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6.9: A Simple Example

( \newcommand{\kernel}{\mathrm{null}\,}\)

For a particle moving in a potential in one dimension, .

Hence

Therefore

(Of course, this is just the total energy, as we expect.)

The Hamiltonian equations of motion are

So, as we’ve said, the second order Lagrangian equation of motion is replaced by two first order Hamiltonian equations. Of course, they amount to the same thing (as they must!): differentiating the first equation and substituting in the second gives immediately , the original Newtonian equation (which we derived earlier from the Lagrange equations).


This page titled 6.9: A Simple Example is shared under a not declared license and was authored, remixed, and/or curated by Michael Fowler.

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