# Perturbation of velocity in Hamilton equations. What do you call it?

Consider a Hamilton function $$H_0(x,p) = \frac{p^2}{2m}+ V(x).$$

The canonical equations then read $$\dot{x}(t) = p/m$$ and $$\dot{p}(t) = -V'(x)$$

Now imagine, we add an additional term $$\dot{x}(t) = p/m + g_0(x,p)\quad\text{and}\quad\dot{p}(t) = -V'(x) + g_1(x,p)$$

Then, we can see $$g_1$$ as an external force, but what is the interpretation of $$g_0$$? Is it meaningful to call it a friction?

$$mg_0$$ is apparently the difference between the kinetic momentum $$m\dot{x}$$ and the canonical/conjugate momentum $$p$$. However if the perturbation $$g_0,g_1$$ does not preserve the canonical structure, i.e. descend from an interaction Hamiltonian, then the notion of canonical/conjugate momentum is not well-defined.
You can't just arbitrarily add anything, Since the evolution equation for $$(x,p)$$ is derived from $$H$$.
In this case, $$\dot{x}=\partial_pH=\frac{p}{m}+g_0(x,p)$$ $$\dot{p}=-\partial_xH=-V'(x)+g_1(x,p)$$ Since you must have, $$dH(q,p)=\partial_qHdq+\partial_pHdp$$ $$\rightarrow -\partial_p\dot{p}=\partial_q\dot{q}$$$$\Rightarrow -\partial_p g_1(x,p)=\partial_qg_0(x,p)$$ If so you can write, $$H=\frac{p^2}{2m}+V(x)+f(x,p)$$ with $$g_0(x,p)=\partial_p f(x,p)\ \ \&\ \ \ g_1(x,p)=-\partial_xf(x,p)$$
One can regard the tern $$f(x,p)$$ as the term that couples particle's velocity and its position. A specific example of such a Hamiltonian can be, Charge particle in a uniform constant EM field. So if $$\vec{B}=B\hat{k}$$, Then you can choose $$\vec{A}=-By\hat{i}$$ so that $$H=\frac{(p_x+eBy)^2}{2m}+\frac{p_y^2}{2m}+\frac{p_z^2}{2m}+e\phi(\vec{r})$$ If you open up the first braket you get a term that will couple the velocity and position.