Canonical coordinates

Canonical coordinates are independent parameters in the Hamiltonian formalism of classical mechanics . They are usually designated as $q_{i}$ ${\ displaystyle q_ {i}}$ $q_ {i}$ and $p_{i}$ ${\ displaystyle p_ {i}}$ $p_ {i}$ .

The canonical coordinates satisfy the fundamental relations expressed in terms of Poisson brackets :

\{q_{i},q_{j}\}=0\qquad \{p_{i},p_{j}\}=0\qquad \{q_{i},p_{j}\}=\delta _{ij}

{\ displaystyle \ {q_ {i}, q_ {j} \} = 0 \ qquad \ {p_ {i}, p_ {j} \} = 0 \ qquad \ {q_ {i}, p_ {j} \} = \ delta _ {ij}}

\ {q_i, q_j \} = 0 \ qquad \ {p_i, p_j \} = 0 \ qquad \ {q_i, p_j \} = \ delta_ {ij}

Canonical coordinates can be obtained from the generalized coordinates of Lagrangian mechanics using Legendre transforms or from another set of canonical coordinates using canonical transformations . If the Hamiltonian is defined on the cotangent bundle, then the generalized coordinates are connected with the canonical coordinates using the Hamilton - Jacobi equations .

Although there may be many options for choosing the canonical coordinates of a physical system, parameters are usually selected that are convenient for describing the configuration of the system and that simplify the solution of the Hamilton equations.

Similar concepts are also used in quantum mechanics , see and canonical commutation relations .

Summary

Since Hamiltonian mechanics in terms of mathematical structure is symplectic geometry , canonical transformations are a special case of contact transformations .

Canonical coordinates are defined as a special set of coordinates on the cotangent bundle of a manifold . They are usually written as many $(q^{i},p_{j})$ ${\ displaystyle (q ^ {i}, p_ {j})}$ or $(x^{i},p_{j})$ ${\ displaystyle (x ^ {i}, p_ {j})}$ , where the letter x or q denotes the coordinates on the manifold, and the letter p denotes the conjugate moment , which is the covariant vector at the point q of the manifold.

The usual definition of canonical coordinates is a coordinate system on a cotangent bundle in which the written as

\sum _{i}p_{i}\,\mathrm {d} q^{i}

{\ displaystyle \ sum _ {i} p_ {i} \, \ mathrm {d} q ^ {i}}

up to the addition of the full differential. A coordinate change that preserves this view is a canonical transformation . This is a special case of , which, in essence, is a change of coordinates on a symplectic manifold .

Formal Research

If a real manifold Q is given , then the vector field X on Q (or, equivalently, the section of the tangent bundle TQ ) can be considered as a function acting on the , due to the duality of the tangent and cotangent spaces. That is a function

P_{X}:T^{*}Q\to \mathbb {R}

{\ displaystyle P_ {X}: T ^ {*} Q \ to \ mathbb {R}}

such that

P_{X}(q,p)=p(X_{q})

{\ displaystyle P_ {X} (q, p) = p (X_ {q})}

preserves all cotangent vectors p in $T_{q}^{*}Q$ ${\ displaystyle T_ {q} ^ {*} Q}$ . Here $X_{q}$ ${\ displaystyle X_ {q}}$ is a vector in $T_{q}Q$ ${\ displaystyle T_ {q} Q}$ , the tangent space of Q at q . Function $P_{X}$ ${\ displaystyle P_ {X}}$ called the moment function corresponding to X.

In local coordinates, the vector field X at q can be written as

X_{q}=\sum _{i}X^{i}(q){\frac {\partial }{\partial q^{i}}}

{\ displaystyle X_ {q} = \ sum _ {i} X ^ {i} (q) {\ frac {\ partial} {\ partial q ^ {i}}}}

,

Where $\partial /\partial q^{i}$ ${\ displaystyle \ partial / \ partial q ^ {i}}$ is the coordinate system in TQ. The conjugate moment is then expressed as

P_{X}(q,p)=\sum _{i}X^{i}(q)\;p_{i}

{\ displaystyle P_ {X} (q, p) = \ sum _ {i} X ^ {i} (q) \; p_ {i}}

,

Where $p_{i}$ ${\ displaystyle p_ {i}}$ are defined as moment functions corresponding to vectors $\partial /\partial q^{i}$ ${\ displaystyle \ partial / \ partial q ^ {i}}$ :

p_{i}=P_{\partial /\partial q^{i}}

{\ displaystyle p_ {i} = P _ {\ partial / \ partial q ^ {i}}}

$q^{i}$ ${\ displaystyle q ^ {i}}$ together with $p_{j}$ ${\ displaystyle p_ {j}}$ form a coordinate system on a cotangent bundle $T^{*}Q$ ${\ displaystyle T ^ {*} Q}$ . These coordinates are called canonical coordinates .

Literature

Herbert Goldstein, Charles P. Poole, Jr., John L. Safko. Classical Mechanics. - 3rd. - San Francisco: Addison Wesley, 2002 .-- S. 347–349. - ISBN 0-201-65702-3 .
Arnold V.I. Mathematical methods of classical mechanics. - 5th ed., Stereotyped. - M .: URSS editorial, 2003 .-- 416 p. - 1,500 copies - ISBN 5-354-00341-5 .

Canonical coordinates

Summary

Formal Research

Literature

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