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into account the constraints is therefore
: where are Lagrange multipliers. By definition, the momentum and the Hamiltonian which is computed to be:
:

Now, we recall that the boundary is free in the direction and is a free parameter. Therefore, we must have:
: At the boundary and , therefore we recover the Young equation.

Non-ideal smooth surfaces and the Young contact angle

The Young equation assumes a perfectly flat and rigid surface often referred to as an ideal surface.

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