1. Hamiltonian

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

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

4. Conjugate Variables  The p j are generalized momenta. Units not always ML/TUnits not always ML/T Product with generalized position has dimensions of actionProduct with generalized position has dimensions of action  The variables q j, p j are conjugate variables Use them to define the Jacobian integralUse them to define the Jacobian integral This is the HamiltonianThis is the Hamiltonian Action: ML 2 /T

5. Legendre Transformation  Line of variable slope f 1 Depends on new variable z  Maximize f 2 to find y*(z). Variable x is passive

6. Incremental Change  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.The variation does not depend on variations in generalized velocity.

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

8. Lagrange vs. Hamilton  Lagrangian system  Number of equations: f Second order diff eqnsSecond order diff eqns Require 2f constantsRequire 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 next