Necessary and Sufficient Condition for First Order System to be Mutually Consistent
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Theorem
Let $\mathbf y$, $\boldsymbol \psi$ be $n$-dimensional vectors.
Let $g$ be a twice differentiable mapping.
Let
- $(1): \quad \map {\boldsymbol \psi} {x, \mathbf y} = \map {\mathbf y'} {x, \mathbf y}$
- $(2): \quad \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } = \map {g_{\mathbf y} } {x, \mathbf y}$
where $\mathbf p$ is a momentum.
Then the boundary conditions defined by $(1)$ are mutually consistent if and only if the mapping $\map g {x, \mathbf y}$ satisfies the Hamilton-Jacobi equation:
- $(3): \quad \dfrac {\partial g} {\partial x} + \map H {x, \mathbf y, \dfrac {\partial g} {\partial \mathbf y} } = 0$
Proof
Necessary condition
Take the partial derivative of $(3)$ with respect to $x$:
- $(4): \quad \frac {\partial^2 \map g {x, \mathbf y} } {\partial \mathbf y \partial x} = -\dfrac \partial {\partial \mathbf y} \map H {x, \mathbf y, \dfrac {\partial g} {\partial \mathbf y} }$
By the Schwarz-Clairaut Theorem, the order of partial derivatives of $g$ can be exchanged:
- $(5): \quad \dfrac {\partial^2 \map g {x, \mathbf y} } {\partial \mathbf y \partial x} = \dfrac {\partial \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial x}$
Since $\map g {x, \mathbf y}$ depends on $\mathbf y$ only in a direct way, it follows that:
- $\dfrac {\partial g} {\partial \mathbf y} = g_{\mathbf y}$
Hence, from $(2)$, $(4)$ and $(5)$ it follows that:
- $\dfrac {\partial \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } } {\partial x} = -\dfrac \partial {\partial \mathbf y} \map H {x, \mathbf y, \mathbf p \sqbrk {x, \mathbf y, \map {\boldsymbol \psi} {x, \mathbf y} } }$
These are the consistency equations.
$\Box$
Sufficient condition
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Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 6.32$: The Field of a Functional