Symplectic System

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_kin(p) + V(q)

which gives the equations of motion:

q'(t) = dH_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.

Notation

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

Valid expressions

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`