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8.09(F14) Chapter 4: Canonical Transformations, Hamilton ...

Chapter 4 Canonical Transformations, Hamilton - jacobi equations , andAction-Angle VariablesWe ve made good use of the Lagrangian formalism. Here we ll study dynamics with theHamiltonian formalism. Problems can be greatly simplified by a good choice of generalizedcoordinates. How far can we push this?Example: Let us imagine that we find coordinatesqithat are all cyclic. Thenp i= 0, sopi= iare all constant. IfHis conserved, then:H=H( 1,.., n)( )is also constant in time. In such a case the remaining equations of motion: Hq i== i( )= i qi it+ i( ) All coordinates are linear in time and the motion becomes very might imagine searching for a variable transformation to make as many coordinates aspossible cyclic.

Chapter 4 Canonical Transformations, Hamilton-Jacobi Equations, and ... Recall the the Euler-Lagrange equations are invariant when: 60. CHAPTER 4. CANONICAL TRANSFORMATIONS, HAMILTON-JACOBI ... where the Hamiltons equations for the evolution of the canonical variables (q;p) are satis ed: @H q_ i= @H and p_ i = @p. i

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  Chapter, Transformation, Equations, Chapter 4, Hamilton, Canonical, Lagrange, S equations, Lagrange equations, Canonical transformations, Chapter 4 canonical transformations, Hamilton jacobi equations, Jacobi

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Transcription of 8.09(F14) Chapter 4: Canonical Transformations, Hamilton ...

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