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Hamiltonian

Hamiltonian. The generalized momentum was defined from the Lagrangian. Euler-Lagrange equations can be written in terms of p. Generalized Momentum. Jacobian Integral. The quantity E is the Jacobian integral of the motion. Constant when L does not contain time. Use. Conjugate Variables.

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Hamiltonian

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  1. Hamiltonian

  2. The generalized momentum was defined from the Lagrangian. Euler-Lagrange equations can be written in terms of p Generalized Momentum

  3. Jacobian Integral • The quantity E is the Jacobian integral of the motion. • Constant when L does not contain time Use

  4. Conjugate Variables • The pj are generalized momenta. • Units not always ML/T • Product with generalized position has dimensions of action • The variables qj, pj are conjugate variables • Use them to define the Jacobian integral • This is the Hamiltonian Action: ML2/T Action: ML2/T

  5. Legendre Transformation • Line of variable slope f1 • Depends on new variable z • Maximize f2 to find y*(z). • Variable x is passive

  6. An incremental change in the Lagrangian can be expanded Express as an incremental change in H. The variation does not depend on variations in generalized velocity. Incremental Change

  7. Canonical Equations • The independence on velocity defines a new function. • The Hamiltonian functional H(q, p, t) • Expand dH and match. • These are canonical conjugate equations

  8. Lagrangian system Number of equations: f Second order diff eqns Require 2f constants Positions and velocities Points are in configuration space and tangent bundle Hamiltonian system Number of equations: 2f +1 First order diff eqns Require 2f +1 constants Velocities come from p Points are in phase space Lagrange vs. Hamilton next

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