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

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Presentation Transcript
el with momentum
The generalized momentum was defined from the Lagrangian.

Euler-Lagrange equations can be written in terms of p.

No dissipative forces

The Hamiltonian can also be expressed with generalized momentum.

EL with Momentum
change of variable
Change of Variable
  • 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
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

Legendre Transformation
hamiltonian variables
The Legendre transformation links the Hamiltonian to the Lagrangian.

Independent variables q, p

Velocity a dependent variable

The Hamiltonian should be written in terms of its independent variables

Replace velocity with momentum

Hamiltonian Variables
incremental change
An incremental change in the Lagrangian can be expanded

Express as an incremental change in H.

Independent of generalized velocity changes

Incremental Change
canonical equations
The Hamiltonian can be directly expanded.

Each differential term matches

These are Hamilton’s canonical equations.

Lagrangian system: f equations

Hamiltonian system: 2f +1 equations

Canonical Equations

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