Mathematical Physics Equations

Newtonian mechanics is a powerful, yet cumbersome tool when a system has constraints or is described using curvilinear coordinates. Imagine a pendulum: in Cartesian coordinates, we must express the tension force as an unknown and then eliminate it. The Lagrangian approach works directly with the ...

Lagrangian mechanics is a reformulation of classical mechanics in the language of functional calculus: instead of forces, we deal with energies, and the equations of motion derive from a single principle—the principle of least action.

Let’s consider a system of N particles. In 3D space, we would need 3N Cartesian coordinates. But if there are constraints (strings, rigid rods), some coordinates depend on the others. Generalized coordinates q = (q₁, ..., qₙ) are any independent set of parameters that uniquely determine the confi...

Examples: a pendulum with length l has one degree of freedom—the angle θ. A double pendulum has two angles (θ₁, θ₂). A water molecule—9 coordinates for three atoms, but 6 degrees of freedom after 3 constraints fixing the bond lengths. Generalized coordinates are the “natural” variables of the pro...

Formulas

Lagrangian mechanics operates in configuration space {q} and velocities {q̇}. Hamiltonian mechanics transitions to phase space {q, p}, where p are the generalized momenta. At first glance, this appears to be merely a change of variables. In reality, it opens access to a deep symmetry of the equat...

The key advantage: Hamilton’s equations are a system of first-order equations (not second order, as in Lagrange). This simplifies numerical integration and stability analysis.

The meaning of the sum Σ pᵢ q̇ᵢ: for a point mass p = mv, q̇ = v, and pq̇ = mv² = 2T. Then H = 2T − L = 2T − (T − V) = T + V — the total mechanical energy! This is valid for conservative systems with natural kinetic energies T = (1/2)Σ mᵢ q̇ᵢ².

Compare with Lagrange’s equations (n second-order equations). Hamilton’s equations are symmetric: q and p enter them on an absolutely equal footing.

Hamilton’s equations are elegant, but solving them in arbitrary coordinates is difficult. The goal of canonical transformations is to find coordinates (Q, P) in which the Hamiltonian takes the simplest form. In the best case — $\tilde{H}(Q, P)$ depends neither on $Q$ nor on $P$, and then the equa...

An analogy from school mathematics: when making a change of variables in integrals, we choose a convenient parametrization. In mechanics, the “convenient parametrization” is a canonical transformation.

The transformation $(q, p) \to (Q, P)$ is called canonical if it preserves the form of Hamilton’s equations, that is, there exists $\tilde{H}(Q, P, t)$ such that $\dot{Q}_i = \partial \tilde{H}/\partial P_i$ and $\dot{P}_i = -\partial \tilde{H}/\partial Q_i$.

An equivalent condition: the transformation preserves the Poisson brackets: $\{Q_i, P_j\}_{q,p} = \delta_{ij}$.

In 1865, James Clerk Maxwell published a system of equations that unified electricity, magnetism, and light into a single theory. Before Maxwell, these seemed like three separate phenomena of nature. After him, it became clear: light is an electromagnetic wave, and the predictions of the theory a...

Maxwell's equations are written with the tools of vector analysis: divergence ∇· and curl ∇×. They allow the laws to be stated locally—in every point of space.

Read as: "the divergence of the electric field E is proportional to the density of electric charge ρ." Physically: electric field lines "flow out" of charges. Charge is the source of the E-field.

Symbols: E — electric field vector (V/m); ρ — volumetric charge density (C/m³); ε₀ = 8.85 × 10⁻¹² F/m — permittivity of vacuum.

Definitions

Proper time

— $d\tau^2 = ds^2/c^2 = dt^2(1 - v^2/c^2)$. This is the time measured by a clock moving with the particle. $\tau$ is invariant. $\gamma = 1/\sqrt{1 - v^2/c^2}$ — Lorentz factor.

Time Dilation

— $\Delta t = \gamma \Delta \tau > \Delta \tau$. Moving clocks run slower. Muons from cosmic rays are created at an altitude of 10 km at a speed of 0.998$c$. Their proper lifetime is 2.2 $\mu$s. Classically: they would travel only $0.998 \times 3 \t...

Length Contraction

— $L = L_0/\gamma$. In the muon’s frame: the distance to Earth contracts from 10 km to $10/15 \approx 670$ m — exactly the distance a muon can travel.

Relativity of Simultaneity

— Events simultaneous in one system ($dt=0$ with $dx \neq 0$) are not simultaneous in another.

4-Momentum

— $p^\mu = m u^\mu = \left(\frac{E}{c}, \vec{p}\right)$. Invariant: $p^\mu p_\mu = m^2 c^2 \rightarrow E^2 = (pc)^2 + (mc^2)^2$.

Formulas

In 1905, Einstein proposed a radically new view of space and time, based on two postulates:

1. Principle of Relativity: The laws of physics are the same in all inertial reference frames 2. Constancy of the Speed of Light: The speed $c = 3 \times 10^8$ m/s is the same in all inertial frames, regardless of the motion of the source or observer

The second postulate seems paradoxical: if I run at speed $v$ towards the light, shouldn’t the light move relative to me at $c + v$? No — and all experiments confirm this. The consequence: our intuitive ideas about time and space are mistaken.

Instead of separate space and time, Minkowski proposed a unified 4-dimensional spacetime. An event is a point $(t, x, y, z)$. The interval between two events:

Definitions

Hawking radiation

— a quantum effect at the event horizon of a black hole—is mathematically analogous to the thermal radiation of an accelerated observer (Unruh effect). The temperature of the radiation: T_H = ℏc³/(8πGMk_B). A black hole of the mass of the Sun radiat...

A static charge creates only the Coulomb field—a static field that decreases as 1/r². But an accelerated charge creates an additional field—the radiation field—which decreases only as 1/r. This is critically important: the energy flux is proportional to E²/r², and if E ~ 1/r, then flux ~ 1/r², an...

The electromagnetic fields from a moving charge arrive with a delay: the potential at point r at time t is determined by the position of the charge at the retarded moment tret, from which the signal propagated at speed c:

Retarded potentials: φ(r, t) = q/(4πε₀) · 1/(κR)|_ret, A = φ v/c², where R = r − r(tret) is the vector from the retarded position, κ = 1 − R̂·β, β = v/c—a parameter that takes into account the "compression" of the field in the direction of motion.

For β → 1 (relativistic motion): κ → 0 in the direction of motion—the fields are strongly "bunched" forward. This is used in synchrotron light sources.

Quantum mechanics arose from a series of experimental discoveries that appeared mutually contradictory: quantization of energy (Planck), wave-particle duality (de Broglie), Heisenberg's uncertainty principle. In the 1930s, Dirac and von Neumann assembled all this into a unified axiomatic formalis...

The axiomatic approach guarantees: we know exactly what we are assuming about nature, and we can systematically derive predictions. Without an axiomatics, quantum theory would be a set of recipes without explanations.

The state of a quantum system at any given moment is completely described by a normalized vector |ψ⟩ in a complex Hilbert space H, ‖|ψ⟩‖ = 1.

What this means: classically, a particle has definite position and momentum. In quantum mechanics, the particle is in "superposition"—not in any one definite state until a measurement is made. The vector |ψ⟩ contains all possible information about the system. Analogy: instead of a specific point ...

Formulas

Most quantum problems are solved numerically or approximately. But two special cases have exact analytic solutions: the quantum harmonic oscillator and the hydrogen atom. They serve as "building blocks" for all of quantum physics—from molecular vibrations to atomic spectra, from photons in a fiel...

Potential: $V(x) = m\omega^2x^2/2$. Hamiltonian: $\hat{H} = \hat{p}^2/(2m) + m\omega^2\hat{x}^2/2$.

$\hat{a} = \sqrt{m\omega/2\hbar}\ (\hat{x} + i\hat{p}/m\omega)$ — annihilation operator $\hat{a}^\dagger = \sqrt{m\omega/2\hbar}\ (\hat{x} - i\hat{p}/m\omega)$ — creation operator

Their commutator: $[\hat{a}, \hat{a}^\dagger] = 1$. Hamiltonian through these: $\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + 1/2) = \hbar\omega(\hat{N} + 1/2)$, where $\hat{N} = \hat{a}^\dagger\hat{a}$—operator of quantum number.

Exact solutions exist only for a few problems: the harmonic oscillator, the hydrogen atom, a particle in a rectangular well. Real-world problems—atoms in fields, molecules, interacting particles—do not have analytical solutions. Perturbation theory is a systematic method for finding approximate s...

Idea: if we know how to solve a problem with Hamiltonian $\hat{H}_0$, and the real Hamiltonian is $\hat{H} = \hat{H}_0 + \lambda\hat{H}'$ ($\lambda$ is small), then solutions are sought in the form of series expansions in $\lambda$.

Formulation: $\hat{H} = \hat{H}_0 + \lambda\hat{H}'$, where $\lambda\hat{H}'$ is a small perturbation. The eigenvectors $|n^0\rangle$ and eigenvalues $E_n^0$ of $\hat{H}_0$ are known. To find: $E_n(\lambda) = E_n^0 + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \ldots$

This is the matrix element of the perturbation evaluated on the unperturbed state. Physically: "the average correction to the energy due to the perturbation."

A gas in a room contains ~10²⁵ molecules. Writing the equations of motion for each one is practically impossible. But we do not need such detailed information: we want to know the temperature, pressure, and heat capacity. Statistical mechanics is the bridge from microscopic degrees of freedom to ...

Boltzmann’s key idea: when we have no information about the microstate, we consider all microstates with equal energy to be equally probable. This is the hypothesis of equal probabilities—the foundation of statistical mechanics.

Conditions: Fixed E (energy), V (volume), N (number of particles). The system is isolated.

Number of microstates: Ω(E, V, N) — the number of ways the system can have the given energy.

Water turns into ice at 0°C — a first-order phase transition. Iron loses magnetism at 770°C (the Curie point) — a second-order phase transition. At first glance, these are ordinary phenomena. In reality: phase transitions are one of the richest areas of theoretical physics, linking symmetry, stat...

Especially intriguing is the critical point: as one approaches it, fluctuations increase at all scales, and the system becomes scale-invariant. Critical exponents turn out to be universal — the same for physically completely different systems!

First Order: Jump of the first derivatives of the free energy (volume, magnetization). Latent heat $L = T\Delta S$. Coexistence of phases. Examples: melting of ice ($\Delta S = L/T \approx 22$ J/(mol·K)); boiling water; transition paramagnet → ferromagnet in an external field.

Second Order (Continuous): Second derivatives of $F$ (heat capacity) diverge or have a discontinuity. No latent heat, no phase coexistence. The order parameter grows continuously from zero at $T_c$. Examples: ferromagnet (magnetization $M \to 0$ as $T \to T_c$); superconductivity; $\lambda$-trans...

Formulas

In quantum mechanics, identical particles are indistinguishable—there is no “particle A” and “particle B”, there are simply “two electrons”. This fact has radical consequences for statistics. By the principle of indistinguishability: the wave function of a system of identical particles must be ei...

Fermions (half-integer spin: 1/2, 3/2, ...): Ψ(...i...j...) = −Ψ(...j...i...). Consequence (Pauli principle): two fermions cannot occupy the same quantum state.

Bosons (integer spin: 0, 1, 2, ...): Ψ(...i...j...) = +Ψ(...j...i...). Bosons “prefer” already occupied states.

Symbols: ε_k—energy of level k; μ—chemical potential (determined from the condition of fixed N); β = 1/(k_BT).

All linear differential equations (DEs) possess the principle of superposition: if φ₁ and φ₂ are solutions to the homogeneous equation, then α₁φ₁ + α₂φ₂ is also a solution. The Green's function utilizes this principle to the fullest: we find the solution for a point source, and then obtain the so...

This is a powerful idea: instead of solving the problem anew every time the source changes, one needs to find the Green's function once and then merely integrate.

For a linear differential operator L (for example, L = −∇² or L = ∂_t − κ∇²), the Green's function G(r, r') is defined as:

with appropriate boundary conditions. Here, δ³(r − r') is the three-dimensional delta function: an infinite "spike" at point r', normalized so that ∫δ³ d³r = 1.

Definitions

Residue

— the coefficient at $(z - z_0)^{-1}$ in the Laurent expansion:

Formulas

Many integrals over the real axis are elegantly computed by closing a contour in the complex plane. The idea: the closed integral along contour C of an analytic function equals 0 (Cauchy's theorem). But if there are singular points (poles) inside C, the integral equals the sum of residues multipl...

In physics, this method appears everywhere: computation of propagators in quantum field theory, dispersion relations in optics, Green's functions in statistical mechanics.

Cauchy Integral: For analytic $f$ in a simply connected domain $D$ and contour $C \subset D$:

\[ f(z) = \dots + \frac{a_{-2}}{(z-z_0)^2} + \frac{a_{-1}}{(z-z_0)} + a_0 + a_1(z-z_0) + \dots \]

Definitions

Advantage of spectral methods: For smooth functions, convergence is exponential

— error ~ e^{−αN} for N terms in the series. Finite-difference schemes provide only algebraic convergence ~ N^{−p}. Therefore, spectral methods are used in atmospheric modeling, computational hydrodynamics (codes ORCL, DEDALE).

Formulas

Differential equations in the "x-t" space contain derivatives—which is not trivial. In the "k-ω" space (space of frequencies and wavenumbers), the derivative ∂/∂t is replaced with multiplication by −iω, and ∂/∂x by ik. A differential equation turns into an algebraic one! This is the magic of inte...

Additional bonus: many physical phenomena are more transparent in frequency space. Oscillation period, bandwidth, dispersion—all these are properties of the frequency spectrum.

Key properties: derivative → multiplication (f'(x) ↔ ikF̂(k)); convolution f*g → product F̂·Ĝ; Parseval’s theorem: ∫|f(x)|²dx = (1/2π)∫|F̂(k)|²dk (norm conservation).

Numerical example: wave equation. ∂²u/∂t² = c² ∂²u/∂x². Fourier in x: ∂²û(k,t)/∂t² = −c²k²û. This is an ordinary ODE in t! Solution: û(k,t) = A(k)e^{ickt} + B(k)e^{−ickt}. Inverse Fourier gives the d'Alembert formula: u(x,t) = f(x+ct) + g(x−ct)—waves propagating in both directions.