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Hamiltonian Formalism

Hamiltonian Formalism. Eric Prebys , FNAL. Motivation. We have focused largely on a kinematics based approach to beam dynamics. Most people find it more intuitive, at least when first learning the material.

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Hamiltonian Formalism

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  1. Hamiltonian Formalism Eric Prebys, FNAL

  2. Motivation Lecture 10 - Hamiltonian Formalism • We have focused largely on a kinematics based approach to beam dynamics. • Most people find it more intuitive, at least when first learning the material. • However, it’s useful to at least become familiar with more formal Lagrangian/Hamiltonian based approach • Can handle problems too complex for kinematic approach • More common in advanced textbooks and papers • Eventually intuitive

  3. Review* Potential energy Kinetic Energy *Nice treatment in Reiser, “Theory and Design of Charged Particle Beams” Lecture 10 - Hamiltonian Formalism The Lagrangian of a body is defined as Hamilton’s variational principle says that the body will follow a trajectory in time (or other independent variable) which minimizes the “action” Generalized force

  4. Demonstration in Cartesian Coordinates Lecture 10 - Hamiltonian Formalism Lagrangian Equations of motion In other words Lagrangian mechanics is really just a turnkey way to do energy conservation in arbitrary coordinate systems.

  5. E&M Lorentz Gauge Homework Lecture 10 - Hamiltonian Formalism Introduce velocity-dependent force: Lagrange’s equations still hold for We describe the magnetic field in terms of the vector potential The Lorentz force now becomes, eg

  6. Relativistic Version Lecture 10 - Hamiltonian Formalism We want to find a relativistically correct Lagrangian. Assume for now In Cartesian coordinates, we have eg.

  7. Lecture 10 - Hamiltonian Formalism Make the substitution Check in Cartesian coordinates for B=0 More generally

  8. Canonical Momentum ordinary momentum canonical momentum Lecture 10 - Hamiltonian Formalism Lagrange’s equations are second order diff. eq. We will find that it will be useful to specify system in term of twice as many first order diff. eqs. We introduce the “conjugate” or “canonical” momentum In Cartesian coordinates

  9. Hamilton’s Equations LHS RHS Hamilton’s Equations of motion Lecture 10 - Hamiltonian Formalism Introduce “Hamiltonian” We take the total differential of both sides Equating the LHS and RHS gives us

  10. Conservation laws Lecture 10 - Hamiltonian Formalism From the last equation, we have In other words, the Hamiltonian is conserved if there is no explicit time dependence of the Lagrangian.

  11. Particle in an Electromagnetic Field Total Energy Lecture 10 - Hamiltonian Formalism Recall In Cartesian coordinates

  12. Hamiltonian in Canonical Momentum Remember this forever! Lecture 10 - Hamiltonian Formalism In order to apply Hamilton’ equations, we must express the Hamiltonian in terms of canonical, rather than mechanical momentum

  13. Change of Coordinates and Generator Functions Lecture 10 - Hamiltonian Formalism We will often find it useful to express the Hamiltonian in other coordinate systems, and need a turnkey way to generate canonical coordinate/momentum pairs. That is We construct the Lagrangian out of the new coordinates We still want the action principle to hold

  14. Lecture 10 - Hamiltonian Formalism This means that the new and old Lagrangians can differ by at most a total time derivative Let’s first consider a function which depends only on the new and old coordinates Then we must have Expand the total time time derivative at the right and combine terms

  15. solve for p and P in terms of q and Q Hamiltonian in terms of new variables In all cases Lecture 10 - Hamiltonian Formalism Because q and Q are independent variables, the coefficients must vanish. F1 is called the “generating function of the canonical transformation. Rather than choosing (q,Q) as variables, we could have chosen (q,P), (Q,p) or (p,P). The convention is:

  16. Example: Harmonic Oscillator We know the Hamiltonian is and change variables to we want the old momentum in terms of the new and old coordinate Lecture 10 - Hamiltonian Formalism

  17. So we have Phase angle J has units of Energy*time“action” These are known as “action-angle” variables. We will see that this will be very useful for studying systems which are perturbed by the addition of small non-linear terms. Lecture 10 - Hamiltonian Formalism

  18. Deviations from a Periodic System Assume we have a system with solutions x0 and y0, which are periodic with period T Now consider an orbit near the periodic orbit Substituting in and expanding, we get These are the equations one obtains with a Hamiltonian of the form (homework) periodic(!) in time rather than constant Lecture 10 - Hamiltonian Formalism

  19. General case We start with a known system We transform to a system which represents small deviations from this system Use a generating function of the second type integrate Lecture 10 - Hamiltonian Formalism

  20. We can calculate the new Hamiltonian and expand for small deviations about the equilibrium No dependence on Q or P, so can be ignored! It’s important to remember that these coefficients are derivatives of the Hamiltonian evaluated at the unperturbed orbit, so in general they are periodic, but not constant in time! Lecture 10 - Hamiltonian Formalism

  21. Particle Motion Revisited Use new symbol Recall we showed that Canonical momentum! We recall our coordinate system from an earlier lecture Particle trajectory Reference trajectory And define canonical s momentum and vector potential as Lecture 10 - Hamiltonian Formalism

  22. We would like to change our independent variable from t to s. Note We can transform this into a partial derivative by setting the total derivative to zero. In general so new Hamiltonian You can show (homework) that Lecture 10 - Hamiltonian Formalism

  23. Consider a system with no E fields and only B fields in the transverse directions, so there is only an s component to the vector potential In this case, H is the total energy, so normal “kinetic” momentum For small deviations Lecture 10 - Hamiltonian Formalism

  24. We showed that the first few terms of the magnetic field are quadrupole sextupole dipole We have You can show (homework) that this is given by We have Lecture 10 - Hamiltonian Formalism

  25. In the case where we have only vertical fields, this becomes Normalize by the design momentum At the nominal momentum ρ=ρ0, so same answer we got before Lecture 10 - Hamiltonian Formalism

  26. By comparing this to the harmonic oscillator, we can write We have a solution of the form Look for action-angle variables Lecture 10 - Hamiltonian Formalism

  27. Look for a generating function such that Integrate to get In an analogy to the harmonic oscillator, the unperturbed Hamiltonian is Lecture 10 - Hamiltonian Formalism

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