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The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the tautological one-form. Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the phase space.
The action principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system. If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry.
Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics . Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom .
The Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field.
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
The extended Hamiltonian gives the most general possible time evolution for any gauge-dependent quantities, and may actually generalize the equations of motion from those of the Lagrangian formalism. For the purposes of introducing the Dirac bracket, of more immediate interest are the second class constraints. Second class constraints are ...
The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle.
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.