A particle's motion is governed by with and ,

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch The Hamiltonian, phase space and the variational principle in phase space

Problem

A particle's motion is governed by

L=\alpha\sqrt{\dot x^2+\dot y^2}-V(x,y)

with

\alpha=3.0\,\mathrm{kg\,m\,s^{-1}}

and

V(x,y)=\beta(x^2+y^2)

\beta=0.50\,\mathrm{N\,m^{-1}}

. Someone argues: "

L

has the form

T-V

with

T=\alpha\sqrt{\dot x^2+\dot y^2}

, and

H

is always the energy

T+V

, so here

H=\alpha\sqrt{\dot x^2+\dot y^2}+\beta(x^2+y^2)

, a fixed number equal to whatever value it has initially." ((a)) Find the error in this argument, naming the relation the momenta obey and what it implies for the Legendre transform. ((b)) Given that the particle starts at

(x,y)=(2.0,0)\,\mathrm{m}

, state what quantity actually stays fixed along the motion (not the claimed

H

), and give its numerical value.

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