SME
Classical Mechanics
Hamiltonian mechanics
Use this for “A 0.5 kg mass on a 50 N/m spring is at x = 0.2 m with p = 1.5 kg·m/s. Find H, ẋ and ṗ from Hamilton’s equations and the area of its phase-space ellipse.”
Diagram
H
=
p
2
2m
+
1
2
kx²
H
=
2
2
2 × 2
+
1
2
×
8
×
0.5
2
=
2
J
Hamilton:
ẋ
=
∂H/∂p
=
p
m
=
1
m/s,
ṗ
=
−∂H/∂x
=
−kx
=
−4
N
ω
=
√(
k
m
)
=
2
rad/s;
area
𝒜
=
2π
H
ω
=
6.283
J·s
H is conserved, so the point runs clockwise round one ellipse.
This is an example. Type your own values to start a new problem.
H
=
(
p (kg·m/s)
)
2
2
×
m (kg)
+
½
k (N/m)
(
x (m)
)
2
=
H (J)
ẋ
Velocity
worked out (not typed) · m/s
ṗ
Force
worked out (not typed) · N
ω
Angular frequency
worked out (not typed) · rad/s
𝒜
Phase-space area
worked out (not typed) · J·s
Formulas
H
=
p
2
2 ×
m
+
1
2
×
k
×
x
2
2
=
2
2
2 × 2
+
1
2
×
8
×
0.5
2
ẋ
=
p
m
1
=
2
2
ṗ
=
−
k
×
x
−4 = −8 × 0.5
ω
=
√(
k
m
)
2
=
√(
8
2
)
𝒜
=
2π
×
H
ω
6.2832
=
2π
×
2
2
Assumptions
•
H = p² ÷ 2m + ½kx² is the total energy (H = pẋ − L for this system) and is conserved, so the point stays on one ellipse.
•
Hamilton’s equations give the flow: ẋ = ∂H/∂p and ṗ = −∂H/∂x; the point goes round clockwise.
•
The ellipse’s half-widths are √(2H ÷ k) along x and √(2mH) along p.
Step-by-step
WE KNOW
m = 2 kg, k = 8 N/m, x = 0.5 m, p = 2 kg·m/s
FIND
energy (H), velocity (ẋ), force (ṗ), angular frequency (ω), phase-space area (𝒜)
Step 1 · Find energy (H)
Use H = p² ÷ (2 × m) + ½ × k × x²
The
kinetic
energy
written
with
momentum,
p
2
2
m
,
plus
the
spring’s
1
2
kx².
H
=
2
2
2 × 2
+
1
2
×
8
×
0.5
2
H
=
2
2
4
+
1
2
×
8
×
0.5
2
H
=
4
4
+
1
2
×
8
×
0.25
H
=
1
+
4
×
0.25
H
=
1
+
1
H
=
2
J
Step 2 · Find velocity (ẋ)
Use ẋ = p ÷ m
Hamilton’s
first
equation:
∂
H
/∂
p
of
p
2
2
m
is
p
m
.
ẋ
=
2
2
ẋ
=
1
m/s
Step 3 · Find force (ṗ)
Use ṗ = −k × x
Hamilton’s
second
equation:
∂
H
/∂
x
of
1
2
kx²
is
kx,
and
ṗ
is
its
negative,
the
spring’s
pull.
ṗ
=
−8
×
0.5
ṗ
=
−4
N
Step 4 · Find angular frequency (ω)
Use ω = √(k ÷ m)
Hamilton’s
equations
together
give
ẍ
=
−(
k
m
)
x
,
so
the
point
goes
round
at
ω
=
√(
k
m
).
ω
=
√(
8
2
)
ω
=
√
4
ω
=
2
rad/s
Step 5 · Find phase-space area (𝒜)
Use 𝒜 = 2π × H ÷ ω
The
ellipse’s
area
π
×
√(
2
H
k
)
×
√
2
mH
=
2πH√(
m
k
),
which
is
2π
H
ω
.
𝒜
=
2π
×
2
2
𝒜
=
6.2832
J·s
Check
Put the numbers back in. Both sides should match.
2
=
2
2
2 × 2
+
1
2
×
8
×
0.5
2
✓
1
=
2
2
✓
−4 = −8 × 0.5 ✓
2
=
√(
8
2
)
✓
6.2832
=
2π
×
2
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.