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Hamiltonian
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Generalized Momentum The generalized momentum was defined from the Lagrangian. Euler-Lagrange equations can be written in terms of p
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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
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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
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Legendre Transformation Line of variable slope f 1 Depends on new variable z Maximize f 2 to find y*(z). Variable x is passive
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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.
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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
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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
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