Difference between "essential boundary conditions" and "natural boundary conditions"? In a boundary value problem, what's the difference between "essential boundary conditions" and "natural boundary conditions"?

In the context of a variational problem for a functional
$$I[q]~:=~\int_{t_i}^{t_f} \! dt ~L(q,\dot{q},t),\qquad \dot{q}~\equiv~ \frac{dq}{dt},$$ 
defined on an interval $[t_i,t_f]\subseteq \mathbb{R}$, the types of boundary conditions (BC) are defined as follows:


*

*Essential/Dirichlet BC: $\quad q(t_i)~=~q_i\quad\text{and}\quad q(t_f)~=~q_f.$

*Natural BC: $\quad p(t_i)~=~0\quad\text{and}\quad p(t_f)~=~0.$
Here $$p~:=~\frac{\partial L}{\partial \dot{q}} $$ is the canonical/conjugate momentum. 
See also e.g. in my related Phys.SE answer here. The types of BC generalize to higher-dimensional regions. 
