Definition:Jacobi's Equation of Functional/Dependent on N Functions

Definition

Let

$\displaystyle\int_a^b \map F {x,\mathbf y,\mathbf y'}\rd x$

be a functional, where $\map {\mathbf y} a=A$ and $\map {\mathbf y} b=B$.

Let

$\displaystyle\int_a^b\paren{\mathbf h'\mathbf P\mathbf h'+\mathbf h\mathbf Q\mathbf h}\rd x$

be a quadratic functional, where $\displaystyle P_{ij}=\frac 1 2 F_{y_i'y_j'}$ and $\displaystyle Q_{ij}=\frac 1 2 \paren{F_{y_iy_j}-\frac \d {\d x} F_{y_iy_j'} }$

Then the Euler's equation of the latter functional:

$-\dfrac \d {\d x} \paren{\mathbf P\mathbf h'}+\mathbf Q\mathbf h=\mathbf 0$

is called Jacobi's Equation of the former functional.

Source of Name

This entry was named for Carl Gustav Jacob Jacobi.