Understanding the Relation of pdt in Mechanics

[Gerenuk]

I noticed that the relations in mechanics can be seen like:

1) assume conservation of angular momentum
2) from this you can mathematically *derive* that energy as defined by dE=v dp is conserved (if all forces are inverse square)

Up to here we haven't specified whether we are dealing with classical or relativistic mechanics, i.e. we do not know the function p(v).

Now instead of writing E=mc^2 you could *equivalently* write
p dt=E ds
from which all of relativity follows.

Now I am wondering if p dt has a physical meaning. I vaguely recall seeing something in a path integral formulation.

What do you think?

 

[Valerie Park]

Yes, p dt does have a physical meaning. In classical mechanics, p dt is the infinitesimal amount of momentum that a particle carries when it travels through a distance dt in time. This is related to the path integral formulation of classical mechanics, where an action is defined as the integral of the Lagrangian over all possible paths of the system. The infinitesimal differences in the action along different paths can be written as p dt, which gives momentum its physical meaning. In relativistic mechanics, p dt is the infinitesimal amount of energy-momentum that a particle carries when it travels through a distance ds in space-time. This is related to the Einstein field equation, where the stress-energy tensor is defined by the integral of the Lagrangian over all possible paths of the system. The infinitesimal differences in the stress-energy tensor along different paths can be written as p dt, which gives energy-momentum its physical meaning.