Difference between "essential boundary conditions" and "natural boundary conditions"?

Question

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

Answer 1

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:

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

2. 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.

Answer 2

From here:

The two types of boundary conditions are used:

1. Essential or geometric boundary conditions which are imposed on the primary variable like displacements, and

2. Natural or force boundary conditions which are imposed on the secondary variable like forces and tractions.