Understanding Inert Bodies: Newton and Archimedes' Equations

[deda]

Newton says: “A body accelerates proportionally with the force it carries and reciprocally to its mass”. His equation is:
$$\frac {F}{m} = \frac {d^2x}{dt^2}$$
So, what Newton does is consider the position changing while time goes by and while the force and mass remain constant. Conclusively, if the stone is lighter I can throw it further away. This way the mass becomes criterion for how inert one body is. But if the lighter and the heavier stone throw each other then they will do it with equal and opposite forces. So:
$$F_1 = -F_2 <=> \frac {M_1}{M_2} = - \frac {a_1}{a_2}$$
$$\frac {F_1}{M_1} <> \frac {F_2}{M_2}<=> a_1 <> a_2$$

Archimedes says: “Magnitudes are in equilibrium on reciprocally proportional distances from the center”. His equation is:
$$\frac {F_1}{F_2} = \frac {D_2}{D_1} = \frac {M_1}{M_2}$$
And if the forces and masses are again constant then:
$$\frac {F_1}{F_2} = \frac {dD_2}{dD_1} = \frac {M_1}{M_2}$$
So, the heavier body will carry more force and will pass smaller distance. Actually the fact that the mass is criterion for how inert one body is comes from Archimedes. Among the differences between the two physics, Archimedes’s one is time - independent.

I tell you: Our physics starts with Archimedes and that physics takes wrong direction when little Newton comes up with his mirage. We cannot blame the kids because they tell the truth even when they lie, yet we cannot consider them seriously.

 

[Antonio Lao]

The key to fully understanding the different principles working behind Archimedes' level and Newton's mechanics is to know the difference between statics of force and dynamics of force.

Statics deals with the equilibrium of forces. This is time independent. Intrinsically, the concept of mass must be described by this method of force equilibrium. The math is the integral calculus which by no coincidence is discovered by Archimedes before it was attributed to Leibniz. This is the quantum nature of the Hamiltonian function.

Dynamics deals with the nonequilibrium of forces. This is time dependent. Intrinsically, the concept of electric charge must be described by this method of force nonequilibrium. The math is the differential calculus and invented by Newton to deal with this method of analysis. This is the beginning of the Lagrangian function and gauge symmetry.

 

[matt grime]

You don't say what D_1 and D_2 are and don't justify why it is that

dD_1/dD_2 = D_1/D_2

which would state that

log(D_1) = log(D_2) + K, ie D_1=kD_2

whatever D_i might be obviously.

 

[deda]

So what?
D_1 = k * D_2
and
k = F_2 / F_1 = M_2 / M_1 = const

D_i is a distance from the center of the lever.