boost/libs/numeric/odeint/doc/concepts/symplectic_system.qbk

[/============================================================================
  Boost.odeint

  Copyright 2011 Mario Mulansky
  Copyright 2011-2012 Karsten Ahnert

  Use, modification and distribution is subject to the Boost Software License,
  Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt)
=============================================================================/]


[section Symplectic System]

[heading Description]

This concept describes how to define a symplectic system written with generalized coordinate `q` and generalized momentum `p`:

[' q'(t) = f(p) ]

[' p'(t) = g(q) ]

Such a situation is typically found for Hamiltonian systems with a separable Hamiltonian:

[' H(p,q) = H[sub kin](p) + V(q) ]

which gives the equations of motion:

[' q'(t) = dH[sub kin] / dp = f(p) ]

[' p'(t) = dV / dq = g(q) ]


The algorithmic implementation of this situation is described by a pair of callable objects for /f/ and /g/ with a specific parameter signature.
Such a system should be implemented as a std::pair of functions or a functors.
Symplectic systems are used in symplectic steppers like `symplectic_rkn_sb3a_mclachlan`.

[heading Notation]

[variablelist
  [[`System`] [A type that is a model of SymplecticSystem]]
  [[`Coor`] [The type of the coordinate  ['q]]]
  [[`Momentum`] [The type of the momentum ['p]]]
  [[`CoorDeriv`] [The type of the derivative of coordinate ['q']]]
  [[`MomentumDeriv`] [The type of the derivative of momentum ['p']]]
  [[`sys`] [An object of the type `System`]]
  [[`q`] [Object of type Coor]]
  [[`p`] [Object of type Momentum]]
  [[`dqdt`] [Object of type CoorDeriv]]
  [[`dpdt`] [Object of type MomentumDeriv]]
]

[heading Valid expressions]

[table
  [[Name] [Expression] [Type] [Semantics]]
  [[Check for pair] [`boost::is_pair< System >::type`] [`boost::mpl::true_`] [Check if System is a pair]]
  [[Calculate ['dq/dt = f(p)]] [`sys.first( p , dqdt )`] [`void`] [Calculates ['f(p)], the result is stored into `dqdt`] ]
  [[Calculate ['dp/dt = g(q)]] [`sys.second( q , dpdt )`] [`void`] [Calculates ['g(q)], the result is stored into `dpdt`] ]
]

[endsect]


[section Simple Symplectic System]

[heading Description]

In most Hamiltonian systems the kinetic term is a quadratic term in the momentum ['H[sub kin] = p^2 / 2m] and in many cases it is possible to rescale coordinates and set /m=1/ which leads to a trivial equation of motion:

[' q'(t) = f(p) = p. ]

while for /p'/ we still have the general form

[' p'(t) = g(q) ]

As this case is very frequent we introduced a concept where only the nontrivial equation for /p'/ has to be provided to the symplectic stepper.
We call this concept ['SimpleSymplecticSystem]

[heading Notation]

[variablelist
  [[System] [A type that is a model of SimpleSymplecticSystem]]
  [[Coor] [The type of the coordinate  ['q]]]
  [[MomentumDeriv] [The type of the derivative of momentum ['p']]]
  [[sys] [An object that models System]]
  [[q] [Object of type Coor]]
  [[dpdt] [Object of type MomentumDeriv]]
]

[heading Valid Expressions]

[table
  [[Name] [Expression] [Type] [Semantics]]
  [[Check for pair] [`boost::is_pair< System >::type`] [`boost::mpl::false_`] [Check if System is a pair, should be evaluated to false in this case.]]
  [[Calculate ['dp/dt = g(q)]] [`sys( q , dpdt )`] [`void`] [Calculates ['g(q)], the result is stored into `dpdt`] ]
]

[endsect]