Book: Methoden der mathematischen Physik

Overview
Published in 1924 and shaped by Hilbert’s Göttingen lectures with Richard Courant as coauthor, Methoden der mathematischen Physik consolidated a new synthesis of analysis tailored to the needs of classical physics. It drew connections among integral equations, the calculus of variations, eigenvalue problems, and partial differential equations, offering a unified language for mechanics, acoustics, elasticity, optics, and heat conduction. The volume is both rigorous and pragmatic: it insists on precise theorems while keeping physical models in view, repeatedly returning to vibrating strings and membranes, steady-state heat flow, and electrostatics as canonical testbeds for methods.

Structure and Themes
The presentation moves from concrete boundary value problems to abstract tools and back again. Boundary value problems for Laplace, Poisson, wave, and heat equations are formulated in energetic form, and their analytical treatment is tied to orthogonal expansions and separation of variables. Integral equation methods, especially the Fredholm and Hilbert–Schmidt theories, are developed to recast PDE boundary problems as compact operator equations on function spaces. Variational principles organize much of the material: the Dirichlet energy underlies potential theory, while the action integral frames mechanics. The approach culminates in spectral decomposition, eigenfunctions of self-adjoint operators provide bases for expanding solutions and quantifying stability and resonance.

Core Methods
A central strand is the rehabilitation and systematization of the Dirichlet principle within a functional-analytic framework. Minimization of energy under boundary constraints is justified via completeness in L2-type settings, turning existence questions into compactness and convergence arguments. The book advances the Rayleigh–Ritz and Courant variational methods for approximating eigenvalues and eigenfunctions, establishing min–max characterizations that tie spectral quantities to extremal energies and immediately yield monotonicity and error bounds.

Integral equation theory is treated through kernels of Hilbert–Schmidt type, with orthogonal expansions of kernels, resolvent construction, and the Fredholm alternative for existence and uniqueness. This kernel–operator viewpoint seamlessly links to Fourier and Sturm–Liouville theory. Orthogonal expansions, Fourier series on intervals, eigenfunction systems for strings and membranes, are presented not merely as formal tools but as convergent representations in the energy norm, legitimizing separation of variables beyond heuristic physics.

Green’s functions appear as a bridge between PDEs and integral equations, encoding boundary data and singular sources. Their symmetry and positivity properties are connected to self-adjointness and the maximum principle, while singular behavior near the source clarifies regularity expectations. Throughout, physical conservation laws (energy and momentum) motivate variational formulations and boundary conditions.

Pedagogical Style
The exposition alternates between canonical examples and general theorems. Prototypes, rectangular and circular domains, one-dimensional strings, axisymmetric membranes, anchor the theory, and approximation schemes are emphasized to show how exact formulae give way to convergent computations. The insistence on orthogonality, completeness, and projection clarifies why expansions work and how truncation yields practical accuracy.

Impact and Legacy
The volume codified the analytical toolkit that would power twentieth-century mathematical physics and analysis. By fusing spectral theory, integral equations, and variational methods, it helped crystallize the concept of Hilbert space and the operator-theoretic treatment of physical problems. Generations of mathematicians and physicists learned from its blend of rigorous proofs and physically grounded examples, and its variational and spectral principles became standard in PDE, elasticity, quantum mechanics, and numerical approximation. The 1924 volume set the stage for the subsequent systematic treatment of partial differential equations and remains a touchstone for the classical methods of analysis in physics.