solutions of a differential equation to solutions.

For smooth functions $u,v$ on a region $V$,

Green’s second identity is the divergence theorem applied to phi grad psi minus psi grad phi.

The wave equation is a second-order linear partial differential equation describing waves with finite propagation speed. In one spatial dimension its homogeneous, constant-coefficient form isHere $x$ is position, $t$ is time, $u(x,t)$ is the disturbance, and subscripts denote partial derivatives. For a wave equation on a string, $u$ is transverse displacement; in linear acoustics, it can be a pressure perturbation. In several spatial dimensions, $u_{xx}$ becomes the Laplacian $\Delta u$. The same mathematical structure connects vibrating strings and acoustic waves, although their physical assumptions and wave speeds differ. ^[ref]

For the whole-line Cauchy problem, prescribe both displacement $u(x,0)=f(x)$ and velocity $u_t(x,0)=g(x)$. A classical solution has continuous partial derivatives through order two; $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ suffice for D'Alembert's formula to produce one. A finite string additionally needs boundary conditions. These statements concern the unforced model with constant $c$; damping, spatially varying coefficients and nonlinear terms change the solution formula.

A travelling wave $F(x-ct)$ carries its entire profile to the right at wave speed $c$; $G(x+ct)$ travels left. Their sum is again a solution by the superposition principle. A standing wave instead has a fixed mode shape whose size and sign oscillate: in $\sin (kx)\cos (ckt)$, each node of a standing wave stays stationary. The distinction is especially clear in the pulse and string diagrams below. For this ideal model, the dispersion relation is ${\omega }^2=c^2{k}^2$: each propagation branch is nondispersive, so a localized profile translates without spreading. This makes the equation a useful reference against which to understand wave dispersion and damping.

The appealing combination is local interaction with global predictability: neighbouring displacements determine acceleration through $u_{xx}$, yet D'Alembert's formula explicitly identifies exactly which initial locations can influence a later observation. Finite propagation speed turns that observation into a domain of dependence, rather than requiring knowledge of the entire initial string.

The Laplace operator is the divergence of the gradient, $\Delta =\nabla \cdot \nabla$.

A smooth bump function is a smooth function with compact support. The flat exponential construction gives bumps supported in prescribed balls.

A harmonic function satisfies Laplace's equation $\Delta u=0$.

A polynomial ansatz determines an unknown polynomial solution by matching coefficients after applying the differential operator.

Separation of variables seeks a product of one-variable factors, reducing a PDE to ordinary differential equations.

A similarity solution combines independent variables into scale-invariant coordinates and reduces a PDE to an ODE.

A diffusion equation describes smoothing caused by a flux down spatial gradients.

A moving-boundary problem determines both a field and part of the domain boundary on which that field is posed. An additional kinematic or conservation law determines the boundary motion.

A nonlinear partial differential equation depends nonlinearly on the unknown function or one of its derivatives.

The modified Helmholtz equation isIt reduces to the Laplace equation when $m=0$.

An elliptic boundary value problem couples an elliptic differential equation in a domain to boundary conditions. Its weak formulation is often an equation for a bounded bilinear form on a Sobolev space.

A transport equation carries a quantity along the flow of a velocity field $b$. For a smooth solution of $u_t+b\cdot \nabla u=0$, the value of $u$ is constant along each characteristic curve. With constant speed $c$, the chain rule gives $\frac{d}{dt}u(x_0+ct,t)=u_t+c{u}_x=0$, hence $u(x,t)=g(x-ct)$. This is exact translation: it preserves the shape rather than smoothing it, as the periodic heat and transport comparison shows.

A solution of a partial differential equation is a function satisfying the equation in a specified sense and obeying its initial or boundary data.