Consider the optimal control problem \begin{gather} \max_{u}\int_{t_{0}}^{t_{1}} f(t,x_{1},x_{2},u)\, dt \nonumber \\ \begin{aligned}[t]\label{sys:bkgrd_OC_problem} &\text{subject to:}\quad x_{1}'(t) = g_{1}(t,x_{1},x_{2},u),\quad x_{1}(t_{0}) = c_{1} \\ &\phantom{\text{subject to:}\quad} x_{2}'(t) = g_{2}(t,x_{1},x_{2},u),\quad x_{2}(t_{0}) = c_{2} \end{aligned} \end{gather}
where $f,g$ are sufficiently smooth. The typical solution via Pontryagin's Maximum Principle involves defining a Hamiltonian $$H = f(t,x_{1},x_{2},u) + \lambda_{1}g_{1}(t,x_{1},x_{2},u) + \lambda_{2}g_{2}(t,x_{1},x_{2},u)$$ where the costates $\lambda_{1}$ and $\lambda_{2}$ satisfy \begin{align*} \lambda_{1}'(t) &= -\frac{\partial H}{\partial x_{1}},\quad \lambda_{1}(t_{1}) = 0\\ \lambda_{2}'(t) &= -\frac{\partial H}{\partial x_{2}},\quad \lambda_{2}(t_{1}) = 0. \end{align*}
A common interpretation of $\lambda_{1}(t)$ and $\lambda_{2}(t)$ are that they represent the marginal values of the states $x_{1}$ and $x_{2}$ at time $t$, respectively.
So, with that in mind, here is my question:
If we modify the problem so that we replace $f(t,x_{1},x_{2},u)$ with $f(t,x_{1},u)$, i.e. remove the explicit dependence of the objective functional on $x_{2}$, but leave everything else the same, does this change how we interpret $\lambda_{1}(t)$ and $\lambda_{2}(t)$? If so, what is the new interpretation? I haven't been able to find a reference that addresses this sort of interpretation question for anything but the simplest one state/one control types of optimal control problems.
Thinking in terms of units: $$[\text{unit of } \lambda_i] = \dfrac{[\text{unit of } f] \times [\text{unit of } t]}{[\text{unit of } x_i]} = \dfrac{[\text{unit of Cost}] }{[\text{unit of } x_i]}, \quad \quad i = 1,2$$
The unit of $f$ does not depend on its inputs. Therefore, changing the number of arguments will not change the interpretation of $\lambda$ in terms of its units.
However, two linearly independent Lagrange terms, $f_a(t, x_1, x_2, u)$ and $f_b(t, x_1, u)$, will always yield different solutions of state, control and costate, even if their units are the same.