Definition:Canonical Variable
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Definition
Let $\map {\mathbf y} x : \R \to \R^n$, $n \in \N $ be a vector-valued function.
Let $F: \R^{2 n + 1} \to \R$ be a differentiable mapping.
Let $J \sqbrk {\mathbf y}$ be a functional of the form:
- $\d J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$
Consider the (real) variables $x, \mathbf y, \mathbf y'$, and the mapping $F$.
Make the following transformation:
- $F_{y_i'} = p_i$
where $F_x$ denotes the partial derivative of $F$ with respect to $x$.
Let $H$ be the Hamiltonian corresponding to $J \sqbrk {\mathbf y}$.
Then the new variables $x, \mathbf y, \mathbf p$, and the mapping $H$ corresponding to $J \sqbrk {\mathbf y}$ are called the canonical variables.
Also known as
By analogy with mechanical problems, the variables $p_i$ are also known as momenta.
Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 3.13$: Derivation of the Basic Formula