development and applications in the nineteenth. Letwhere . Then is extremized if it satisfies the Euler–Lagrange equationsBut if depends on higher-order derivatives of , that is, if then must satisfy the Euler–Poisson equation,Poisson's Traité de mécanique (2 vols. 8vo, 1811 and 1833) was written in the style of Laplace and Lagrange and was long a standard work. Let be the position, be the kinetic energy, the potential energy, both independent of time . Lagrange's equation of motion readsHere, the dot notation for the time derivative is used, . Poisson set . He argued that if is independent of , he could writegiving He introduced an explicit formula for momenta,Thus, from the equation of motion, he gotPoisson's text influenced the work of William Rowan Hamilton and Carl Gustav Jacob Jacobi. A translation of Poisson's Treatise on Mechanics was published in London in 1842. Let and be functions of the canonical variables of motion and . Then their Poisson bracket is given byEvidently, the operation anti-commutes. More precisely, . By Hamilton's equations of motion, the total time derivative of iswhere is the Hamiltonian. In terms of Poisson brackets, then, Hamilton's equations can be written as and . Suppose is a constant of motion, then it must satisfyMoreover, Poisson's theorem states the Poisson bracket of any two constants of motion is also a constant of motion. In September 1925, Paul Dirac received proofs of a seminal paper by Werner Heisenberg on the new branch of physics known as quantum mechanics. Soon he realized that the key idea in Heisenberg's paper was the anti-commutativity of dynamical variables and remembered that the analogous mathematical construction in classical mechanics was Poisson brackets. He found the treatment he needed in E. T. Whittaker's Analytical Dynamics of Particles and Rigid Bodies. Continuum mechanics and fluid flow In 1821, using an analogy with elastic bodies, Claude-Louis Navier arrived at the basic equations of motion for viscous fluids, now identified as the Navier–Stokes equations. In 1829 Poisson independently obtained the same result. George Gabriel Stokes re-derived them in 1845 using continuum mechanics. Poisson, Augustin-Louis Cauchy, and Sophie Germain were the main contributors to the theory of elasticity in the nineteenth century. The calculus of variations was frequently used to solve problems. Wave propagation Poisson also published a memoir on the theory of waves (Mém. ft. l'acad., 1825). Thermodynamics In his work on heat conduction, Joseph Fourier maintained that the arbitrary