ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: of the solution operator is Note that the exponential in equation ( 6 ) is simply a formal way to write down an operator. It is acting on the function to its right, in the above case this is the identity Id. To see that this construct indeed provides a solution to the differential equation with initial conditions y 0 , note that the series expansion of the exponential results in a series of differential operators acting on Id( y 0 ). This series is simply the Taylor expansion of the solution φ around t = 0. Since it is not possible to write down an explicit solution for the N -body problem, we turn to operator splitting methods. One way to view an operator splitting method is to take a differential equation of the form and rewrite it as Note that due to the linearity of the Poisson bracket, we have . This implies that for Hamiltonian systems we can split the equations of motion by splitting the Hamiltonian H into two parts, i.e. H = A + B . Depending on the splitting, we get two new sets of differential equations that we can study independently: In practice, we will choose a splitting that allows us to easily solve these new differential equations independently. In other words, we want to be able to explicitly write down the solution operators and . Using the notation from above, we have but because these exponentials are operators, not scalars, we unfortunately cannot use standard rules for exponentials. Specifically, we typically have For illustration purposes, imagine the above equation were true. We could then solve the differential equation in equation ( 7 ) by first applying the solution operator φ [ B ] to our initial conditions for a time t (remember, operators act to the right), then take the result and apply the solution operator φ [ A ] on it, again for time t . The result of the last operation would then correspond to the solution φ [ H ] at time t . But because the equation above is not true, we need to rely on the Baker–Campbell–Hausdorff (BCH) identity, 2 to get a similar expression where is a commutator. So if we apply the splitting procedure described above (corresponding to the left-hand side of equation 12 ), we do not solve the differential equation , but some other differential equation (given by the argument in the exponential on the right-hand side of equation 12 ). Formally, we can write down the Hamiltonian corresponding to the differential equation that we are actually solving as where we have used which follows from the Jacobi identity. Although we do not end up solving the Hamiltonian system we were trying to solve, the fact that we can write down a different Hamiltonian system that we seem to solve exactly is often quoted as why symplectic integrators have superior properties. 3 We can study the properties of this ‘nearby’ Hamiltonian to see how far our approximation is from the true solution. As we can see from the equation above, for small t , the Hamiltonian is close to the original Hamiltonian and only