# Explaining friction using Hamiltonian mechanics

I have heard the opinion that it is a good assumption that microscopically all forces are actually conservative so in principle all classical mechanics problems could be solved using Lagrangian / Hamiltonian dynamics. Then, I was wondering how to account for friction. One thing that I have heard is that if you want to consider friction on system parametrized by coordinates $$(q_1, ..., q_n)$$ and momenta $$(p_1, ..., p_n)$$ then you should increase your system to include environment until the total system can be assumed to not interact with anything else. Then you consider Hamiltonian dynamics for coordinates $$(q_1, ..., q_n, ..., q_m)$$ and momenta $$(p_1, ..., p_n, ..., p_m)$$. But I am still not sure what is the intuition behind obtaining the friction effect. What kind of approximations we are making? What kind of averaging are we making? I am not asking to specifically calculate the friction coefficient or give any numerical values. I am interested in what the theoretical idea is that would convince me that by considering larger system you can actually get that effective motion would be, for example, if $$n=1$$ then $$q_1''(t) = -k q_1'(t)$$ (friction proportional to velocity) or $$q_1'' = -\nu$$ (static friction).

Please let me know if my question is unclear and I would appreciate any references or opinions.

## 2 Answers

Macroscopically, friction corresponds to dissipation of kinetic energy into heating system and environment.

From a microscopic point of view, it is the huge difference in the number of degrees of freedom corresponding to microscopic dynamics of molecules of the system and of the environment as compared with the few degrees of freedom describing the macroscopic motion of the system which accounts for the apparent non conservation of mechanical energy observed at macroscopic scale.

As approximation, there are cases where such situation corresponds to perform a kind of coarse graining such that the (usually) unaccessible microscopic degrees of freedom are eliminated (in a sense averaged out) and replaced by dissipative terms connected to thermodynamic quantities.

An example of the kind of coarse graining you are looking for is the way one can justify the passage from the conservative dynamics describing a system of big spheres embedded in a sea of small spheres (a model for brownian motion). There is a nice paper by De Grooth on Am. J. Phys. showing in a pedagogical way the transition. In the following, I'll summarize briefly the outline of the calculations.

For simplicity let's work in 1D. We have a system of heavy spheres of mass $$M$$ and small spheres of mass $$m$$. The spheres interact through elastic collisions. The analysis of the conservation of energy and momentum in a single collision between a small sphere of velocity $$v$$ and a a big sphere of velocity $$V$$ allows to get the velocities after collision $$v^{\prime},V^{\prime}$$ as: $$\begin{eqnarray} V^{\prime} & = & \frac{M-m}{M+m} V + \frac{2m}{M+m} v \\ v^{\prime} & = & \frac{m-M}{m+M} v + \frac{2M}{M+m}V \end{eqnarray}$$ At the first order in $$\frac{m}{M}$$ the first of the previous equations becomes $$V^{\prime}=\left( 1-\frac{2m}{M} \right)V + \frac{2m}{M}v$$ which implies a change of momentum of the heavy sphere, after one collision $$\Delta P = 2m v - 2m V.$$ In an interval of time $$\Delta t$$, small enough that the velocity of the heavy particle changes weakly, we may have many collisions, say $$N$$, with a collision rate $$n=N/\Delta t$$. The total change of momentum after $$N$$ collisions is $$2m \sum_{i=0}^{N-1}v_i - 2m \sum_{i=0}^{N-1} V_i$$ Since $$V_i$$ are not too different in the interval of time $$\Delta t$$, the second term may be rewritten as $$-2mnV(t)\Delta t$$. By dividing it by $$\Delta t$$, we find that the second term corresponds to a friction force $$-\gamma V$$ ($$\gamma>0$$). The first term can be treated as a random force, but, for the present argument this is a side remark, although interesting in the context of the analysis of the brownian motion. Here, the interesting point is the emergence of the friction from a coarse graining process.

Notice that this is just an example of deriving dissipative effects out of a conservative fundamental dynamics. Other possibilities, ad more rigorous treatments exist.

In terms of energy conservation, friction is essentially heat being dissipated in the environment. To account for friction in a Lagrangian one would need to include the terms corresponding to the kinetic energy of the molecules constituting the objects of the system. This would result in a mix of coordinates and momenta for the macroscopic objects (i.e. the object rubbing against the surface), and microscopic object (the molecules on the surface of the object and of the surface).

A simpler approach would be to consider the kinetic energy of the macroscopic objects, and approximate the kinetic energy of the microscopic objects with their thermal energy (~~temperature).