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99 lines
3.3 KiB
Plaintext
99 lines
3.3 KiB
Plaintext
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[/============================================================================
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Boost.odeint
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Copyright 2011 Mario Mulansky
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Copyright 2011-2012 Karsten Ahnert
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Use, modification and distribution is subject to the Boost Software License,
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Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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=============================================================================/]
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[section Symplectic System]
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[heading Description]
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This concept describes how to define a symplectic system written with generalized coordinate `q` and generalized momentum `p`:
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[' q'(t) = f(p) ]
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[' p'(t) = g(q) ]
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Such a situation is typically found for Hamiltonian systems with a separable Hamiltonian:
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[' H(p,q) = H[sub kin](p) + V(q) ]
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which gives the equations of motion:
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[' q'(t) = dH[sub kin] / dp = f(p) ]
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[' p'(t) = dV / dq = g(q) ]
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The algorithmic implementation of this situation is described by a pair of callable objects for /f/ and /g/ with a specific parameter signature.
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Such a system should be implemented as a std::pair of functions or a functors.
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Symplectic systems are used in symplectic steppers like `symplectic_rkn_sb3a_mclachlan`.
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[heading Notation]
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[variablelist
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[[`System`] [A type that is a model of SymplecticSystem]]
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[[`Coor`] [The type of the coordinate ['q]]]
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[[`Momentum`] [The type of the momentum ['p]]]
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[[`CoorDeriv`] [The type of the derivative of coordinate ['q']]]
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[[`MomentumDeriv`] [The type of the derivative of momentum ['p']]]
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[[`sys`] [An object of the type `System`]]
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[[`q`] [Object of type Coor]]
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[[`p`] [Object of type Momentum]]
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[[`dqdt`] [Object of type CoorDeriv]]
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[[`dpdt`] [Object of type MomentumDeriv]]
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]
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[heading Valid expressions]
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[table
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[[Name] [Expression] [Type] [Semantics]]
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[[Check for pair] [`boost::is_pair< System >::type`] [`boost::mpl::true_`] [Check if System is a pair]]
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[[Calculate ['dq/dt = f(p)]] [`sys.first( p , dqdt )`] [`void`] [Calculates ['f(p)], the result is stored into `dqdt`] ]
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[[Calculate ['dp/dt = g(q)]] [`sys.second( q , dpdt )`] [`void`] [Calculates ['g(q)], the result is stored into `dpdt`] ]
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]
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[endsect]
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[section Simple Symplectic System]
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[heading Description]
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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:
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[' q'(t) = f(p) = p. ]
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while for /p'/ we still have the general form
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[' p'(t) = g(q) ]
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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.
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We call this concept ['SimpleSymplecticSystem]
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[heading Notation]
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[variablelist
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[[System] [A type that is a model of SimpleSymplecticSystem]]
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[[Coor] [The type of the coordinate ['q]]]
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[[MomentumDeriv] [The type of the derivative of momentum ['p']]]
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[[sys] [An object that models System]]
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[[q] [Object of type Coor]]
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[[dpdt] [Object of type MomentumDeriv]]
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]
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[heading Valid Expressions]
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[table
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[[Name] [Expression] [Type] [Semantics]]
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[[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.]]
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[[Calculate ['dp/dt = g(q)]] [`sys( q , dpdt )`] [`void`] [Calculates ['g(q)], the result is stored into `dpdt`] ]
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]
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[endsect]
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