Hamiltonian mechanics
Introduced by Sir William Rowan Hamilton,[1] Hamiltonian mechanics replaces (generalized) velocities
Both theories provide interpretations of classical mechanics and describe the same physical phenomena.
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
A spherical pendulum consists of a mass m moving without friction on the surface of a sphere.
Spherical coordinates are used to describe the position of the mass in terms of (r, θ, φ), where r is fixed, r = ℓ.The Lagrangian for this system is[2]
are independent coordinates in phase space, not constrained to follow any equations of motion (in particular,
Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but important theoretical results can be derived from them, because coordinates and momenta are independent variables with nearly symmetric roles.
is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set.
In the Lagrangian framework, the conservation of momentum also follows immediately, however all the generalized velocities
[4] The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in classical mechanics, and suggest analogous formulations in quantum mechanics: the path integral formulation and the Schrödinger equation.
The relation holds true for nonrelativistic systems when all of the following conditions are satisfied[5][6]
Preliminary to this proof, it is important to address an ambiguity in the related mathematical notation.
To perform a change of variables inside of a partial derivative, the multivariable chain rule should be used.
Hence, to avoid ambiguity, the function arguments of any term inside of a partial derivative should be stated.
In Cartesian coordinates the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):
This Lagrangian, combined with Euler–Lagrange equation, produces the Lorentz force law
According to the Darboux's theorem, in a small neighbourhood around any point on M there exist suitable local coordinates
A Hamiltonian system may be understood as a fiber bundle E over time R, with the fiber Et being the position space at time t ∈ R. The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T∗Et, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian.
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.
This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called "the time"); in other words, an isotopy of symplectomorphisms, starting with the identity.
By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space.
The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra.
The transformed Hamiltonian depends only on the Gi, and hence the equations of motion have the simple form
for some function F.[9] There is an entire field focusing on small deviations from integrable systems governed by the KAM theorem.
In general, Hamiltonian systems are chaotic; concepts of measure, completeness, integrability and stability are poorly defined.
The continuous, real-valued Heisenberg group provides a simple example of a sub-Riemannian manifold.
Hamilton's equations above work well for classical mechanics, but not for quantum mechanics, since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time.
These Poisson brackets can then be extended to Moyal brackets comporting to an inequivalent Lie algebra, as proven by Hilbrand J. Groenewold, and thereby describe quantum mechanical diffusion in phase space (See Phase space formulation and Wigner–Weyl transform).
This more algebraic approach not only permits ultimately extending probability distributions in phase space to Wigner quasi-probability distributions, but, at the mere Poisson bracket classical setting, also provides more power in helping analyze the relevant conserved quantities in a system.
