number of problems. The first was constructed in 1788 by Joseph Louis Lagrange, an Italian-French mathematician. In Lagrangian mechanics the solution uses the path of least action and follows the calculus of variations. William Rowan Hamilton re-formulated Lagrangian mechanics in 1833. The advantage of Hamiltonian mechanics was that its framework allowed a more in-depth look at the underlying principles. Most of the framework of Hamiltonian mechanics can be seen in quantum mechanics however the exact meanings of the terms differ due to quantum effects. Although classical mechanics is largely compatible with other "classical physics" theories such as classical electrodynamics and thermodynamics, some difficulties were discovered in the late 19th century that could only be resolved by more modern physics. When combined with classical thermodynamics, classical mechanics leads to the Gibbs paradox in which entropy is not a well-defined quantity. As experiments reached the atomic level, classical mechanics failed to explain, even approximately, such basic things as the energy levels and sizes of atoms. The effort at resolving these problems led to the development of quantum mechanics. Similarly, the different behaviour of classical electromagnetism and classical mechanics under velocity transformations led to the theory of relativity. Classical mechanics in the contemporary era By the end of the 20th century, classical mechanics in physics was no longer an independent theory. Along with classical electromagnetism, it has become imbedded in relativistic quantum mechanics or quantum field theory. It defines the non-relativistic, non-quantum mechanical limit for massive particles. Classical mechanics has also been a source of inspiration for mathematicians. The realization that the phase space in classical mechanics admits a natural description as a symplectic manifold (indeed a cotangent bundle in most cases of physical interest), and symplectic topology, which can be thought of as the study of global issues of Hamiltonian mechanics, has been a fertile area of mathematics research since the 1980s. See also Mechanics Timeline of classical mechanics Notes References Classical mechanics Classical mechanics Isaac Newton