Dynamics

1. Dynamics

2. 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

3. 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

4. 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

5. 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

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

7. 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 next