Dynamics. EL with Momentum  The generalized momentum was defined from the Lagrangian.  Euler-Lagrange equations can be written in terms of p. No dissipative.

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Presentation transcript:

Dynamics

EL with Momentum  The generalized momentum was defined from the Lagrangian.  Euler-Lagrange equations can be written in terms of p. No dissipative forcesNo dissipative forces  The Hamiltonian can also be expressed with generalized momentum.

 A curve f(x) can be defined in terms of its derivatives. Slope and intercept of tangent  Find a new function g in terms of new variable z. Function at maximum Change of Variable

Legendre Transformation  The Legendre transform replaces one variable with another based on the derivative. Transform is own inverse Partial derivatives for multiple variables  Thermodynamics uses the transform for energy. Enthalpy H Internal energy U

Hamiltonian Variables  The Legendre transformation links the Hamiltonian to the Lagrangian. Independent variables q, pIndependent variables q, p Velocity a dependent variableVelocity a dependent variable  The Hamiltonian should be written in terms of its independent variables Replace velocity with momentumReplace velocity with momentum

Incremental Change  An incremental change in the Lagrangian can be expanded  Express as an incremental change in H. Independent of generalized velocity changes

Canonical Equations  The Hamiltonian can be directly expanded. Each differential term matchesEach differential term matches  These are Hamilton’s canonical equations. Lagrangian system: f equationsLagrangian system: f equations Hamiltonian system: 2f +1 equationsHamiltonian system: 2f +1 equations next