Analytical mechanics

Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics, constructed from the eighteenth century onwards as a formulation of the subject as founded by Isaac Newton.

Related Topics:
Classical mechanics - Eighteenth century - Isaac Newton

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It began with d'Alembert's principle. By analogy with Fermat's principle, which is the variational principle in geometric optics, Maupertuis' principle was discovered in classical mechanics.

Related Topics:
D'Alembert's principle - Fermat's principle - Variational principle - Geometric optics - Maupertuis' principle

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Using generalized coordinates, we obtain Lagrange's equations. Using the Legendre transformation, we obtain

Related Topics:
Coordinate - Lagrange's equations - Legendre transformation

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generalized momentum and the Hamiltonian.

Related Topics:
Generalized momentum - Hamiltonian

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Hamilton's canonical equations provides integral, while Lagrange's equation provides differential equations. Finally we may derive the Hamilton-Jacobi equation.

Related Topics:
Hamilton's canonical equations - Integral - Lagrange's equation - Differential equations - Hamilton-Jacobi equation

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The study of the solutions of the Hamilton-Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology. In this formulation, the solutions of the Hamilton-Jacobi equations are the integral curves of Hamiltonian vector fields.

Related Topics:
Symplectic manifold - Symplectic topology - Integral curve - Hamiltonian vector field

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