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1 Conservation of Energy
Today we will talk about what conservation of energy is good for — how it can help us solve problems in classical mechanics. If we have a particle q: R → Rn^ satisfying F = ma where F is conservative:
F (t) = − ∇V (q(t))
where V : Rn^ → R is called the potential, then energy is conserved. Let
E(t) =
m q˙(t)^2 + V (q(t)).
Then
d dt E(t) = m q˙(t) · ¨q(t) + ∇V (q(t)) · q˙(t) = F (q(t)) · q˙(t) + ∇V (q(t)) · q˙(t) = 0
What good is this? It helps understand the motion of the particle: for any solution of Newton’s second law 1 2
m q˙(t)^2 + V (q(t)) + E
|| q˙(t) || =
m
(E − V (q(t)))
so we know the particle’s speed given its position. This is especially powerful for a particle on the line (n=1).
Example: A particle on a line. In this case, suppose the force depends only on position:
F (t) = f (q(t))
for f : R → R. Then automatically F is conservative:
f = −∇V
= −
dV dx
where
V (x) =
∫ (^) x
x 0
f (s)ds.
Note: we can add any constant to V. Also: the fact that any f is −∇V for some V is special to 1 dimension. So we have:
| q˙(t) | =
m
(E − V (q(t)))
so
q˙(t) = ±
m
(E − V (q(t)))
For example:
graph of some f unction V (x) on plane with chosen energy value V = E
The particle’s position, say x, must have
V (x) ≤ E.
This is called the classically allowed region - in our example, [x 0 , x 1 ]. The set of x ∈ R where V (x) > E is the classically forbidden. A particle at a local maximum can go one of the two possible directions. If the potential increases all the way up to V = E, the particle stops for moment and then Newton’s second law demands that the particle goes back down the graph. In our example the particle must oscillate between x 0 and x 1 , moving faster where V is smaller.
Example: A particle in R^3 in a central force.
picture of a central f orce f ield
A central force depends only on position, so it’s given by f : R^3 − { 0 } → R^3 , but where f is spherically symmetric:
f (x) = φ(|| x ||)
x || x ||
where φ: [0, ∞) → R. (We’ll worry about the origin in R^3 when necessary.) We’ll write
|| q(t) || = r(t)
q(t) || q(t) || = rˆ(t)
so Newton’s 2nd law says mq¨(t) = φ(r(t))ˆr(t).
Kepler started thinking about planetary motion - this is motion in a central force
φ(r) = −
k r^2
He noted that planets sweep out equal area in equal time:
picture of planet going around sun with area f rom t 0 to t 0 + ∆t and f rom t 1 to t 1 + ∆t
This is secretly “conservation of angular momentum”. This will let us understand motion in any central force. First, a central force is automatically conservative: if
f (x) = φ(|| x ||)ˆx, (ˆx =
x || x ||
then f (x) = − ∇V (x)
where V (x) = v(|| x ||)
