Physics 321
Hour 20
Hamiltonians
The Hamiltonian
• 𝐻 = 𝐻(𝑥, 𝑝, 𝑡) whereas ℒ = ℒ(𝑥, 𝑥, 𝑡)
• For simple systems, H = T + U
• There are two first order equations of motion for
each variable:
𝜕𝐻
𝜕ℒ
= −𝑝
=𝑝
𝜕𝑥
𝜕𝑥
𝜕𝐻
𝜕ℒ
= +𝑥
=𝑝
𝜕𝑝
𝜕𝑥
• The Lagrangian method gives one second order
equation for each variable.
The Hamiltonian - Origins
Take ℒ = ℒ(𝑞1 , 𝑞1 , 𝑞2 , 𝑞2 , 𝑡)
𝑑ℒ
𝜕ℒ
𝜕ℒ
𝜕ℒ
𝜕ℒ
𝜕ℒ
=
𝑞1 +
𝑞1 +
𝑞2 +
𝑞2 +
𝑑𝑡 𝜕𝑞1
𝜕𝑞1
𝜕𝑞2
𝜕𝑞2
𝜕𝑡
𝜕ℒ
= 𝑝1 𝑞1 + 𝑝1 𝑞1 + 𝑝2 𝑞2 + 𝑝2 𝑞2 +
𝜕𝑡
𝑑
𝜕ℒ
= (𝑝1 𝑞1 + 𝑝2 𝑞2 ) +
𝑑𝑡
𝜕𝑡
Therefore
𝑑
𝜕ℒ
(𝑝1 𝑞1 + 𝑝2 𝑞2 − ℒ) +
=0
𝑑𝑡
𝜕𝑡
The Hamiltonian - Origins
Let
𝑑
𝜕ℒ
(𝑝1 𝑞1 + 𝑝2 𝑞2 − ℒ) +
=0
𝑑𝑡
𝜕𝑡
𝑝1 𝑞1 + 𝑝2 𝑞2 − ℒ = 𝐻
If ℒ has no explicit time dependence, then H is a
conserved quantity.
The Hamiltonian - Notes
• The Hamiltonian is a function of p and q. But p is
not ‘the momentum,’ it is the generalized
momentum conjugate to q.
𝜕ℒ
𝑝𝑖 =
𝜕𝑞𝑖
• The general expression is:
𝐻=
𝑝𝑖 𝑞𝑖 − ℒ
𝑖
• That means you generally have to find the
Lagrangian before you can find he Hamiltonian!
The Hamiltonian - Notes
• The Hamiltonian is H=T+U unless
• The Lagrangian has explicit time dependence
• The transformation between the coordinates 𝑞𝑖
and Cartesian coordinates have explicit time
dependence
• But … you have to write the Hamiltonian in terms
of the correct generalized momenta – so you
usually still need the Lagrangian first!
Example
hamilton.nb