In classical statistical mechanics, the ensemble is a probability distribution over phase points ( as opposed to a single phase point in ordinary mechanics ), usually represented as a distribution in a phase space with canonical coordinates.
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In ordinary differential geometry, there is no canonical coordinate system on the manifold; thus, typically, all discussion must be with regard to an atlas, that is, with regard to functions on the manifold.
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Where x and t are canonical coordinates and p and p _ t are their conjugate momenta respectively and represent our extended phase space ( we will show that we can recover the usual Newton's equations from this expression ).
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First, to accommodate the square root, we will wish to require that the scalar " square " mass " m " 2 commute with the canonical coordinates " x i ", which we write as:
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Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential ( exact form ), may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations.
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Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential ( exact form ), may be called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations.
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As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of a manifold.
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The extension to higher dimensional systems is analogous, and the extension to higher derivatives simply mean that the phase space is of even higher dimension than the configuration space, which exacerbates the instability ( since the Hamiltonian is linear in even more canonical coordinates ).
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:I don't know the name of the equation, but in classical mechanics, the left hand side is a Poisson bracket for u and v if x and y were canonical coordinates .-- talk ) 22 : 28, 28 February 2015 ( UTC)
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The Heisenberg group is a central extension of such a commutative Lie group / algebra : the symplectic form defines the commutation, analogously to the canonical commutation relations ( CCR ), and a Darboux basis corresponds to canonical coordinates in physics terms, to momentum operators and position operators.
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