Modified Equations for Variational Integrators

Transcription

1 Modified Equations for Variational Integrators Mats Vermeeren Technische Universität Berlin Groningen December 18, 2018 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

2 Table Of Contents 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

3 Table Of Contents 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

4 Continuous Lagrangian Mechanics Lagrange function: L : R 2N = TQ R : (q, q) L(q, q). Solutions are curves q(t) that minimize (or are critical points of) the action S = t1 t 0 L(q(t), q(t)) dt where the integration interval [t 0, t 1 ] and the boundary values q(t 0 ) and q(t 1 ) are fixed. 0 = δs = = t1 t 0 t1 t 0 L L δq + δ q dt q q ( L q d L dt q ) ( ) L t 1 δq dt + q δq t 0 Euler-Lagrange Equation: L q d L dt q = 0. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

5 Continuous Lagrangian Mechanics Lagrange function: L : R 2N = TQ R : (q, q) L(q, q). Solutions are curves q(t) that minimize (or are critical points of) the action S = t1 t 0 L(q(t), q(t)) dt where the integration interval [t 0, t 1 ] and the boundary values q(t 0 ) and q(t 1 ) are fixed. 0 = δs = = t1 t 0 t1 t 0 L L δq + δ q dt q q ( L q d L dt q ) ( ) L t 1 δq dt + q δq t 0 Euler-Lagrange Equation: L q d L dt q = 0. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

6 Legendre transformation Relates Hamiltonian and Lagrangian formalism: Differentiating w.r.t. q, p and q, p = L q q = H p 0 = H q + L q p q = H(q, p) + L(q, q). = ( ) ( H L q + ṗ + q d ) L, dt q establishes equivalence between Lagrangian and Hamiltonian equations of motion. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

7 Legendre transformation Relates Hamiltonian and Lagrangian formalism: Differentiating w.r.t. q, p and q, p = L q q = H p 0 = H q + L q p q = H(q, p) + L(q, q). = ( ) ( H L q + ṗ + q d ) L, dt q establishes equivalence between Lagrangian and Hamiltonian equations of motion. Only works if the Lagrangian is nondegenerate: 2 L q 2 0 Hamiltonian systems preserve the symplectic 2-form ω = i dp i dq i. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

8 Symplectic structure Let Φ t be the flow of a Hamiltonian system, i.e. Φ 0 (q, p) = (q, p) and ( d H dt Φ t(q, p) = p (Φ t(q, p)), H ) q (Φ t(q, p)). Then for each t, Φ t is a symplectic map, Φ t ω = ω, where ω is the canonical symplectic form ω = i dq i dp i. Definition A symplectic integrator is a discretization (in time) of a Hamiltonian systems, such that each discrete time-step is given by a symplectic map. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

9 Discrete Lagrangian mechanics Lagrange function: L : R N R N R : (x, x) L(x, x). Solutions are discrete curves x = (x 0, x 1,..., x n ) that are critical points of the action n S disc = hl disc (x j 1, x j ) Euler-Lagrange equation: j=1 D 2 L disc (x j 1, x j ) + D 1 L disc (x j, x j+1 ) = 0, where D 1, D 2 denote the partial derivatives of L disc. Definition A variational integrator for a continuous system with Lagrangian L is a discrete Lagrangian system with and hence S disc S. L disc ( x(t h), x(t)) L(x(t), ẋ(t)), Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

10 Equivalence Theorem If the Lagrangian/Hamiltonian is regular, variational and symplectic integrators are equivalent. Proof. The discrete Lagrangian is a generation function of the symplectic map describing one time step, p j = hd 1 L disc (x j, x j+1 ) p j+1 = hd 2 L disc (x j, x j+1 ) Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

11 Example: Störmer-Verlet method Consider a mechanical system ẍ = U (x) with Lagrangian L(x, ẋ) = 1 ẋ, ẋ U(x) 2 The Störmer-Verlet discretization is given by the discrete Lagrangian L disc (x j, x j+1 ) = 1 xj+1 x j, x j+1 x j 1 2 h h 2 U (x j) 1 2 U (x j+1) Its discrete Euler-Lagrange equation is x j+1 2x j + x j 1 h 2 = U (x j ) Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

12 Example: Störmer-Verlet method Consider a mechanical system ẍ = U (x) with Lagrangian L(x, ẋ) = 1 ẋ, ẋ U(x) 2 The Störmer-Verlet discretization is given by the discrete Lagrangian L disc (x j, x j+1 ) = 1 xj+1 x j, x j+1 x j 1 2 h h 2 U (x j) 1 2 U (x j+1) Its discrete Euler-Lagrange equation is x j+1 2x j + x j 1 h 2 = U (x j ) Abstract notation: ψ(x j 1, x j, x j+1 ; h) = 0 with ψ(x j 1, x j, x j+1 ; h) = x j+1 2x j + x j 1 h 2 + U (x j ) Symplectic equivalent: x j+1 = x j + hp j h2 2 U (x j ) p j+1 = p j h 2 U (x j ) h 2 U (x j+1 ) Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

13 Modified Equations Exact solution of a differential equation: Numerical solution with a variational integrator: Notice conservation of Energy: Easy to prove for (continuous) Hamiltonian systems Follows by Noether s theorem from invariance under time-translation of the Lagrangian Symplectic/variational integrators very nearly preserve energy. Why? Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

14 Modified Equations Conservation of Energy: Easy to prove for (continuous) Hamiltonian systems Follows by Noether s theorem from invariance under time-translation of the Lagrangian Symplectic/variational integrators very nearly preserve energy. Why? Idea of proof: find a modified equation, a differential equation with solutions that interpolate the numerical solutions: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

15 Modified Equations Modified equations are usually given by power series. Often they do not converge. Definition The differential equation ẍ = f (x, ẋ; h), where f (x, ẋ; h) f 0 (x, ẋ) + hf 1 (x, ẋ) + h 2 f 2 (x, ẋ) +... is a modified equation for the second order difference equation Ψ(x j 1, x j, x j+1 ; h) = 0 if, for every k, every solution of the truncated differential equation ẍ = T k ( fh (x, ẋ) ) = f 0 (x, ẋ) + hf 1 (x, ẋ) + h 2 f 2 (x, ẋ) h k f k (x, ẋ). satisfies Ψ ( x(t h), x(t), x(t + h); h ) = O ( h k+1). Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

16 Modified Equations for symplectic integrators Symplectic integrators are known the very nearly preserve energy, because Theorem The modified equation for a symplectic integrator is a Hamiltonian equation. Can we arrive at a similar result purely on the Lagrangian side? Are modified equations for variational integrators Lagrangian? Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

17 Table Of Contents 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

18 General idea Look for a modified Lagrangian L mod (x, ẋ) such that the discrete Lagrangian L disc is its exact discrete Lagrangian, i.e. jh (j 1)h L mod (x(t), ẋ(t))dt = hl disc ( x((j 1)h), x(jh) ). The Euler-Lagrange equation of L mod will then be the modified equation. The best we can hope for is to find such a modified Lagrangian up to an error of arbitrarily high order in h. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

19 The discrete Lagrangian evaluated on a continuous curve We can write the discrete Lagrangian as a function of x and its derivatives, all evaluated at the point jh h 2, L disc [x] = L disc ( x h 2 ẋ Here and in the following: x + h 2 ẋ = L disc (x j 1, x j ; h) ( ) h 2 ẍ..., 2 ( h 2 ) 2 ẍ +..., h). [x] denotes dependence on x and any number of its derivatives, we evaluate at t = jh h 2 whenever we omit the variable t, i.e. x = x ( jh h 2), x j = x(jh) and x j 1 = x((j 1)h). Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

20 A truly continuous Lagrangian We want to write the discrete action n S disc = hl disc (x j 1, x j ) = as an integral. j=1 Lemma (Euler-MacLaurin formula) n j=1 For any smooth function f : R R N we have n j=1 ( hf jh h ) nh 2 0 = nh 0 where B i are the Bernoulli numbers. [ ( )] hl disc x jh h 2 h 2i ( 2 1 2i 1 ) B 2i (2i)! f (2i) (t) dt i=0 (f (t) h2 24 f (t) + 7h f (4) (t) +... The symbol indicates that this is an asymptotic series. ) dt, Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

21 A truly continuous Lagrangian Definition We call L mesh [x(t)] = L disc [x(t)] + i=1 d 2 ( 2 1 2i 1 ) h 2i B 2i (2i)! d 2i dt 2i L disc[x(t)] = L disc [x(t)] h2 24 dt 2 L disc[x(t)] + 7h dt 4 L disc[x(t)] +... the meshed modified Lagrangian of L disc. Formally, the meshed modified Lagrangian satisfies L mesh [x(t)] dt = hl disc (x j, x j+1 ) where x j = x(jh). d 4 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

22 A truly continuous Lagrangian Definition We call L mesh [x(t)] = L disc [x(t)] + i=1 d 2 ( 2 1 2i 1 ) h 2i B 2i (2i)! d 2i dt 2i L disc[x(t)] = L disc [x(t)] h2 24 dt 2 L disc[x(t)] + 7h dt 4 L disc[x(t)] +... the meshed modified Lagrangian of L disc. Formally, the meshed modified Lagrangian satisfies L mesh [x(t)] dt = hl disc (x j, x j+1 ) where x j = x(jh). Are we finished? L mesh [x] depends on many more derivatives than the original L(x, ẋ). Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56 d 4

23 The meshed variational problem Definition classical variational problem: find critical curves of some action b a L[x(t)] dt in the set of smooth curves C. meshed variational problem: find critical curves of some action b a L[x(t)] dt in the set of piecewise smooth curves that are consistent with a mesh of size h, C M,h = {x C 0 ([a, b]) t 0 [a, b] : t [a, b] : x not smooth at t t t 0 hn}. x t Classical variational problem x Meshed variational problem t Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

24 The meshed variational problem Criticality conditions of a meshed variational problem: δl Euler-Lagrange equations: δx = 0, where Natural interior conditions: j 2 : k=0 δl = 0, δx (j) or equivalently: j 2 : δl δx (j) = ( 1) k dk L dt k x (j+k). L = 0, x (j) If L is a non-convergent power series, these equations are formal. x t x Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56 t

25 Exploiting the natural interior conditions S disc (x(0), x(h),...) = L mesh ([x], h) dt, hence S disc critical δl mesh δx = 0. Variations that are supported on a single mesh interval do not change the discrete action do not change L mesh ([x], h) dt. are the variations that produce natural interior conditions It follows that for L mesh the NIC are automatically satisfied: δl mesh δx = 0 j 2 : L mesh x (j) = 0 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

26 Definition The modified Lagrangian is the formal power series L mod (x, ẋ) = L mesh [x] ẍ=fh (x,ẋ), x (3) = d dt f h(x,ẋ),..., where ẍ = f h (x, ẋ) is the modified equation. The k-th truncation of the modified Lagrangian is ) L mod,k = T k (L mod (x, ẋ)) = T k (L mesh [x], x (j) =F j k 2 (x,ẋ) where T k denotes truncation after the h k -term and ẍ = Fk 2 (x, ẋ; h) + O(hk+1 ) = F k (x, ẋ; h) + O(h k+1 ), x (3) = Fk 3 (x, ẋ; h) + O(hk+1 ), x (4) = Fk 4 (x, ẋ; h) + O(hk+1 ),.... are the k-th truncation of the modified equation and its derivatives. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

27 Lemma The meshed modified Lagrangian L mesh [x] and the modified Lagrangian L mod (x, ẋ) have the same critical curves. Proof. L mod,k x = L mesh x + L mesh ẍ Fk 2 x + L mesh F k x (3) x +... x (j) =F j k 1 (x,ẋ) Also, L mod,k ẋ = L mesh x = L mesh ẋ + O(h k+1 ), + L mesh ẍ Fk 2 ẋ + L mesh Fk 3 x (3) ẋ +... x (j) =F j k 1 (x,ẋ) L mod,k x = L mesh ẋ d dt + O(h k+1 ), L mod,k ẋ = j=0 ( 1) j d j dt j L mesh x (j) + O(h k+1 ). Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

28 Main result Theorem For a discrete Lagrangian L disc that is a consistent discretization of some L, the k-th truncation of the Euler-Lagrange equation of L mod,k (x, ẋ) is the k-th truncation of the modified equation. Proof. Let x be a solution of the Euler-Lagrange equation for L mod (x, ẋ). Consider the discrete curve x j = x(jh). x is critical for the action L mod (x, ẋ) dt. By the Lemma, x is critical for the action L mesh [x] dt. By construction, the actions S disc = j L disc(y(jh), y((j + 1)h)) and S = b a L mod[y(t)] dt are equal for any smooth curve y. Therefore the discrete curve (x(jh)) j is critical for the discrete action S disc. Hence D 2 L disc (x(t h), x(t)) + D 1 L disc (x(t), x(t + h)) = 0. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

29 Table Of Contents 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

30 Example: Störmer-Verlet discretization L(x, ẋ) = 1 2 ẋ 2 U(x), L disc (x j, x j+1 ) = 1 x j+1 x j 2 h Its Euler-Lagrange equation is We have L disc [x] = x j+1 2x j + x j 1 h 2 = U (x j ). ẋ + h2 24 x (3) +..., ẋ + h U (x j) 1 2 U (x j+1). 24 x (3) +... ( h 2) 2 ẍ... ) 1 2 U ( x + h 2ẋ (x U h 2ẋ = 1 2 ẋ 2 U + h2 ( ẋ, x (3) 3U ẍ 3U (ẋ, ẋ)) + O(h 4 ) 24 ( h 2) 2 ẍ +... ) Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

31 Example: Störmer-Verlet discretization L disc [x] = 1 2 ẋ 2 U + h2 ( ẋ, x (3) 3U ẍ 3U (ẋ, ẋ)) + O(h 4 ), 24 From this we calculate the meshed modified Lagrangian, d 2 L mesh [x] = L disc [x] h2 24 dt 2 L disc[x] + O(h 4 ) = 1 2 ẋ 2 U + h2 ( ẋ, x (3) 3U ẍ 3U (ẋ, ẋ)) 24 h2 ( ẍ, ẍ + ẋ, x (3) ) U ẍ U (ẋ, ẋ) + O(h 4 ) 24 = 1 2 ẋ 2 U + h2 ( ẍ, ẍ 2U ẍ 2U (ẋ, ẋ) ) + O(h 4 ). 24 Eliminate second derivatives using ẍ = U + O(h 2 ), L mod,3 (x, ẋ) = 1 2 ẋ 2 U + h2 ( U 2 2U (ẋ, ẋ) ). 24 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

32 Example: Störmer-Verlet discretization The modified Lagrangian is L mod,3 (x, ẋ) = 1 2 h2 ( ẋ, ẋ U + U 2 2U (ẋ, ẋ) ). 24 Observe that this Lagrangian is not separable for general U. The corresponding Euler-Lagrange equation is ẍ U + h2 ( 2U U 2U (ẋ, ẋ) + 4U (ẋ, ẋ) + 4U ẍ ) = Solving this for ẍ we find the modified equation ẍ = U + h2 ( U (ẋ, ẋ) U U ) + O(h 4 ). 12 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

33 The Kepler problem Potential: U(x) = 1 x. Lagrangian: L = 1 1 ẋ, ẋ + 2 x. Equation of motion ẍ = x x Störmer-Verlet discretization: x j+1 2x j + x j 1 h 2 = U (x j ). 2 3 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

34 The Kepler problem Potential: U(x) = 1 x. Lagrangian: L = 1 1 ẋ, ẋ + 2 x. Equation of motion ẍ = x x Störmer-Verlet discretization: x j+1 2x j + x j 1 h 2 = U (x j ). Midpoint discretization: x j+1 2x j + x j 1 h 2 = 1 2 U ( xj 1 + x j 2 ) 1 ( ) xj + x j+1 2 U Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

35 Störmer-Verlet discretization of the Kepler problem The modified Lagrangian of the Störmer-Verlet discretization is L mod,3 (x, ẋ) = 1 2 ẋ 2 U + h2 ( U U 2U (ẋ, ẋ) ). 24 For the Kepler problem we have U(x) = 1 x, hence ( ( ) 1 L mod,3 x, ẋ = 2 ẋ ẋ, 2 ) x + h2 1 ẋ x, ẋ 24 x 4 2 x x 5. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

36 Laplace-Runge-Lenz vector The shape and orientation of an orbit for the kepler Problem is determined by the Laplace-Runge-Lenz vector, which is the Noether integral for a generalized variational symmetry. For the modified Lagrangian this is only an approximate symmetry. Lemma (From a perturbative version of Noether s theorem) The precession rate for the perturbed Lagrangian L = 1 2 is in first order approximation 1 ẋ, ẋ + + U(x, ẋ), x 2 U(x, ẋ) 2πa b radians per period, where a and b are the semimajor and semiminor axes, and denotes the time-average along the unperturbed orbit. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

37 Störmer-Verlet discretization of the Kepler problem Proposition The numerical precession rate of the Störmer-Verlet method is π (15 a3 24 b 6 3 a ) b 4 h 2 + O(h 4 ) Predicted: rad per revolution. Measured: rad per revolution. 3 3 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

38 Midpoint discretization of the Kepler problem Proposition The numerical precession rate of the midpoint rule is π (15 a3 12 b 6 3 a ) b 4 h 2 + O(h 4 ) Predicted: rad per revolution. Measured: rad per revolution. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

39 New methods Precession rate Störmer-Verlet: Precession rate Midpoint rule: π 12 π (15 a3 24 b 6 3 a b 4 (15 a3 b 6 3 a b 4 ) h 2 + O(h 4 ) ) h 2 + O(h 4 ) This allows us to construct new integrators with precession of order h 4. Mixed Lagrangian L(x j, x j+1 ) = 2 3 L SV (x j, x j+1 ) L MP(x j, x j+1 ) Lagrangian composition L j (x k, x k+1 ) = { L MP (x k, x k+1 ) L SV (x k, x k+1 ) if 3 j, otherwise. Composition { of difference equations ( ) ( ) xj+1 2x j + x j 1 = h2 2 U xj 1 +x j 2 h2 2 U xj +x j+1 2 x j+1 2x j + x j 1 = h 2 U (x j ) Is this a variational integrator? if j 2 mod 3, otherwise. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

40 Qualitative analysis of new methods Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

41 Precession rates Precession rate MP SV FR LC ML DEC C h 2 reference h 4 reference MP,SV: old methods LC, ML, DEC: new methods FR: Forest, Ruth. Fourth-order symplectic integration, C: Chin. Symplectic integrators from composite operator factorizations, Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

42 Table Of Contents 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

43 Lagrangians linear in velocities L : T R N = R 2N R of the form L(q, q) = α(q), q H(q), where α : R N R N, H : R N R, and the brackets, denote the standard scalar product. Let ( ) A(q) = α αi (q) (q) = q j i,j=1,...,n and A skew (q) = A(q) T A(q) We assume that A skew (q) is invertible, then the Euler-Lagrange equation for L is given by q = A skew (q) 1 H (q) T Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

44 Examples of Lagrangians linear in velocities Dynamics of point vortices in the (complex) plane L(z, z, ż, ż) = N Γ j Im(z j ż j ) 1 π j=1 ż j = i 2π k j Variational formulation in phase space N j 1 Γ j Γ k log z j z k, j=1 k=1 Γ k z j z k for j = 1,..., N. L(p, q, ṗ, q) = p, q H(p, q). ( ) H T ( ) H T q = and ṗ =. p q Guiding centre motion (plasma physics) Many PDEs, e.g. nonlinear Schrödinger equation. (But modified equations are not so useful for PDEs) Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

45 Possible discretization of L(q, q) = q T A q H(q) ( ) qj + q T ( j+1 qj+1 q j L disc (q j, q j+1, h) = A 2 h ) 1 2 H(q j) 1 2 H(q j+1) Discrete EL equation q j+1 q j 1 2h = (A T A) 1 H (q j ) T. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

46 Possible discretization of L(q, q) = q T A q H(q) ( ) qj + q T ( j+1 qj+1 q j L disc (q j, q j+1, h) = A 2 h Discrete EL equation q j+1 q j 1 2h = (A T A) 1 H (q j ) T. ) 1 2 H(q j) 1 2 H(q j+1) The EL equation involves 3 points needs 2 points of initial data. The differential equation is of 1st order needs only 1 point of initial data. This means we are dealing with a 2-step method. We have a multi-step method, so parasitic oscillations may occur. Solution: double the dimension of the system: principal and parasitic part. The enlarged system is still Lagrangian. We cannot replace q in the Lagrangian. Reason: NIC only involve q, q (3),. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

47 Modified equations for 2-step methods Principal modified equation satisfies q = f (q) + hf 1 (q) + h 2 f 2 (q) h k f k (q) a 0 q(t) + a 1 q(t + h) + a 2 q(t + 2h) h = b 0 f (q(t)) + b 1 f (q(t + h)) + b 2 f (q(t + 2h)) + O(h k+1 ). Full system of modified equations ẋ = f 0 (x, y) + hf 1 (x, y) h k f k (x, y) ẏ = g 0 (x, y) + hg 1 (x, y) h k g k (x, y), such that the discrete curve q j = x(t + jh) + ( 1) j y(t + jh) satisfies a 0 q j + a 1 q j+1 + a 2 q j+2 h = b 0 f (q j ) + b 1 f (q j+1 ) + b 2 f (q j+2 ) + O(h k+1 ) Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

48 Doubling the dimension The discrete curve (x j, y j ) j Z is critical for L(x j, y j, x j+1, y j+1, h) = 1 2 L(x j+y j, x j+1 y j+1, h)+ 1 2 L(x j y j, x j+1 +y j+1, h), if and only if the discrete curves (q + j ) j Z and (q j ) j Z, defined by are critical for L(q j, q j+1, h). q ± j = x j ± ( 1) j y j, Lagrangian for the full system of modified equations = Lagrangian for the principal modified equation of the extended system. Hence we can calculate a Lagrangian for the full system of modified equations with the tools we already have. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

49 Example 1 For L disc (q j, q j+1, h) = we find 1 2 Aq j Aq j+1, q j+1 q j 1 h 2 H(q j) 1 2 H(q j+1) L mod,0 (x, y, ẋ, ẏ, h) = Ax, ẋ + Aẏ, y 1 2 H(x + y) 1 H(x y). 2 Its Euler-Lagrange equations are ( 1 ẋ = A 1 skew 2 H (x + y) T + 1 ) 2 H (x y) T + O(h), ẏ = A 1 skew ( 12 H (x + y) T + 12 ) H (x y) T + O(h). Linearize the second equation around y = 0 ẏ = A 1 skew H (x)y + O( y 2 + h) Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

50 Example 1 Magnitude of oscillations satisfies ẏ = A 1 skew H (x)y + O( y 2 + h) Unless the matrix A 1 skew H (x) is exceptionally friendly, we expect growing parasitic oscillations. (Note that an eigenvalue analysis does not apply because A 1 skew H (x) is not constant) Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

51 Example 2 For we find L disc (q j, q j+1, h) = A q j + q j+1, q ( ) j+1 q j qj + q j+1 H 2 h 2 L mod,0 (x, y, ẋ, ẏ, h) = Ax, ẋ + Aẏ, y H(x). Its Euler-Lagrange equations are ẋ = A 1 skew H (x) T + O(h), ẏ = 0 + O(h). Even better, ẏ = 0 to any order no growing oscillations Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

52 Table Of Contents 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

53 Contact Hamiltonian dynamics Contact geometry is an odd-dimensional counterpart to symplectic geometry. In mechanics, we typically work on T Q R, a Hamilton function H : T Q R R : (q, p, z) H(q, p, z) generates dynamics by q = H p ṗ = H q p H z ż = p H p H Example. system: H(q, p, z) = 1 2 p2 + V (q) + αz describes a damped mechanical q = p ṗ = V (q) αp ż = p 2 H Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

54 Contact variational principle Given a Lagrange function consider the ode L : TQ R R, Given a discrete Lagrange function L : Q Q R R, consider the ode ż = L(q, q, z) on a fixed interval [0, T ], with z(0), q(0) and q(t ) prescribed. Herglotz variational principle: z(t ) critical under variations of the curve q. z j+1 z j h = L(q j, q j+1, z j ) for j {0,..., N} with z 0, q 0 and q N prescribed. Discrete Herglotz variational principle: z N critical under variations of the discrete curve q. In this framework, we can adapt the theory of variational integrators to contact systems. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

55 Table Of Contents 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

56 Integrable systems An integrable system is (a system of) nonlinear differential or difference equation(s), that to some extend behaves like a linear system. Our perspective An equation is integrable if it is part of a sufficiently large system of compatible equations. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

57 Integrable systems An integrable system is (a system of) nonlinear differential or difference equation(s), that to some extend behaves like a linear system. Our perspective An equation is integrable if it is part of a sufficiently large system of compatible equations. In Mechanics: A Hamiltonian system with Hamilton function H : T Q R 2N R is Liouville-Arnold integrable if there exist N functionally independent Hamilton functions H = H 1, H 2,... H N in involution: {H i, H j } = 0. (1+1)-dimensional PDEs: Infinite sequence H = H 1, H 2,... of Hamiltonians in involution. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

58 Integrable systems An integrable system is (a system of) nonlinear differential or difference equation(s), that to some extend behaves like a linear system. Our perspective An equation is integrable if it is part of a sufficiently large system of compatible equations. In Mechanics: A Hamiltonian system with Hamilton function H : T Q R 2N R is Liouville-Arnold integrable if there exist N functionally independent Hamilton functions H = H 1, H 2,... H N in involution: {H i, H j } = 0. (1+1)-dimensional PDEs: Infinite sequence H = H 1, H 2,... of Hamiltonians in involution. Variational description of an integrable hierarchy: pluri-lagrangian or Lagrangian multiform systems. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

59 Quad equations Quad equation: U 2 U 12 Q(U, U 1, U 2, U 12, α 1, α 2 ) = 0 α 2 Subscripts of U denote lattice shifts, α 1, α 2 are parameters. Invariant under symmetries of the square, affine in each of U, U 1, U 2, U 12. U U 23 α 1 U 1 U 123 Integrability for systems quad equations: Multi-dimensional consistency of U 3 U 13 Q(U, U i, U j, U ij, α i, α j ) = 0, α 3 U 2 i.e. the three ways of calculating U 123 give the same result. U α 2 α 1 U 1 U 12 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

60 Pluri-Lagrangian structure for quad equations For some discrete 2-form L( ij ) = L(U, U i, U j, U ij, α i, α j ), the action L( ) is critical on all 2-surfaces Γ in N N simultaneously. Γ [Lobb, Nijhoff. Lagrangian multiforms and multidimensional consistency. J. Phys. A. 2009] Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

61 Continuous Pluri-Lagrangian systems A field u : R N Q is a solution of the pluri-lagrangian problem for the Lagrangian 2-form, L = L ij [u] dt i dt j. i,j if the action Γ L is critical on all smooth surfaces Γ in RN. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

62 Continuous Pluri-Lagrangian systems A field u : R N Q is a solution of the pluri-lagrangian problem for the Lagrangian 2-form, L = L ij [u] dt i dt j. i,j if the action Γ L is critical on all smooth surfaces Γ in RN. Continuum limits of pluri-lagrangian systems use many of the ideas presented today. Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

63 Summary 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

64 Summary Obtaining a high-order modified Lagrangian L mesh [x] is relatively straightforward, but its interpretation is not. From L mesh [x] a first order Lagrangian L mod,k (x, ẋ) can be found using the meshed variational principle. (This might be simpler in the contact formulation.) If the Lagrangian is nondegenerate, the modified Lagrangian can also be obtained by Legendre transform from the modified Hamiltonian. Our approach extends to degenerate Lagrangians that are linear in velocities. Similar ideas apply to continuum limits of integrable systems. Can we get improved error estimates from the Lagrangian perspective? What about nonholonomic constraints? What about PDEs? Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56

65 Thank you for your attention! 1 Basics of Geometric Numerical Integration [1] 2 Modified Lagrangians [2] 3 Variational integration of the Kepler problem [3] 4 A class of degenerate Lagrangians [4] 5 Variational integrators for Contact Hamiltonian systems [5] 6 Related ideas in integrable systems [6] [1] Hairer, Lubich, Wanner. Geometric Numerical Integration. Springer, [2] V. Modified equations for variational integrators. Numer. Math. 137:1001, [3] V. Numerical precession in variational discretizations of the Kepler problem. In: Discrete Mechanics, Geometric Integration and Lie Butcher Series, Springer, [4] V. Modified equations for variational integrators applied to Lagrangians linear in velocities. arxiv: [5] Work in progress with Alessandro Bravetti and Marcello Seri. [6] V. Continuum limits of Pluri-Lagrangian systems. arxiv: Mats Vermeeren (TU Berlin) Modified equations for variational integrators December 18, / 56