Is there some geometrical interpretation of force from Newton's Laws?

[OlegKmechak]

dP = F dt
dE = F dr

or if we introduce ds = (dt, dr)

(dP, dE) = F ds

And both dP and dE are constant in closed system.

Some questions:
- How does its implies on definition of Force?
- Is there some clever geometrical interpretation of Force?
- Why P and E seems almost interchengable?

 

[Nugatory]

Both momentum and energy are conserved in a closed system, but if we’re applying a net force to an object it’s not a closed system. The dP and dE equations tell us how the momentum and energy change in this non-closed system as a result of applying the force.

A quick look at the equations will tell you that the change in energy is proportional to the distance across which the force is applied while the change in momentum is proportional to the time the force is applied; they are different things not interchangeable. The difference will be more apparent if you write the equations out more precisely to reflect that $P$ and $r$ are vectors while $E$ and $t$ are scalars.

And what is that (dP,dE) formalism in the third equation? It looks like the four-vector formalism of special relativity? If that’s what it is, you might want to hold off on that until you’ve thoroughly nailed down your understanding of the Newtonian model.

 

[dx]

Force is the gradient of the potential energy. Working in one dimension, we have F = -dU/dx. The rate of change of potential energy with time is

$$\frac{dU}{dx} \frac{dx}{dt} = -Fv$$

The rate of change of kinetic energy T is

$$\frac{d}{dt}\frac{1}{2}mv^2 = \frac{dT}{dv} \frac{dv}{dt} =mv \frac{dv}{dt} = mva$$

Because energy is conserved, the change in kinetic energy must compensate the change in potential energy, so we must have Fv = mva, i.e. we have

$$F = ma$$

 

[A.T.]

Force is the gradient of the potential energy.

This is only true for conservative forces, not for all forces.

 

[OlegKmechak]

And what is that (dP,dE) formalism in the third equation? It looks like the four-vector formalism of special relativity? If that’s what it is, you might want to hold off on that until you’ve thoroughly nailed down your understanding of the Newtonian model.

In special relativity time and space are same things(with precission to signature). So from this point of view momentum P and energy E are same things(vector in 4-d space) or isn't it?
Sory, it is hard to me to explain what I want to find out :) I will take a little break