*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 \[ \frac{\diff \mathbf{\widetilde{C}}}{\diff t} = \mathbf{\widetilde{K}} \, \mathbf{\widetilde{C}} - \frac{1}{2} (\delta\mathbf{K}) \Big [ \mathbf{\widetilde{C}} - \mathbf{C} \Big ] . \]