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Eric Prebys, FNAL

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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 USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 2

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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 USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 3 *Nice treatment in Reiser, “Theory and Design of Charged Particle Beams” Potential energy Kinetic Energy

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Lagrangian Equations of motion In other words Lagrangian mechanics is really just a turnkey way to do energy conservation in arbitrary coordinate systems. USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 4

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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 USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 5 Lorentz Gauge Homework

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We want to find a relativistically correct Lagrangian. Assume for now In Cartesian coordinates, we have eg. USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 6

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Make the substitution Check in Cartesian coordinates for B=0 More generally USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 7

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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 USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 8 canonical momentum ordinary momentum

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Introduce “Hamiltonian” We take the total differential of both sides Equating the LHS and RHS gives us USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 9 LHS RHS Hamilton’s Equations of motion

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From the last equation, we have In other words, the Hamiltonian is conserved if there is no explicit time dependence of the Lagrangian. USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 10

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Recall In Cartesian coordinates USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 11 Total Energy

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In order to apply Hamilton’ equations, we must express the Hamiltonian in terms of canonical, rather than mechanical momentum USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 12 Remember this forever!

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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 USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 13

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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 USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 14

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Because q and Q are independent variables, the coefficients must vanish. F 1 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: USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 15 solve for p and P in terms of q and Q Hamiltonian in terms of new variables In all cases

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USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 16 We know the Hamiltonian is and change variables to we want the old momentum in terms of the new and old coordinate

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USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 17 So we have J has units of Energy*time “action” Phase angle 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.

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USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 18 Assume we have a system with solutions x 0 and y 0, 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

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USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 19 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

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USPAS, Knoxville, TN, Jan , 2014 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!

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USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 21 Recall we showed that Canonical momentum! We recall our coordinate system from an earlier lecture Reference trajectory Particle trajectory And define canonical s momentum and vector potential as Use new symbol

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USPAS, Knoxville, TN, Jan , 2014 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

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USPAS, Knoxville, TN, Jan , 2014 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

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USPAS, Knoxville, TN, Jan , 2014 Lecture 10 - Hamiltonian Formalism 24 We showed that the first few terms of the magnetic field are dipole quadrupole sextupole We have You can show (homework) that this is given by We have

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USPAS, Knoxville, TN, Jan , 2014 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= ρ 0, so same answer we got before

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USPAS, Knoxville, TN, Jan , 2014 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

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USPAS, Knoxville, TN, Jan , 2014 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

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