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This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E. Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.

Differentiaalvergelijkingen [Hyperbolische ] --- Differential equations [Hyperbolic] --- Equations différentielles hyperboliques --- Symplectic manifolds --- Differential equations, Hyperbolic. --- Symplectic manifolds. --- Variétés symplectiques --- Equations différentielles hyperboliques --- Variétés symplectiques --- Manifolds, Symplectic --- Geometry, Differential --- Manifolds (Mathematics) --- Hyperbolic differential equations --- Differential equations, Partial --- Action (physics). --- Boundary value problem. --- Canonical form. --- Causal structure. --- Classical mechanics. --- Complex analysis. --- Configuration space. --- Conservative vector field. --- Conserved current. --- Conserved quantity. --- Continuum mechanics. --- Derivative. --- Diffeomorphism. --- Differentiable manifold. --- Differential geometry. --- Dimension. --- Dimensional analysis. --- Dirichlet's principle. --- Einstein field equations. --- Electromagnetic field. --- Equation. --- Equations of motion. --- Equivalence class. --- Error term. --- Euclidean space. --- Euler system. --- Euler's equations (rigid body dynamics). --- Euler–Lagrange equation. --- Existence theorem. --- Existential quantification. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Foliation. --- Fritz John. --- General relativity. --- Hamiltonian mechanics. --- Hamilton–Jacobi equation. --- Harmonic map. --- Hessian matrix. --- Holomorphic function. --- Hyperbolic partial differential equation. --- Hyperplane. --- Hypersurface. --- Identity element. --- Iteration. --- Iterative method. --- Lagrangian (field theory). --- Lagrangian. --- Legendre transformation. --- Lie algebra. --- Linear approximation. --- Linear differential equation. --- Linear map. --- Linear span. --- Linearity. --- Linearization. --- Maximum principle. --- Maxwell's equations. --- Nonlinear system. --- Open set. --- Ordinary differential equation. --- Orthogonal complement. --- Parameter. --- Partial differential equation. --- Phase space. --- Pointwise. --- Poisson bracket. --- Polynomial. --- Principal part. --- Principle of least action. --- Probability. --- Pullback bundle. --- Pullback. --- Quadratic form. --- Quantity. --- Requirement. --- Riemannian manifold. --- Second derivative. --- Simultaneous equations. --- Special case. --- State function. --- Stokes' theorem. --- Subset. --- Surjective function. --- Symplectic geometry. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Theoretical physics. --- Theory. --- Underdetermined system. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Zero of a function. --- Zero set.