Definition:Conjugate Point (Calculus of Variations)

Definition 1
Let


 * $ \displaystyle - \frac{ \mathrm d }{ \mathrm d x} \left ( { Ph' } \right ) + Qh = 0$

with boundary conditions


 * $ h \left ( { a } \right ) = 0, \quad h \left ( { c } \right ) = 0, \quad a < c \le b$

Suppose


 * $ h \left ( { x } \right ) =0 \quad \neg \forall x \in \left [ { a \,. \,. \,b } \right ] $

Suppose


 * $ h \left ( { a } \right ) = 0, \quad h \left ( { \tilde { a } } \right ) = 0, \quad a \ne \tilde { a } $

Then the point $ \tilde { a } $ is called conjugate to the point $ a $ solution to the aforementioned differential equation.

Definition 2
Let $ y = y \left ( { x } \right) $ and $ y^* = y^* \left ( { x } \right) $ be extremal functions.

Let


 * $ M = \left ( { a, y \left ( { a } \right) } \right )$


 * $ \tilde M = \left ( { \tilde a, y \left ( { \tilde a } \right) } \right )$

Let $ y $ and $ y^* $ both pass through the point $ M $.

Let


 * $ y^* \left ( { x - \tilde a } \right) - y \left ( { x - \tilde a } \right) = \epsilon \left \vert y^* \left ( { x - \tilde a } \right) - y \left ( { x - \tilde a } \right) \right \vert_1 $

where


 * $ \displaystyle \left \vert y^* \left ( { x - \tilde a } \right) - y \left ( { x - \tilde a } \right) \right \vert_1 \to 0 \implies \epsilon \to 0 $

Then $ \tilde M $ is conjugate to $ M $.

Definition 3
Let $ y = y \left ( { x } \right )$ and $ y = \tilde y \left ( { x } \right )$ be extremal functions.

Let


 * $ \displaystyle M = \left ( { a, y \left ( { a } \right ) } \right ) $


 * $ \displaystyle \tilde M = \left ( { \tilde a, y \left ( { \tilde a } \right ) } \right ) $

Let both $ y = y \left ( { x } \right )$ and $ y = \tilde y \left ( { x } \right )$ pass through the point $ M $.

Let


 * $ \displaystyle \lim_{ \left \vert y \left ( { x } \right ) - \tilde y \left ( { x } \right ) \right \vert_1 \to 0 } \left [ { \left ( { x, y \left ( { x } \right ) } \right ) : y \left ( { x } \right ) - \tilde y \left ( { x } \right ) = 0  } \right ] = \tilde M $

In other words, let $ \tilde M $ be the limit of points of intersection of $ y = y \left ( { x } \right ) $ and $ y = \tilde y \left ( { x } \right ) $ as $ \left \vert y \left ( { x } \right ) - \tilde y \left ( { x } \right ) \right \vert_1 \to 0 $

Then $ \tilde M $ is conjugate to $ M $.

Also defined as
In the context of Calculus of Variations, functionals are one of the most important concepts.

Therefore, instead of a function, a functional which is minimised by the given function is used as a concept of reference.

Then, if $ \tilde a $ is conjugate to $ a $ solution of $ \displaystyle \left [ { - \frac{ \mathrm d }{ \mathrm d x} \left ( { Ph'  } \right ) + Qh = 0 } \right ] $, then it is also conjugate  $ \displaystyle \int_a^b \left ( { Ph'^2 + Qh^2  } \right ) \mathrm d x $.