Degenerate Quadratic Forms of the Calculus of Variations

Abstract

When solving problems of the classical calculus of variations and examining the solution of the Euler equation obtained via second-order conditions, there arises the problem of verifying the positive semi-definitness of the integral quadxatic form. This form looks as follows:

$$U(x) = \int_0^1 {\left\langle {A(t)\dot x(t),\dot x(t)} \right\rangle \quad + \left\langle {B(t)x(t),x(t)} \right\rangle \quad + 2\left\langle {C(t)\dot x(t),x(t)} \right\rangle dt\quad + \left\langle {\Omega (x(0),x(1)),(x(0),x(1))} \right\rangle } $$

((1.1))