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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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
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.
Internal energy ULegendre Transformation
Independent variables q, p
Velocity a dependent variable
The Hamiltonian should be written in terms of its independent variables
Replace velocity with momentumHamiltonian Variables
Express as an incremental change in H.
Independent of generalized velocity changesIncremental Change
The Hamiltonian can be directly expanded. Lagrangian.
Each differential term matches
These are Hamilton’s canonical equations.
Lagrangian system: f equations
Hamiltonian system: 2f +1 equationsCanonical Equations