  Many systems in biology, physics, and engineering are modeled by nonlinear
dynamical systems where the states are usually unknown and only a subset of the
state variables can be physically measured. Can we understand the full system
from what we measure? In the mathematics literature, this question is framed as
the observability problem. It has to do with recovering information about the
state variables from the observed states (the measurements). In this paper, we
relate the observability problem to another structural feature of many models
relevant in the physical and biological sciences: the conserved quantity. For
models based on systems of differential equations, conserved quantities offer
desirable properties such as dimension reduction which simplifies model
analysis. Here, we use differential embeddings to show that conserved
quantities involving a set of special variables provide more flexibility in
what can be measured to address the observability problem for systems of
interest in biology. Specifically, we provide conditions under which a
collection of conserved quantities make the system observable. We apply our
methods to provide alternate measurable variables in models where conserved
quantities have been used for model analysis historically in biological
contexts.
