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


2 Answers 2


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.

  • $\begingroup$ What will be the natural Boundary Conditions if instead of $L(q,\dot{q},t)$ we have $L(q,\dot{q},\ddot{q},t)$? $\endgroup$
    – TheStudent
    Jul 29, 2021 at 10:27
  • $\begingroup$ Hi @honey kumar. You might be interested in my Phys.SE answer here. $\endgroup$
    – Qmechanic
    Jul 31, 2021 at 10:48
  • $\begingroup$ Thankyou @Qmechanic $\endgroup$
    – TheStudent
    Jul 31, 2021 at 14:25

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.

  • 1
    $\begingroup$ Link says: No preview available. $\endgroup$
    – Qmechanic
    Mar 21, 2016 at 20:32

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