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- [/============================================================================
- 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]
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