Newton’s Second from a

potential

D. Craig, WTAMU

2007–02–02

Suppose we have a particle moving in a sys-

tem characterized by a potential

V(~

r).

Its total

energy is the sum of the kinetic energy and

potential energy:

1

2

mv

2

+ V(~

r) = E.

Assume that

E

is conserved, so

dE

dt

= 0.

We want to take the time derivative of both

sides of the first equation.

First term is not

too hard:

d

dt

1

2

mv

2

!

=

m

2

d

dt

(v

2

x

+ v

2

y

+ v

2

z

)

=

m

2

(2v

x

dv

x

dt

+ 2v

y

dv

y

dt

+ 2v

z

dv

z

dt

)

= m

~

v

·

d~

v

dt

!

.

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For the time derivative of the potential energy,

we have to remember that even though

V

itself

doesn’t change in time,

~r(t)

does. So we have

dV(~

r)

dt

=

lim

δt

→0

V(~

r(t + δt)) − V(~r(t))

δt

,

with

δV = V(~

r(t+δt))−V(~r(t))

. Now

δV =

∇

V

·

δ~

r

so

dV(~

r)

dt

=

lim

δt

→0

∇

V

·

δr

δt

= (~

v

· ∇

V).

Putting these results together:

~

v

·

m

d~

v

dt

+

∇

V

!

= 0.

~

v

is any arbitrary velocity, so the term in paren-

theses must always be zero:

m

d~

v

dt

= −

∇

V = ~

F,

which is Newton’s second law.

2

So Newton’s second law can be derived from

energy conservation. We have also shown that

~

F = −

∇

V,

which is an important general idea: forces arise

from changes in the potential energy function

acting on a particle.

In two or three dimensions the change along a

path can be characterized by the gradient.

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