Understanding Hamiltonians: From Basics to Conservation in Physics
This lesson covers the fundamentals of Hamiltonians in physics, exploring their role in mechanical systems. It explains how the Hamiltonian ( H ) is formulated as ( H = T + U ) and its relation to the Lagrangian method, which yields equations of motion. The session discusses the significance of generalized momentum and the conservation of ( H ) under conditions of explicit time dependence. The interplay between the Lagrangian and the Hamiltonian is emphasized, highlighting that one often must derive the Lagrangian to determine the Hamiltonian effectively.
Understanding Hamiltonians: From Basics to Conservation in Physics
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Presentation Transcript
Physics 321 Hour 20 Hamiltonians
whereas • For simple systems, H = T + U • There are two first order equations of motion for each variable: • The Lagrangian method gives one second order equation for each variable. The Hamiltonian
Take Therefore The Hamiltonian - Origins
Let If has no explicit time dependence, then H is a conserved quantity. The Hamiltonian - Origins
The Hamiltonian is a function of p and q. But p is not ‘the momentum,’ it is the generalized momentum conjugate to q. • The general expression is: • That means you generally have to find the Lagrangian before you can find he Hamiltonian! The Hamiltonian - Notes
The Hamiltonian is H=T+U unless • The Lagrangian has explicit time dependence • The transormation between the coordinatiates and Cartesian coordinates have explicit time dependence • But … you have to write the Hamiltonian in terms of the correct generalized momenta – so you usually still need the Lagrangianfirst! The Hamiltonian - Notes
hamilton.nb Example
