iSoul In the beginning is reality

Work and energy, exertion and verve

Here we show the work and energy in the linear motion of a particle in space-time (see J.M. Knudsen and P.G. Hjorth’s Elements of Newtonian Mechanics, 1995, p.51). Consider a particle of mass m moving along the r axis so all quantities are scalars. Newton’s second law is then

mr/dt² = F, or

m dv/dt = F,

with mass m, force F, and velocity v. Multiply both sides by v = dr/dt:

m (dv/dt) v = F (dr/dt), or

mv dv = F dr := dW,

where W is called the work done by the force F over the segment dr. Define T as

T = mv²/2,

which is called the kinetic energy of the particle. Then

dT = dW.

That is, the change in the kinetic energy of the particle over the segment dr equals the work done by the force F.

If F = F(r) does not depend on time, then define the potential energy U = U(r) through

dU(r) := –dW = –F(r)dr.

That is, the change in the potential energy U(r) over the segment dr is equal to minus the work done by the external force F. Since

dT = –dU(r),

and upon integrating,

T + U(r) = E,

where the constant E is called the total mechanical energy of the system.


Here we show the exertion and verve in the linear motion of an moticle (point vass) in time-space. Consider an moticle of vass n moving along the t axis so all quantities are scalars. Newton’s second law for time-space is then

nt/dr² = Γ, or

n du/dr = Γ,

with vass ℓ, surge Γ, and allegrity u. Multiply both sides by u = dt/dr:

n (du/dr) uΓ (dt/dr), or

nu duΓ dt := dX,

where X is called the exertion done by the surge Γ over the time segment dt. Define V as

V = nu²/2,

which is called the kinetic verve of the moticle. Then

dV = dX.

That is, the change in the kinetic verve of the moticle over the segment dt equals the exertion done by the surge Γ.

If Γ = Γ(t) does not depend on position, then define the potential verve Y = Y(t) through

dY(t) := –dX = –Γ(t)dt.

That is, the change in the potential verve V(t) over the time segment dt is equal to minus the exertion done by the external surge Γ. Since

dV = –dY(t),

and upon integrating,

V + Y(t) = Z,

where the constant Z is called the total mechanical verve of the system.

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