Lagrangian and Hamiltonian Mechanics as Dual Analytic Formulations
AnalyticalMechanics LagrangianMechanics HamiltonianMechanics
02:00 Abide by Reason frames Lagrangian and Hamiltonian mechanics for learners who have seen Newton’s force laws and energy conservation but not the analytic reformulations based on scalar quantities.
Legendre Transform as Bridge Between Lagrangian and Hamiltonian
LegendreTransform LagrangianToHamiltonian DualDescriptions
10:00 This note is for learners trying to understand the mathematical relationship between (L(q,\dot{q})) and (H(q,p)), beyond memorizing the formula (H = p\dot{q} - L).
Generalized Momentum Beyond mv and Relativistic Example
GeneralizedMomentum RelativisticMechanics NonNewtonianMomentum
17:00 Abide by Reason addresses the misconception that momentum is always (p = mv), showing how generalized momentum arises from the Lagrangian.
Hamilton’s Equations, Phase-Space Flows, and Energy Level Curves
HamiltonsEquations PhaseSpace EnergyLevels
22:30 Abide by Reason explains Hamilton’s equations and phase-space geometry for learners moving beyond configuration-space pictures.
When to Prefer Lagrangian vs Hamiltonian Formulations in Practice
MethodChoice AnalyticalTradeoffs NumericalConsiderations
28:00 Abide by Reason offers practical guidance for students deciding which analytic framework to use on real mechanics problems.