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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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.
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 U
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
An incremental change in the Lagrangian can be expanded
Express as an incremental change in H.
Independent of generalized velocity changes
The Hamiltonian can be directly expanded.
Each differential term matches
These are Hamiltonâ€™s canonical equations.
Lagrangian system: f equations
Hamiltonian system: 2f +1 equations