Dynamics
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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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Dynamics

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Dynamics

Dynamics


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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