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Concept
Hamiltonian mechanics
Summary
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.
Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.
Let be a mechanical system with the configuration space and the smooth Lagrangian Select a standard coordinate system on The quantities are called momenta. (Also generalized momenta, conjugate momenta, and canonical momenta). For a time instant the Legendre transformation of is defined as the map which is assumed to have a smooth inverse For a system with degrees of freedom, the Lagrangian mechanics defines the energy function
The Legendre transform of turns into a function known as the . The Hamiltonian satisfies
which implies that
where the velocities are found from the (-dimensional) equation which, by assumption, is uniquely solvable for The (-dimensional) pair is called phase space coordinates. (Also canonical coordinates).
In phase space coordinates the (-dimensional) Euler–Lagrange equation
becomes Hamilton's equations in dimensions
Indeed, since the Hamiltonian is the Legendre transform of the Lagrangian , one has
and, since
the Euler-Lagrange equations yield
Let be the set of smooth paths for which and The action functional is defined via
where and (see above). A path is a stationary point of (and hence is an equation of motion) if and only if the path in phase space coordinates obeys the Hamilton's equations.
A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one nonrelativistic particle of mass m. The value of the Hamiltonian is the total energy of the system, in this case the sum of kinetic and potential energy, traditionally denoted T and V, respectively.
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