# Definition:Canonical Variable

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## Definition

Let $\mathbf y=\langle y_i\rangle_{1\le i\le n}$ be a vector-valued function.

Let $J\sqbrk{\mathbf y}$ be a functional of the form:

- $\displaystyle J\sqbrk{\mathbf y}=\int_{x_0}^{x_1} \map F {x,\mathbf y,\mathbf y'} \rd x$

Consider the variables $x,\mathbf y,\mathbf y',F$.

Now, make a transformation:

- $F_{y_i'}=p_i$

Let $H$ be the Hamiltonian corresponding to $J\sqbrk{\mathbf y}$.

The new variables $x,\mathbf y,\mathbf p,H$ corresponding to $J\sqbrk{\mathbf y}$ are called **the canonical variables**.

## Also known as

By analogy with mechanical problems, 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