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.

Also defined as
In the context of, 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 $.