Conservation of Energy
Let F be a conservative force field, having potential φ.
Suppose that an object, having mass m, moves along a curve C from point A to B.
The work done is
Z
φ(B) − φ(A) =
F · dr
(Fundamental Theorem of Line Integrals)
ZC
ma · dr
=
(Newton’s second law)
C
Z b
=
=
=
=
=
=
dr
ma ·
dt
(Parametrize from t = a to t = b)
dt
a
Z b
dv
m
· v dt
a dt
Z b
v · dv
m
a
Z
m b
d(v · v)
2 a
Z
m b
d(v 2 )
2 a
m
(v(B)2 − v(A)2 ).
2
Here we write v(B) instead of v(b), considering v as a function of the point.1
We conclude that
mv(A)2
mv(B)2
= φ(A) −
.
φ(B) −
2
2
Hence if we define the quantity
mv 2
E =φ−
2
to be the energy, then we have shown that the energy is constant on the entire path. This
is the principle of Conservation of Energy.2
1
2
For this to be well-defined, we assume that the path does not intersect itself.
In physics, one usually changes the sign of φ, so that the energy E can be written as a sum φ +
mv 2
2 .