Geometric Numerical Integration 1. Hamiltonian mechanics and numerical methods Mario Fernandez-Pend´ as´ Universidad de Oviedo/Universida´ d’Uvieu´ BCAM, May 20-24 2019 Motivation I This course is devoted to the study of numerical techniques used for simulating dynamical systems, especially conservative systems. The emphasis of this course is on Hamiltonian systems I In a dynamical simulation, an integrator replaces a differential equation in continuous time by a difference equation defining approximate captures of the solution at discrete time steps. I More precisely, geometric integrators are numerical methods that preserve geometric properties of the exact flow of a differential equation. 2 Objectives I Summary of Hamiltonian mechanics, and some well-known numerical methods and concepts related. I Discussion of the geometric structure of the Hamiltonians systems and why symplectic integrators are interesting. I Introduction to modified equations as a basic tool to study symplectic integrators. I Summary of constrained mechanical systems relating them to realistic applications. I Presentation of some recent developments in the field on molecular dynamics using adaptive integrators. 3 Index course 1. Hamiltonian mechanics and numerical methods 2. Symplectic integration 3. Modified equations 4. Constrained mechanical systems 5. Adaptive geometric integrators 4 Preliminaries 5 Preliminaries We are concerned with autonomous systems of differential D equations in R d x = f (x); (1) dt where the function (vector field) f is assumed throughout to be D defined in the whole of R and to be sufficiently smooth. 6 This is not the most general autonomous system of second order d differential equations in R because the derivative dq=dt does not appear in the right-hand side. Preliminaries An important role is played by the particular case d d q = M−1p; p = F(q); (2) dt dt D d d where x = (p; q) 2 R , D = 2d, q 2 R , p 2 R and M is a constant invertible matrix. So that, f = (M−1p; F(q)). By eliminating p, (2) is seen to be equivalent to d2 M q = F(q): dt2 7 Preliminaries An important role is played by the particular case d d q = M−1p; p = F(q); (2) dt dt D d d where x = (p; q) 2 R , D = 2d, q 2 R , p 2 R and M is a constant invertible matrix. So that, f = (M−1p; F(q)). By eliminating p, (2) is seen to be equivalent to d2 M q = F(q): dt2 This is not the most general autonomous system of second order d differential equations in R because the derivative dq=dt does not appear in the right-hand side. 7 These are called Hamilton equations of motion. Preliminaries When the forces depend only on the positions, Newton’s second law for a mechanical system gives rise to differential equations of the form (2). Then, q, (d=dt)q, p, F are respectively the vectors of coordinates, velocities, momenta, and forces, and M is the matrix of masses. 8 Preliminaries When the forces depend only on the positions, Newton’s second law for a mechanical system gives rise to differential equations of the form (2). Then, q, (d=dt)q, p, F are respectively the vectors of coordinates, velocities, momenta, and forces, and M is the matrix of masses. These are called Hamilton equations of motion. 8 Preliminaries Definition D D For a fixed real t, 't : R ! R is the t-flow of the system under consideration d x = f (x); (3) dt D and 't is the map that associates with each α 2 R the value at time t of the solution of (3) that at the initial time 0 takes the initial value α. 9 Preliminaries Example. The harmonic oscillator As a very simple but important example, we consider the standard harmonic oscillator, the system in R2 of the special form (2) given by d d q = p; p = −q: (4) dt dt For future reference, we note that, with matrix notation, the solutions satisfy1: q(t) q(0) cos t sin t = M ; M = : (5) p(t) t p(0) t − sin t cos t 1Sanz-Serna and Calvo, Numerical Hamiltonian problems, 1994 10 Preliminaries Example. The harmonic oscillator Thus, the flow has the expression 't(ξ; η) = (ξ cos t + η sin t; −ξ sin t + η cos t) : (6) When α = (ξ; η) 2 R2 is fixed and t varies, the right-hand side of (6) yields the solution that at t = 0 takes the initial value (ξ; η). The notation 't(ξ; η) emphasizes that, in the flow, it is the parameter t that is seen as fixed, while (ξ; η) 2 R2 is regarded as a variable. Geometrically, 't is the clockwise rotation of angle t around the origin of the (q; p)-plane. 11 For simplicity, we shall assume hereafter that 't(α) is always defined. Preliminaries For a given system d x = f (x); (7) dt D it is possible that for some choices of α and t, the vector 't(α) 2 R is not defined; this will happen if t is outside the interval in which the solution of (7) with initial value α exists. 12 Preliminaries For a given system d x = f (x); (7) dt D it is possible that for some choices of α and t, the vector 't(α) 2 R is not defined; this will happen if t is outside the interval in which the solution of (7) with initial value α exists. For simplicity, we shall assume hereafter that 't(α) is always defined. 12 Preliminaries Flows possess the group property: I For arbitrary real s and t, 't ◦ 's = 's+t; I for arbitrary real s, t and u, ('t ◦ 's) ◦ 'u = 'u+(s+t) = '(u+s)+t = 't ◦ ('s ◦ 'u); I '0 is the indentity map; I and, for each t, −1 ('t) = '−t i.e., '−t is the inverse of the map 't. 13 Hamiltonian systems 14 Hamiltonian systems Let us assume that the dimension D of d x = f (x) (8) dt d is even, D = 2d, and let us write x = (q; p) with q; p 2 R . Then, the system (8) is said to be Hamiltonian if there is a function d i H : R2 ! R such that, for i = 1;:::; d, the scalar components f of f are given by @H @H f i(q; p) = (q; p); f d+i(q; p) = − (q; p): @pi @qi 15 Hamiltonian systems Thus, the system is d @H d @H qi = (q; p); pi = − (q; p); dt @pi dt @qi or, in vector notation2, d q = J−1rH(q; p); (9) dt p where @H @H @H @H T rH = ;:::; ; ;:::; @qi @qd @p1 @pd and 0 −I J = d×d d×d : Id×d 0d×d 2Sanz-Serna and Calvo, Numerical Hamiltonian problems, 1994 16 Theorem3 A system of the special form d d q = M−1p; p = F(q); dt dt is Hamiltonian if and only if F = −∇U(q) for a suitable real-valued function U, i.e., d d q = M−1p; p = −∇U(q): dt dt Hamiltonian systems d The function H is called the Hamiltonian, R2 is the phase space, and d is the number of degrees of fredoom. 3Bou-Rabee and Sanz-Serna, “Geometric integrators and the Hamiltonian Monte Carlo method”, 2018 17 Hamiltonian systems d The function H is called the Hamiltonian, R2 is the phase space, and d is the number of degrees of fredoom. Theorem3 A system of the special form d d q = M−1p; p = F(q); dt dt is Hamiltonian if and only if F = −∇U(q) for a suitable real-valued function U, i.e., d d q = M−1p; p = −∇U(q): dt dt 3Bou-Rabee and Sanz-Serna, “Geometric integrators and the Hamiltonian Monte Carlo method”, 2018 17 Hamiltonian systems In the case of the theorem before, H(q; p) = K(p) + U(q); (10) where 1 K(p) = pT M−1p: 2 In mechanics, K and U are respectively the kinetic and potential energy, and H represents the total energy in the system. The harmonic oscillator (4) provides the simplest example: 1 1 K = p2; U = q2: 2 2 18 Hamiltonian systems Example. The Lennard-Jones oscillator4 Another problem of interest is the anharmonic Lennard-Jones oscillators with Hamiltonians of the form H(q; p) = p2 + U(q), where " # σ 12 σ 6 U(q) = 4 − ; q q where is the depth of the potential well, σ is the finite distance at which the inter-particle potential is zero (in this case σ = 1), and q is the distance between the particles. In general, the Lennard-Jones potential approximates the interaction between a pair of neutral atoms or molecules. 4Leimkuhler and Reich, Simulating Hamiltonian dynamics, 2004. 19 Hamiltonian systems Example. N-body problem This model describes an homogeneous system of N bodies 3 moving in R with masses mi, i = 1;:::; N, interacting through a particle-particle interaction potential (pair-potential) U(r), with r the distance between two particles. The corresponding equations of motion are d −1 d X rU(rij) qi = mi pi; pi = − (qi − qj); i = 1; 2;:::; N; dt dt rij i6=j where rij = kqi − qjk. The Hamiltonian is N 2 N−1 N 1 X kpik X X H(q; p) = + U(rij): (11) 2 mi i=1 i=1 j=i+1 20 Hamiltonian systems Example. Molecular dynamics5 Molecular dynamics simulations require the solution of Hamiltonian systems with N degrees of freedom d d q = M−1p; p = −∇U(q); dt dt where the total energy is given by (11) and U are given potential energies.