Uniqueness of type of functional which is able to produce E-L equation in the form of $-\Delta u+D \varphi \cdot D u=f$.

Question

I want to ask about the uniqueness of functional which is able to produce E-L equation in the form of $$-\Delta u+D \varphi \cdot D u=f.$$

The answer here said that the energy functional of $$ -\Delta u+D \varphi \cdot D u=f $$ is $$\tag{1} I[w]:=\int_U L(D w, w, x) \,\mathrm d x $$ with $L(p, z, x)=e^{-\varphi(x)}\left(\frac{1}{2}|p|^2-f(x) z\right)$.

My question is: Is this the only way (in a certain sense, regarded as a weighted functional (by multiplying $e^{-\varphi}$)) to produce $D \varphi \cdot D u$ term? For example, if we have a PDE
$$ -\Delta u+h \cdot D u=f， $$ then the only way it can be written as the E-L equation of some functional is when $h=D \varphi$, and the functional is in the form of (1).

Answer 1

Actually, Lagrangian mechanics has difficulty to handle velocity-dependent forces. A few strategies can be adopted in order to generate damping / dissipative terms like $h \cdot D\phi$.

Time-dependent Lagrangian. The exponential prefactor you mention in your porst turns out to be the usual "trick" and perhaps the "cleanest" way to produce such a damping term. It is to be noted that the time-dependence of the Lagrangian comes from the fact the energy is not conserved because of the said dissipative term.

This is the only way to generate the term $D\phi \cdot Du$ alone that I am aware of; this term may originate from Lagrangians with different structures, but it will come along with other unwanted terms or prefactors.

Auxiliary variable. You can introduce a new degree of freedom through an ancillary variable $v$, which will help you to adjust and design the equations of motion. For example, the bivariate Lagrangian $$ L = Du \cdot Dv - \frac{1}{2}h \cdot (uDv - vDu) - f(x)(u+v) - \frac{1}{2}(u-v) \operatorname{div}h $$ leads to the following system of differential equations : $$ \begin{cases} -\Delta u + h \cdot Du = f \\ -\Delta v - h \cdot Dv = f \end{cases} $$ Note that the obtained equations of motion are disentangled already, so that you can deal with the first one while ignoring the other one. Also, the field $h$ needn't derive from a potential $\phi$ in that case.

Let's highlight too that the auxiliary variable $v$ represents the environment of the system described by $u$. Indeed, this Lagrangian is time-independent, thus the energy is conserved in such a way that the energy gain/loss of the system due to the term $h \cdot Du$ is compensated by its environment through $-h \cdot Dv$.

Dissipation function. Generalized forces involving "velocities" are tackled sometimes with the help of a so-called dissipation function $Q$ and a modified Euler-Lagrange equation, namely $$ \frac{\partial L}{\partial u} - \operatorname{div}_x\frac{\partial L}{\partial Du} = \operatorname{div}_{Du}Q, $$ with $L = \frac{1}{2}(Du)^2 - fu$ and $Q = \frac{1}{2}h(Du)^2$ in the present case. Unfortunately, the right-hand side doesn't derive from a variational principle anymore, but it comes from ad hoc corrections.