The idea of totality of motion is a rather interesting and would come to human minds as common sense. It is however, very different. This would be my concern in my development of mechanics with arc length as parameter. Let me share this idea I have, that has been in my mind for some time now. Consider a dissipative force. It is, of course, always act opposite of the direction of motion. Now try to integrate the force through the region where it retraces its path. Because we have said the force acts opposite the direction of motion, the sign must change whenever the direction changes. Thus the integral that we are doing is separated into two parts, one with positive sign and the other negative. That is
F = - sgn(v) F(q)
F = - sgn(v)F(dq/dt)
The work done by
W = - (int( sgn(v) F(q)dq)
W = - (int(sgn(v)F(dq/dt)d(dq/dt)dt).
The presence of sgn(v) implies the separated parts of the integration in accordance to sign. Now I present this idea of mine, which I am not sure of its truth. If we parametrized this function F with the arc length, s , and put restriction such that W(a) > W(b) for a>b, we will have a irreversible work function and we may replace the integral mentioned with a single integral. Furthermore, the bijectivity of the arc length function tells us that F may be a function of s ( F that is function of ds/dt may be mapped as F(s)).
Now comes the issue of the totality of motion. If the motion can be determined by its initial conditions, that is q = f(t) and dq/dt = f'(t) (which implies s = g(t) and s'= g'(t)), can the force be parametrized as above? Can it be related to such deterministic mapping? What happens if this force is seen from the quantum mechanical perspective? Can we assume the solution exist, and work with the known form of the solution, not the equation itself? These are questions that remain for me to answer, which to some degree, philosophical. I do hope readers will enlight me and provide insights that supports or reject my proposition.