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Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their…

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Overview

Notation

Conditions for restricted canonical transformation

Liouville's theorem

Generating function approach

Infinitesimal canonical transformation

One parameter subgroup of Canonical transformations

Examples

Modern mathematical description

History

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Canonical transformation

Nodes58
Edges57
Triples50
Avg. degree1.97
Density0.034483
Components1

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Canonical transformation

Top relations

related to Examples · 9
Canonical transformation → However, Indeed, Jacobian, JI, Keeping, Note, Set, SO, The
related to history · 7
Canonical transformation → Charles Delaunay, Earth-Moon-Sun, French Academy, Mémoires, Sciences, The, This
related to Canonical transformation relations · 4
Canonical transformation → From, K-H, Since, This
related to Generalized use of generating functions · 4
Canonical transformation → Hence, However, In, Jacobian
see also · 4
Canonical transformation → Hamiltonian, Jacobi, Mathieu, SymplectomorphismHamilton
related to Conditions for restricted canonical transformation · 3
Canonical transformation → Hamilton's, Restricted, The
related to Modern mathematical description · 3
Canonical transformation → Further, In, The
related to Motion as canonical transformation · 3
Canonical transformation → Hamilton's, If, Motion
related to Bilinear invariance conditions · 2
Canonical transformation → Consider, These
related to Liouville's theorem · 2
Canonical transformation → Liouville's, The

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Important terminology

canonical displaystyle textstyle transformation frac partial coordinates mathbf transformations new form function generalized left dot right equations begin end eta

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Canonical transformationis achange of canonical coordinates0.90text
cotangent bundlesinstance ofModern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical pre…0.80text
exterior derivativesinstance ofModern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical pre…0.80text
symplectic manifoldsinstance ofModern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical pre…0.80text
q represent a list of N generalized coordinates that need not transform like a vector under rotationinstance ofNotationBoldface variables0.80text
similarly p represents the corresponding generalized momentuminstance ofNotationBoldface variables0.80text
e.g.instance ofNotationBoldface variables0.80text
qinstance ofNotationBoldface variables0.80text
Canonical transformationrelated to Bilinear invariance conditionsThese0.60section
Canonical transformationrelated to Bilinear invariance conditionsConsider0.60section
Canonical transformationrelated to Canonical transformation relationsFrom0.60section
Canonical transformationrelated to Canonical transformation relationsK-H0.60section

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