Chapter 8: The Hamilton Equations of Motion. Section 8.1: Legendre Transformations and the Hamilton Equations of Motion, Part I

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Ch. 8 Hamilton Equations of MotionSect. 8.1
Legendre Transformations

• Lagrange Eqtns of motion n degrees of freedom
• (d/dt)(?L/?qi) - (?L/?qi) 0 (i 1,2,3,
  n)
• n 2nd order, time dependent, differential
  equations.
• ? The system motion is determined for all time
  when 2n initial values are specified n qis n
  qis
• We can represent the state of the system motion
  by the time dependent motion of a point in an
  abstract n-dimensional configuration space
  (coords n generalized coords qi).
• PHYSICS In the Lagrangian Formulation of
  Mechanics, a system with n degrees of freedom a
  problem in n independent variables qi(t). The
  generalized velocities, qi(t) are simply
  determined by taking the time derivatives of the
  qi(t). The velocities are not independent
  variables.

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Hamiltonian Formulation of Mechanics

• Hamiltonian Mechanics A fundamentally different
  picture!
• Describes the system motion in terms of 1st
  order, time dependent equations of motion. The
  number of initial conditions is, of course, still
  2n.
• We must describe the system motion with 2n
  independent 1st order, time dependent,
  differential equations expressed in terms of 2n
  independent variables.
• We choose n of these n generalized coordinates
  qi.
• We choose the other n n generalized (conjugate)
  momenta pi.

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• Hamiltonian Mechanics
• Describes the system motion in terms of n
  generalized coordinates qj n generalized
  momenta pi. It gets 2n 1st order, time dependent
  equations of motion.
• Recall that by DEFINITION The generalized
  Momentum associated with the generalized
  coordinate qj
• pi ? (?L/?qi)
• (q,p) ? conjugate or canonical variables.
• See footnote, p 338, which discusses the
  historical origin of the word canonical.

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Legendre Transformations

• Physically, the Lagrange formulation assumes the
  coordinates qi are independent variables the
  velocities qi are dependent variables only
  obtained by taking time derivatives of the qi
  once the problem is solved!
• Mathematically, the Lagrange formalism treats qi
  qi as independent variables. e.g., in
  Lagranges equations, (?L/?qi) means take the
  partial derivative of L with respect to qi
  keeping all other qs ALSO all qs constant.
  Similarly (?L/?qi) means take the partial
  derivative of L with respect to qi keeping all
  other qs ALSO all qs constant.
• Treated as a pure math problem, changing from the
  Lagrange formulation to the Hamilton formulation
  corresponds to changing variables from (q,q,t)
  (q,q, independent) to (q,p,t) (q,p independent)

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• To change from the Lagrange to the Hamilton
  formulation
• Change (or transform) variables from (q,q,t)
  (q,q, independent) to (q,p,t) (q,p independent).
• Mathematicians call such a procedure a Legendre
  Transformation. Pure math for a while
• Consider a function f(x,y) of 2 independent
  variables (x,y)
• The exact differential of f df ? u dx v dy
• Obviously u ? (?f/?x) v ? (?f/?y)
• Now, change variables to u y, so that the
  differential quantities are expressed in terms of
  du dy.
• Let g g(u,y) be a function defined by g ? f -
  ux

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• Change from f(x,y) ? df ? u dx v dy
• u ? (?f/?x) v ? (?f/?y)
• To g(u,y) ? f - ux. The exact differential of
  g
• dg ? df - u dx - x du v dy x du
• Obviously v ? (?g/?x) x ? - (?g/?u)
• This is a Legendre Transformation. Such
  transformations are used often in thermodynamics.
  See examples in Goldstein, pp 336 337.

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• Change from the Lagrange to the Hamilton
  formulation.
• ? Changing variables from (q,q,t) (q,q,
  independent) to
• (q,p,t) (q,p independent) is a Legendre
  Transformation.
• However, its one where many variables are
  involved instead of just 2.
• Consider the Lagrangian L L(q,q,t) (n qs, n
  qs)
• The exact differential of L (sum on i)
• dL ? (?L/?qi)dqi (?L/?qi)dqi (?L/?t)dt
  (1)
• Canonical Momentum is defined (d/dt)(?L/?qi) -
  (?L/?qi) 0
• pi ? (?L/?qi) ? pi ? (?L/?qi)
  (2)
• Put (2) into (1) ? dL pi dqi pi dqi
  (?L/?t)dt (3)

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• dL pi dqi pi dqi (?L/?t)dt
  (3)
• Define the Hamiltonian H by the Legendre
  Transformation (sum on i)
• H(q,p,t) ? qipi L(q,q,t) (4)
• ? dH qidpi pidqi dL (5)
• Combining (3) (5)
• ? dH qidpi - p dqi - (?L/?t)dt
  (6)
• Since H H(q,p,t) we can also write
• dH ? (?H/?qi)dqi (?H/?pi)dpi (?H/?t)dt
  (7)
• Directly comparing (6) (7)
• ? qi ? (?H/?pi), - pi ? (?H/?qi), - (?L/?t) ?
  (?H/?t)

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• Hamiltonian H (sum on i)
• H(q,p,t) ? qi pi L(q,q,t)
  (a)
• ? qi ? (?H/?pi)
  (b)
• - pi ? (?H/?qi)
  (c)
• - (?L/?t) ? (?H/?t)
  (d)
• (b) (c) together
• ? Hamiltons Equations of Motion
• or the Canonical Equations of Hamilton
• 2n 1st order, time dependent equations of motion
  replacing the n 2nd order Lagrange Equations of
  motion

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Discussion of Hamiltons Eqtns

• Hamiltonian H(q,p,t) qipi L(q,q,t)
  (a)
• Hamiltons Equations of Motion
• qi (?H/?pi) (b), - pi (?H/?qi) (c), -(?L/?t)
  (?H/?t) (d)
• 2n 1st order, time dependent equations of motion
  replacing the n 2nd order Lagrange Eqtns of
  motion.
• (a) A formal definition of the Hamiltonian H in
  terms of the Lagrangian L. However, as well see,
  in practice, we neednt know L first to be able
  to construct H.
• (b) qi (?H/?pi) Gives qis as functions of
  (q,p,t).
• ? Given initial values, integrate to get qi
  qi(q,p,t) ? They form the inverse relations of
  the equations
• pi (?L/?qi) which give pi pi(q,q,t). ?
  No new information.

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• Hamiltonian H(q,p,t) qipi L(q,q,t)
  (a)
• Hamiltons Equations of Motion
• qi (?H/?pi) (b), -pi (?H/?qi) (c), - (?L/?t)
  (?H/?t) (d)
• (b) qi (?H/?pi) ? qi qi(q,p,t). ? No new
  info.
• This is true in terms of SOLVING mechanics
  problems. However, within the Hamiltonian picture
  of mechanics, where H H(q,p,t) is obtained NO
  MATTER HOW (not necessarily by (a)), this has
  equal footing ( contains equally important
  information as (c)).
• (c) pi - (?H/?qi)
• ? Given the initial values, integrate to get pi
  pi(q,p,t)
• (d) -(?L/?t) (?H/?t) This is obviously only
  important in time dependent problems!

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• Recall the energy function h from Ch. 2 (Eq.
  (2.53) Define the Energy Function h
• h ? qi(?L/?qi) - L h(q1,..qn,q1,..qn,t)
• The Hamiltonian H the energy function h have
  identical (numerical) values. However, they are
  functions of different variables!
• h ? h(q,q,t) while H H(q,p,t)
• NOTE!!!!! A proper Hamiltonian (for use in
• Hamiltonian dynamics) is ALWAYS (!!!!) written as
  a function
• of the generalized coordinates momenta H ?
  H(q,p,t).
• Similarly, a proper Lagrangian (for use in
  Lagrangian
• dynamics) is ALWAYS (!) written as a function of
  the
• generalized coordinates velocities L ? L(q,q,t)

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• TO EMPHASIZE THIS
• Consider a single free particle (p mv)
• Energy KE T (½)mv2 only.
• So, h T and H T
• But, if it is a PROPER HAMILTONIAN (!!!), can it
  be written H (½)mv2 ?
• NO!!!!!! H MUST be expressed in terms of the
  momentum p, NOT the velocity v!
• So the PROPER HAMILTONIAN(!!!) is
• H p2/(2m) !!!!!

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Recipe for Hamiltonian Mechanics

• Hamiltonian H(q,p,t) qipi L(q,q,t)
  (a)
• Hamiltons Equations of
  Motion
• qi (?H/?pi) (b), - pi (?H/?qi) (c), -
  (?L/?t) (?H/?t) (d)
• Recipe (CONSERVATIVE FORCES!)
• 1. Set up the Lagrangian, L T V L(q,q,t)
  
• 2. Compute n conjugate momenta using pi ?
  (?L/?qi)
• 3. Form the Hamiltonian H from (a).
• This is of the mixed form H H(q,q,p,t)
• 4. Invert the n pi ? (?L/?qi) to get qi ?
  qi(q,p,t).
• 5. Apply the results of 4 to eliminate the qi
  from H to
• get a proper Hamiltonian H H(q,p,t).
• Then only then can you properly
  correctly use (b) (c) to get the equations of
  motion!

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• If you think that this is a long, tedious
  process, you arent alone! Personally, this is
  why I prefer the Lagrange method!
• This requires that you set up the Lagrangian
  first!
• If you already have the Lagrangian, why not go
  ahead do Lagrangian dynamics instead of going
  through all of this to do Hamiltonian dynamics?
• Further, combining the 2n 1st order differential
  equations of motion
• qi (?H/?pi) (b) - pi (?H/?qi) (c)
• gives the SAME n 2nd order differential
  equations of
• motion that Lagrangian dynamics gives!
• However, for many physical systems of interest,
  it is fortunately possible to considerably
  shorten this procedure, even eliminating many
  steps completely!

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Hamiltonian Mechanics (In Most Cases of
Interest!)

• Weve seen (in Ch. 2) that in many cases The
  Lagrangian a sum of functions which are
  homogeneous in the generalized velocities of
  degree 0, 1, 2. That is (schematically)
• L L0(q,t) L1(q,t)qk
  L2(q,t)qkqm
• Use this form to construct the Hamiltonian
• H qipi L(q,q,t)
• ? H qipi L0(q,t) L1(q,t)qk
  L2(q,t)qkqm
• Weve also seen (in Ch. 2) that in many cases
  The equations defining the generalized
  coordinates dont depend on the time explicitly
• ? L2(q)qkqm T (the kinetic
  energy) L1 0
• Weve also seen (in Ch. 2) that in many cases
  The forces are conservative a potential V
  exists ? L0 - V

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• Weve seen (in Ch. 2) that in many cases All of
  the conditions on the previous slide hold
  simultaneously.
• ? H T V
• That is, in this case, the Hamiltonian is
  automatically the total mechanical energy E
• If that is the case, we can skip many steps of
  the recipe and write H T V immediately.
  Express T in terms of the MOMENTA pi (not the
  velocities qi !)! Often it is easy to see how the
  pi depend on the qi thus its easy to do this.
  Once this is done, we can go ahead do
  Hamiltonian Dynamics without ever having written
  the Lagrangian down!!

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• Often, we can go further! For large classes of
  problems, the 1st 2nd degree Lagrangian terms
  can be written (sum on i)
• L1(q,t)qk L2(q,t)qkqm qiai(q,t)
  (qi)2Ti(q,t)
• So
• L L0(q,t) qiai(q,t) (qi)2Ti(q,t) (A)
• If the Lagrangian can be written in the form of
  (A), we can do the algebraic manipulations in
  steps 2-5 in the recipe in general, once for
  all. Do this by matrix manipulation!