{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# p09fl27 - Lecture 27 Hamiltons equations(4 Nov 09 Review 2...

This preview shows pages 1–3. Sign up to view the full content.

Lecture 27: Hamilton’s equations (4 Nov 09) Review 2 Nov homework A. Review – Hamiltonian 1. FW Secs. 18 and 20 and L11 2. In the case that the Lagrangian has no explicit time dependence, L = L ( { q j } , { ˙ q j } ), form the total time derivative, with variation through the q j and ˙ q j : dL dt = X j ∂L ∂q j ˙ q j + ∂L ˙ q j ¨ q j 3. Lagrange equations give ∂L ∂q j = d dt ∂L ˙ q j = d dt p j ; p j ∂L ˙ q j 4. so dL dt = X j ˙ q j d dt ∂L ˙ q j + ∂L ˙ q j ¨ q j = d dt X j ˙ q j p j 5. Thus H j ˙ q j p j - L is a conserved quantity (the energy): dH dt = 0 6. Lorentz force from L10 (Sec. 33 of FW). The Lagrangian that generates the equation of motion for particle of mass m and charge q in electric and magnetic fields E and B (potential and vector potential φ and A ) is L = m 2 v 2 - + q v · A The canonical momentum is then ( v = ˙ r ) p = m v + q A 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(mechanical momentum m v ) and the Hamiltonian is H = v · p - m 2 v 2 + - q v · A = m 2 v 2 + H = 1 2 m ( p - q A ) 2 +
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

p09fl27 - Lecture 27 Hamiltons equations(4 Nov 09 Review 2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online