### State space evolution beyond mechanics

Our starting point will be the assumption that the state of our system (biological, physical, chemical, or otherwise) at a given time $$t$$ is represented by a collection of $$D$$ generic coordinates joined in a vector $$\mathbf{C}(t) = (C_1(t), C_2(t), C_i(t), \ldots C_D(t))$$. Note this vector depends on time implicitly.

Next, we postulate the existence of a conserved property, named energy, as a function of the state variables $$E = E(\mathbf{C}(t))$$. Differentiating the energy yields $\frac{\diff E}{\diff t} = \sum_j \frac{\partial E}{\partial C_j} \frac{\diff C_j}{\diff t} ,$ which provides an exact expression of the rate of change of the generic coordinate $$C_i$$ $\frac{\diff C_i}{\diff t} = \left( \frac{\partial E}{\partial C_i} \right)^{-2} \frac{\diff E}{\diff t} \frac{\partial E}{\partial C_i} - \sum_{j\ne i} \frac{\diff C_j}{\diff t} \left( \frac{\partial E}{\partial C_i} \right)^{-1} \frac{\partial E}{\partial C_j} = \sum_j L_{ij} \frac{\partial E}{\partial C_j} .$ We can write the above equation in vector-matrix form $\frac{\diff \mathbf{C}}{\diff t} = \mathbf{L} \frac{\diff E}{\diff \mathbf{C}} = \mathbf{K} \mathbf{C} .$ This is a general equation for the deterministic evolution of any system whose state is given by a non-stochastic vector $$\mathbf{C}$$. The scope of this equation of evolution is beyond mechanics because $$\mathbf{C}$$ is not limited to the positions and velocities (or momenta) of particles. Note that even if we restrict the vector to $$\mathbf{C} = (\mathbf{p}, \mathbf{q})$$ the description is still more general than Hamiltonian mechanics because the equation (3) can deal with dissipative systems.

#### Uncertainty and stability

The above expressions are deterministic. To introduce fluctuations we can use a variational scheme to find the rate of the deviations from the average deterministic evolution $\frac{\diff (\delta\mathbf{C})}{\diff t} = (\delta\mathbf{K}) \mathbf{C} + \mathbf{K} (\delta\mathbf{C}) + \frac{1}{2} (\delta\mathbf{K}) (\delta\mathbf{C}) .$ Combining this expression with (3) yields for the random vector $$\mathbf{\widetilde{C}} = \mathbf{C} + \delta\mathbf{C}$$ $\frac{\diff \mathbf{\widetilde{C}}}{\diff t} = \mathbf{K} \mathbf{\widetilde{C}} + \mathbf{\widetilde{f}} ,$ for a fluctuation component of the rate given by $$\mathbf{\widetilde{f}} = (\delta\mathbf{K}) \mathbf{C} + (1/2) (\delta\mathbf{K}) (\delta\mathbf{C})$$. Using $$\mathbf{\widetilde{K}} = \mathbf{K} + \delta\mathbf{K}$$ the rate can be rewriten in the suggestive form $\frac{\diff \mathbf{\widetilde{C}}}{\diff t} = \mathbf{\widetilde{K}} \, \mathbf{\widetilde{C}} - \frac{1}{2} (\delta\mathbf{K}) \Big [ \mathbf{\widetilde{C}} - \mathbf{C} \Big ] .$