Discretizations of Lagrangian Mechanics

Transcription

1 Discretizations of Lagrangian Mechanics George W. Patrick Applied Mathematics and Mathematical Physics Mathematics and Statistics, University of Saskatchewan, Canada August 2007 With Charles Cuell

2 Discretizations of Lagrangian Mechanics 1 Continuous Lagrangian system: Q, L: T Q R, S = L ( q (t) ) dt, δs = 0, q(a) = q 1, q(b) = q 2

3 Discretizations of Lagrangian Mechanics 2 Continuous Lagrangian system: Q, L: T Q R, S = L ( q (t) ) dt, δs = 0, q(a) = q 1, q(b) = q 2 Discrete means discrete time: q(t) q i.

4 Discretizations of Lagrangian Mechanics 3 Continuous Lagrangian system: Q, L: T Q R, S = L ( q (t) ) dt, δs = 0, q(a) = q 1, q(b) = q 2 Discrete means discrete time: q(t) q i. Moser-Veselov discretization: Q, L d : Q Q R, S d = L d ( qi+1, q i ), dsd = 0

5 Discretizations of Lagrangian Mechanics 4 Continuous Lagrangian system: Q, L: T Q R, S = L ( q (t) ) dt, δs = 0, q(a) = q 1, q(b) = q 2 Discrete means discrete time: q(t) q i. Moser-Veselov discretization: Q, L d : Q Q R, S d = L d ( qi+1, q i ), dsd = 0 New framework: new discretizations = Moser-Veselov discretizations, but = not explicit Moser-Veselov discretizations new discretizations

6 Discretizations of Lagrangian Mechanics 5 Equvalent framework to Moser-Veselov. But:

7 Discretizations of Lagrangian Mechanics 6 Equvalent framework to Moser-Veselov. But: more geometrically appealing

8 Discretizations of Lagrangian Mechanics 7 Equvalent framework to Moser-Veselov. But: more geometrically appealing provides a self-contained Lagrangian formalism

9 Discretizations of Lagrangian Mechanics 8 Equvalent framework to Moser-Veselov. But: more geometrically appealing provides a self-contained Lagrangian formalism helps in the formal analysis of the discrete systems

10 Discretizations of Lagrangian Mechanics 9 Equvalent framework to Moser-Veselov. But: more geometrically appealing provides a self-contained Lagrangian formalism helps in the formal analysis of the discrete systems helps in the construction of numerical integrators

11 Discretizations of Lagrangian Mechanics 10 Equvalent framework to Moser-Veselov. But: more geometrically appealing provides a self-contained Lagrangian formalism helps in the formal analysis of the discrete systems helps in the construction of numerical integrators The new framework makes it easier to create sophisticated new discretizations, all the way through nonlinear, nonholonomic systems. It clarifies discretizations of Lagrangian systems, and it should help wherever such discretizations are used.

12 Contents 11 1 Introduction 2 Basics curve segments for as discrete tangent vectors lift principle to phase space 3 Discretizations of tangent bundles Discretizations of tangent bundles Discrete tangent bundles 4 Discretizations of Lagrangian systems Discretizations of Lagrangian systems Discrete Lagrangian systems 5 Automatic variationalization 6 Existence, uniqueness, and accuracy of the discrete flow

13 Basics 12 Q (v q) q v q + (v q) The discrete representation of a tangent vector is a curve segment. Replace T Q with a set V of curve segments in Q. There should be one curve segment for every v q, so usually V = T Q.

14 Basics 13 v(t) = q (t) S = R L(v(t))) dt 1 0 {q(t)} {v(t)} S = R L(q (t)) dt -1 Lifting the continuous variational principle from curves in the configuration space Q to curves in the phase space TQ. v(t) = dq dt, S = L ( v(t) )

15 Basics 14 First order constraint: { q (t) } = So the variational principle { v(t) : d ( τq v(t) ) } = v(t) dt is equivalent to

16 Basics 15 First order constraint: { q (t) } = { v(t) : d ( τq v(t) ) } = v(t) dt So the variational principle L: T Q R, S = L ( q (t) ) dt, δs = 0, is equivalent to q(a) = q 1, q(b) = q 2

17 Basics 16 First order constraint: { q (t) } = { v(t) : d ( τq v(t) ) } = v(t) dt So the variational principle L: T Q R, S = L ( q (t) ) dt, δs = 0, is equivalent to q(a) = q 1, q(b) = q 2 L: T Q R, S = L ( v(t) ) dt, δs δv = 0, T τ Q (δv(a)) = 0, T τ Q (δv(b)) = 0, d ( τq v(t) ) = v(t) dt

18 Basics 17 Given a discrete tangent bundle and a discrete Lagrangian V, + : V R, : V R L d : V R we use the action S = L d (v i ) To find the discrete analogue of the Lagrangian system, only it is required discrete analoue of the first order constraint.

19 Basics 18 A discrete tangent bundle for { q } = R n is { q, v } = R n R n with (q, v) = q, + (q, v) = q + hv

20 Basics 19 A discrete tangent bundle for { q } = R n is { q, v } = R n R n with (q, v) = q, + (q, v) = q + hv The equations q 1 = (q, v) = q, q 2 = + (q, v) = q + hv have solution q = q 1, v = q 2 q 1 h

21 Basics 20 A discrete tangent bundle for { q } = R n is { q, v } = R n R n with (q, v) = q, + (q, v) = q + hv The equations q 1 = (q, v) = q, q 2 = + (q, v) = q + hv have solution q = q 1, v = q 2 q 1 h The derivative of the sequence q i is v i = ( ± ) 1 (q i+1, q i ), ± ( +, )

22 Basics 21 v i = ( ± ) 1 (q i+1, q i ), ± ( +, )

23 Basics 22 v i = ( ± ) 1 (q i+1, q i ), ± ( +, ) The derivative v i of q i satisfies + (v i 1 ) = (v i )

24 Basics 23 v i = ( ± ) 1 (q i+1, q i ), ± ( +, ) The derivative v i of q i satisfies + (v i 1 ) = (v i ) If v i satisfies (v i+1 ) = + (v i ) then it is the derivative of q i defined by q i = + (v i 1 ) = (v i )

25 Basics 24 v i = ( ± ) 1 (q i+1, q i ), ± ( +, ) The derivative v i of q i satisfies + (v i 1 ) = (v i ) If v i satisfies (v i+1 ) = + (v i ) then it is the derivative of q i defined by q i = + (v i 1 ) = (v i ) The discrete first order condition is + (v i ) = (v i+1 ) Same as: the curve segments connect

26 Basics 25 v 3 v1 v 2 q 0 q 1 q 2 q 3

27 Basics 26 v 3 v1 v 2 q 0 q 1 q 2 q 3 S d = N L d (v i ), ds d δv i = 0 i=1

28 Basics 27 v 3 v1 v 2 q 0 q 1 q 2 q 3 N S d = L d (v i ), ds d δv i = 0 i=1 + (v i ) = (v i+1 ), T + (δv i ) = T (δv i+1 )

29 Basics 28 v 3 v1 v 2 q 0 q 1 q 2 q 3 N S d = L d (v i ), ds d δv i = 0 i=1 + (v i ) = (v i+1 ), T + (δv i ) = T (δv i+1 ) T (δv 0 ) = 0, T + (δv N ) = 0

30 Discretizations of tangent bundles 29

31 Discretizations of tangent bundles 30 A discretization of T M is a tuple (ψ, α +, α ), where ψ : U R 2 M M, α + : [0, a) R 0, α : [0, a) R 0, are such that 1 U is open and { 0 } { 0 } M U; 2 α +, α are C 1, α + (0) = α (0) = 0, and α + dα+ dh (0), α dα+ dh (0), satisfy α + α = 1; 3 ψ(h, 0, v m ) = m, and ψ t (h, 0, v m) = v m ; 4 the boundary maps defined by h (v m) ψ ( h, α (h), v m ), + h (v m) ψ ( h, α + (h), v m ), are C 1 and d dh + h (v m) = α + v m, h=0 d dh h (v m) = α v m. h=0

32 Discretizations of tangent bundles 31 Let M be a manifold. A discrete tangent bundle of M is a tuple (V, +, ), where V is a manifold, dim V = 2 dim M and + : V M and : V M satisfy 1 + and are submersions such that ker T + ker T = 0; and 2 for all m M, the backward fiber V + m ( + ) 1 (m) and the forward fiber V m ( ) 1 (m) meet in exactly one point, denoted 0 m. The discrete zero section is 0 V ( ± ) 1 (M M), where (M M) is the diagonal of M M.

33 Discretizations of tangent bundles 32 vert v V V + + (v) vert + v V v V (v) vert + v V v vert + ṽ V ṽ (v) + (v) τṽ,v M M

34 Discretizations of tangent bundles 33 M = R k, ψ ( h, t, (m, v) ) m + tv + O(h 2 )

35 Discretizations of tangent bundles 34 M = R k, ψ ( h, t, (m, v) ) m + tv + O(h 2 ) m = m + α (h)v + O(h 2 ), m + = m + α + (h)v + O(h 2 )

36 Discretizations of tangent bundles 35 M = R k, ψ ( h, t, (m, v) ) m + tv + O(h 2 ) m = m + α (h)v + O(h 2 ), m + = m + α + (h)v + O(h 2 ) h = 0 : m = m, m + = m

37 Discretizations of tangent bundles 36 M = R k, ψ ( h, t, (m, v) ) m + tv + O(h 2 ) m = m + α (h)v + O(h 2 ), m + = m + α + (h)v + O(h 2 ) h = 0 : m = m, m + = m m = m+ + m, z = m+ m 2 h

38 Discretizations of tangent bundles 37 M = R k, ψ ( h, t, (m, v) ) m + tv + O(h 2 ) m = m + α (h)v + O(h 2 ), m + = m + α + (h)v + O(h 2 ) h = 0 : m = m, m + = m m = m+ + m, z = m+ m 2 h m = m + α+ (h) + α (h) v + O(h 2 ), 2 z = v + O(h)

39 Discretizations of tangent bundles 38 M = R k, ψ ( h, t, (m, v) ) m + tv + O(h 2 ) m = m + α (h)v + O(h 2 ), m + = m + α + (h)v + O(h 2 ) h = 0 : m = m, m + = m m = m+ + m, z = m+ m 2 h m = m + α+ (h) + α (h) v + O(h 2 ), 2 z = v + O(h) h = 0 : m = m, z = v

40 Discretizations of tangent bundles 39 global local S. Lang. Differential manifolds. Addison-Wesley, C. Cuell and G. W. Patrick [2007]. Skew critical problems. Submitted Regul. Chaotic Dyn., 17pp.

41 Discretizations of Lagrangian systems 40 A discretization of a Lagrangian system L: T Q R is a tuple (L h, ψ, α +, α ) where (ψ, α +, α ) is a discretization of T Q and L h : T Q R is a function such that L h (v q ) = α + (h) α (h) L ψ t (h, t, v q) dt + O(h 2 ). A discrete Lagrangian system is a tuple (L d, +,, V, Q) where L d : V R and (V, +, ) is a discrete tangent bundle on Q.

42 Discretizations of Lagrangian systems 41 The variational technology in J. E. Marsden, G. W. Patrick, and S. Shkoller [1998]. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199,

43 Discretizations of Lagrangian systems 42 The variational technology in J. E. Marsden, G. W. Patrick, and S. Shkoller [1998]. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199, identifies the symplectic form

44 Discretizations of Lagrangian systems 43 The variational technology in J. E. Marsden, G. W. Patrick, and S. Shkoller [1998]. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199, identifies the symplectic form proves symplecticity of the discrete flow

45 Discretizations of Lagrangian systems 44 The variational technology in J. E. Marsden, G. W. Patrick, and S. Shkoller [1998]. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199, identifies the symplectic form proves symplecticity of the discrete flow identifies the momentum

46 Discretizations of Lagrangian systems 45 The variational technology in J. E. Marsden, G. W. Patrick, and S. Shkoller [1998]. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199, identifies the symplectic form proves symplecticity of the discrete flow identifies the momentum proves conservation of momentum

47 Discretizations of Lagrangian systems 46 The variational technology in J. E. Marsden, G. W. Patrick, and S. Shkoller [1998]. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199, identifies the symplectic form proves symplecticity of the discrete flow identifies the momentum proves conservation of momentum θ L d (v) = dl(v)t π, θ + L d (v) = dl(v)t π +, dθ + L d = dθ L d = ω Ld J ξ = ι ξv θ + L d

48 Automatic Variationalization 47 Given w 1 = (q 1, v 1 ) TQ solve, for w 2 = (q 2, v 2 ) T ˆQ, the equations DL h (w 1 ) = λ D ˆ h (w 1) + µd ˆ + h (w 1) + ν 1 ˆDg(q1 ), 0 + ν 2 ˆD2 g(q1 )v 1, Dg(q 1 ) DL h (w 2 ) = λ + D ˆ + h (w 2) µd ˆ h (w 2) + ν + 1 ˆDg(q2 ), 0 + ν + 2 ˆD2 g(q2 )v 2, Dg(q 2 ) 2N 2N q = ˆ + h (w 1) + Dg( q) T θ1, q = ˆ h (w 2) + Dg( q) T θ2 2N g( q) = 0 r λ Dg(q 1 )T = 0, λ + Dg(q + 2 )T = 0 µdg( q) T = 0 3r g(q 2 ) = 0, Dg(q 2 )v 2 = 0 2r g(q 1 ) = 0, g(q+ 2 ) = 0 2r q 1 = ˆ h (w 1) + Dg(q 1 )T θ, q + 2 = ˆ + h (w 2) + Dg(q + 2 )T θ + 2N 8N + 8r Lagrange multipliers λ, λ +, µ Lagrange multipliers ν 1 +, ν+ 2, ν 1, ν 2 time advanced state (q 2, v 2 ) TQ midpoint q variables θ 1, θ 2, θ, θ + variables q 1, q+ 2 3N 4r 2N N 4r 2N 8N + 8r

49 Discrete existence and uniqueness 48 L: T Q R, ψ(h, t, v q ), L d = α + (h) α (h) ( ) ψ L t (h, t, v q) dt+o(h 2 )

50 Discrete existence and uniqueness 49 L: T Q R, ψ(h, t, v q ), L d = α + (h) α (h) ( ) ψ L t (h, t, v q) dt+o(h 2 ) Want: for small enough h, for all v there is a unique ṽ such that dl d (v)δv + dl d (ṽ)δv = 0, + (v) = (ṽ), T δv = 0, T + δṽ = 0, T + δv = T δṽ

51 Discrete existence and uniqueness 50 L: T Q R, ψ(h, t, v q ), L d = α + (h) α (h) ( ) ψ L t (h, t, v q) dt+o(h 2 ) Want: for small enough h, for all v there is a unique ṽ such that dl d (v)δv + dl d (ṽ)δv = 0, + (v) = (ṽ), T δv = 0, T + δṽ = 0, T + δv = T δṽ Must be a perturbative proof at h = 0.

52 Discrete existence and uniqueness 51 L: T Q R, ψ(h, t, v q ), L d = α + (h) α (h) ( ) ψ L t (h, t, v q) dt+o(h 2 ) Want: for small enough h, for all v there is a unique ṽ such that dl d (v)δv + dl d (ṽ)δv = 0, + (v) = (ṽ), T δv = 0, T + δṽ = 0, T + δv = T δṽ Must be a perturbative proof at h = 0. But h = 0 this problem is badly degenerate: h = 0 : (v q ) = + (v q ) = q, L d = 0

53 Discrete existence and uniqueness 52 Blow up the variational principle at h = 0. S d (v, ṽ) = L(v) + L(ṽ), v, ṽ T q Q, v + ṽ = z q 2 Nondegenerate if L is hyperregular and the solution is v q = ṽ q = z q.

54 Discrete existence and uniqueness 53 Blow up the variational principle at h = 0. S d (v, ṽ) = L(v) + L(ṽ), v, ṽ T q Q, v + ṽ = z q 2 Nondegenerate if L is hyperregular and the solution is v q = ṽ q = z q. Semiglobal inverse function theorem; tubular neighbourhood.

55 Accuracy 54 Want: if ψ, ψ and L d, L d match to order r, then F, F match to order ψ.

56 Accuracy 55 Want: if ψ, ψ and L d, L d match to order r, then F, F match to order ψ. Easy: F = F + O(h r 1 ). Its r 1 because of a division by h in the blow-up.

57 Accuracy 56 Want: if ψ, ψ and L d, L d match to order r, then F, F match to order ψ. Easy: F = F + O(h r 1 ). Its r 1 because of a division by h in the blow-up. Difficult: F = F + O(h r ).

58 Accuracy 57 Want: if ψ, ψ and L d, L d match to order r, then F, F match to order ψ. Easy: F = F + O(h r 1 ). Its r 1 because of a division by h in the blow-up. Difficult: F = F + O(h r ). F, F are obtained from graphs Γ, Γ.

59 Accuracy 58 Want: if ψ, ψ and L d, L d match to order r, then F, F match to order ψ. Easy: F = F + O(h r 1 ). Its r 1 because of a division by h in the blow-up. Difficult: F = F + O(h r ). F, F are obtained from graphs Γ, Γ. Γ = Γ + O(h r 1 )

60 Accuracy 59 Want: if ψ, ψ and L d, L d match to order r, then F, F match to order ψ. Easy: F = F + O(h r 1 ). Its r 1 because of a division by h in the blow-up. Difficult: F = F + O(h r ). F, F are obtained from graphs Γ, Γ. Γ = Γ + O(h r 1 ) Γ = Γ + rsd r (v, ṽ)

61 Accuracy 60 Want: if ψ, ψ and L d, L d match to order r, then F, F match to order ψ. Easy: F = F + O(h r 1 ). Its r 1 because of a division by h in the blow-up. Difficult: F = F + O(h r ). F, F are obtained from graphs Γ, Γ. Γ = Γ + O(h r 1 ) Γ = Γ + rsd r (v, ṽ) rsd r (v, ṽ) is symmetric. It affects the graphs symetrically. That does not affect the maps.