Uncovering the Lagrangian of a system from discrete observations

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1 Uncovering the Lagrangian of a system from discrete observations Yakov Berchenko-Kogan Massachusetts Institute of Technology 7 January, 2012 Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

2 Background The Problem Given discrete measurements of a Lagrangian system, can we recover the Lagrangian? Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

3 Background The Problem Given discrete measurements of a Lagrangian system, can we recover the Lagrangian? The Discrete Euler-Lagrange Equations Given a system with Lagrangian L(x, v), we can discretize the action with time step τ by summing the discrete Lagrangian ( x + y L d (x, y) = τ L, y x ). 2 τ Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

4 Background The Problem Given discrete measurements of a Lagrangian system, can we recover the Lagrangian? The Discrete Euler-Lagrange Equations Given a system with Lagrangian L(x, v), we can discretize the action with time step τ by summing the discrete Lagrangian ( x + y L d (x, y) = τ L, y x ). 2 τ The principle of stationary action yields the discrete Euler-Lagrange equations D 2 L d (x, y) + D 1 L d (y, z) = 0, relating any three consecutive points x, y, and z on a discrete trajectory. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

5 Recovering the Discrete Lagrangian The Problem Given a pair of points (x 0, y 0 ), we would like to use data points on trajectories that pass nearby to estimate the Taylor expansion of the discrete Lagrangian L d at (x 0, y 0 ). Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

6 Recovering the Discrete Lagrangian The Problem Given a pair of points (x 0, y 0 ), we would like to use data points on trajectories that pass nearby to estimate the Taylor expansion of the discrete Lagrangian L d at (x 0, y 0 ). A Caveat Two Lagrangians might be equivalent in the sense that they yield the same equations of motion. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

7 Recovering the Discrete Lagrangian The Problem Given a pair of points (x 0, y 0 ), we would like to use data points on trajectories that pass nearby to estimate the Taylor expansion of the discrete Lagrangian L d at (x 0, y 0 ). A Caveat Two Lagrangians might be equivalent in the sense that they yield the same equations of motion. For example, if L(x, y) is a discrete Lagrangian, then the Lagrangian L (x, y) = αl(x, y) + β(y 2 x 2 ) + γ(y x) + δ produces the same discrete Euler-Lagrange equations, for any choice of α, β, γ, and δ. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

8 Recovering the Discrete Lagrangian The Problem Given a pair of points (x 0, y 0 ), we would like to use data points on trajectories that pass nearby to estimate the Taylor expansion of the discrete Lagrangian L d at (x 0, y 0 ). A Caveat Two Lagrangians might be equivalent in the sense that they yield the same equations of motion. For example, if L(x, y) is a discrete Lagrangian, then the Lagrangian L (x, y) = αl(x, y) + β(y 2 x 2 ) + γ(y x) + δ produces the same discrete Euler-Lagrange equations, for any choice of α, β, γ, and δ. Trajectory data can t distinguish between equivalent Lagrangians, nor would it be useful to do so. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

9 A Second-Order Approximation Taylor Expansion of the Discrete Lagrangian Given a pair of points (x 0, y 0 ), we approximate L d with its second-degree Taylor polynomial at (x 0, y 0 ). Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

10 A Second-Order Approximation Taylor Expansion of the Discrete Lagrangian Given a pair of points (x 0, y 0 ), we approximate L d with its second-degree Taylor polynomial at (x 0, y 0 ). We rewrite it in the form L d a(x p) 2 + 2b(x p)(y p) + c(y p) 2 + d p (x p) + e p (y p) + f p, where p = (x 0 + y 0 )/2. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

11 A Second-Order Approximation Taylor Expansion of the Discrete Lagrangian Given a pair of points (x 0, y 0 ), we approximate L d with its second-degree Taylor polynomial at (x 0, y 0 ). We rewrite it in the form L d a(x p) 2 + 2b(x p)(y p) + c(y p) 2 + d p (x p) + e p (y p) + f p, where p = (x 0 + y 0 )/2. The expression in higher dimensions is analogous. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

12 A Second-Order Approximation Taylor Expansion of the Discrete Lagrangian Given a pair of points (x 0, y 0 ), we approximate L d with its second-degree Taylor polynomial at (x 0, y 0 ). We rewrite it in the form L d a(x p) 2 + 2b(x p)(y p) + c(y p) 2 + d p (x p) + e p (y p) + f p, where p = (x 0 + y 0 )/2. The expression in higher dimensions is analogous. Discrete Euler-Lagrange Equations For three consecutive points x, y, and z on a trajectory, we can apply the discrete Euler-Lagrange equations D 2 L d (x, y) + D 1 L d (y, z) = 0 to find 0 2(a + c)(y p) + 2b(x p + z p) + (d p + e p ). Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

13 A Second-Order Approximation Taylor Expansion of the Discrete Lagrangian Given a pair of points (x 0, y 0 ), we approximate L d with its second-degree Taylor polynomial at (x 0, y 0 ). We rewrite it in the form L d a(x p) 2 + 2b(x p)(y p) + c(y p) 2 + d p (x p) + e p (y p) + f p, where p = (x 0 + y 0 )/2. The expression in higher dimensions is analogous. Discrete Euler-Lagrange Equations For three consecutive points x, y, and z on a trajectory, we can apply the discrete Euler-Lagrange equations D 2 L d (x, y) + D 1 L d (y, z) = 0 to find 0 2(a + c)(y p) + 2b(x p + z p) + (d p + e p ). We will use nearby triplets (x, y, z) from our trajectory measurements to estimate a + c, b, and d p + e p. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

14 New Parameters for the Lagrangian Scaling the Parameters In order to estimate the parameters using many data points, we need to assign weights to the parameters appropriately. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

15 New Parameters for the Lagrangian Scaling the Parameters In order to estimate the parameters using many data points, we need to assign weights to the parameters appropriately. The parameter d p + e p has different units from a + c and b. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

16 New Parameters for the Lagrangian Scaling the Parameters In order to estimate the parameters using many data points, we need to assign weights to the parameters appropriately. The parameter d p + e p has different units from a + c and b. Taylor approximations of the discrete Euler-Lagrange equations suggest that an appropriate rescaling of the parameters at (x 0, y 0 ) is A := (a + c) y 0 x 0, B := 2b y 0 x 0, D := d p + e p. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

17 New Parameters for the Lagrangian Scaling the Parameters In order to estimate the parameters using many data points, we need to assign weights to the parameters appropriately. The parameter d p + e p has different units from a + c and b. Taylor approximations of the discrete Euler-Lagrange equations suggest that an appropriate rescaling of the parameters at (x 0, y 0 ) is A := (a + c) y 0 x 0, B := 2b y 0 x 0, D := d p + e p. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

18 New Parameters for the Lagrangian Scaling the Parameters In order to estimate the parameters using many data points, we need to assign weights to the parameters appropriately. The parameter d p + e p has different units from a + c and b. Taylor approximations of the discrete Euler-Lagrange equations suggest that an appropriate rescaling of the parameters at (x 0, y 0 ) is A := (a + c) y 0 x 0, B := 2b y 0 x 0, D := d p + e p. Discrete Euler-Lagrange Equations For three consecutive points (x, y, z) on a trajectory, we have 0 A(2y x 0 y 0 ) + B(x + z x 0 y 0 ) + D y 0 x 0. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

19 Estimating the Parameters Discrete Euler-Lagrange Equations For three consecutive points (x, y, z) on a trajectory, we have 0 A(2y x 0 y 0 ) + B(x + z x 0 y 0 ) + D y 0 x 0. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

20 Estimating the Parameters Discrete Euler-Lagrange Equations For three consecutive points (x, y, z) on a trajectory, we have 0 A(2y x 0 y 0 ) + B(x + z x 0 y 0 ) + D y 0 x 0. Estimating Lagrangian Parameters with Several Data Points Given data of consecutive triplets (x i, y i, z i ), we estimate A, B, and D up to scaling at a point (x 0, y 0 ) as follows. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

21 Estimating the Parameters Discrete Euler-Lagrange Equations For three consecutive points (x, y, z) on a trajectory, we have 0 A(2y x 0 y 0 ) + B(x + z x 0 y 0 ) + D y 0 x 0. Estimating Lagrangian Parameters with Several Data Points Given data of consecutive triplets (x i, y i, z i ), we estimate A, B, and D up to scaling at a point (x 0, y 0 ) as follows. Construct a matrix M whose rows are w i (2y i (x 0 + y 0 ) x i + z i (x 0 + y 0 ) y 0 x 0 ). Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

22 Estimating the Parameters Discrete Euler-Lagrange Equations For three consecutive points (x, y, z) on a trajectory, we have 0 A(2y x 0 y 0 ) + B(x + z x 0 y 0 ) + D y 0 x 0. Estimating Lagrangian Parameters with Several Data Points Given data of consecutive triplets (x i, y i, z i ), we estimate A, B, and D up to scaling at a point (x 0, y 0 ) as follows. Construct a matrix M whose rows are w i (2y i (x 0 + y 0 ) x i + z i (x 0 + y 0 ) y 0 x 0 ). ( AB ) For the correct values of the parameters, we will have M 0. D Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

23 Estimating the Parameters Discrete Euler-Lagrange Equations For three consecutive points (x, y, z) on a trajectory, we have 0 A(2y x 0 y 0 ) + B(x + z x 0 y 0 ) + D y 0 x 0. Estimating Lagrangian Parameters with Several Data Points Given data of consecutive triplets (x i, y i, z i ), we estimate A, B, and D up to scaling at a point (x 0, y 0 ) as follows. Construct a matrix M whose rows are w i (2y i (x 0 + y 0 ) x i + z i (x 0 + y 0 ) y 0 x 0 ). ( AB ) For the correct values of the parameters, we will have M 0. Estimate A, B, and D by the eigenvector corresponding to the least eigenvalue of M T M. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13 D

24 Assigning Weights to the Data Points The matrix of coefficients The ith row of M is w i (2y i (x 0 + y 0 ) x i + z i (x 0 + y 0 ) y 0 x 0 ). Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

25 Assigning Weights to the Data Points The matrix of coefficients The ith row of M is Distance w i (2y i (x 0 + y 0 ) x i + z i (x 0 + y 0 ) y 0 x 0 ). We define the distance bewteen (x 0, y 0 ) and (x, y) to be δ((x 0, y 0 ), (x, y)) 2 = x+y 2 x 0+y τs 2 where τ s is a parameter and τ is the timestep. y x τ y 0 x 0 τ 2, Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

26 Assigning Weights to the Data Points The matrix of coefficients The ith row of M is Distance w i (2y i (x 0 + y 0 ) x i + z i (x 0 + y 0 ) y 0 x 0 ). We define the distance bewteen (x 0, y 0 ) and (x, y) to be δ((x 0, y 0 ), (x, y)) 2 = x+y 2 x 0+y τs 2 where τ s is a parameter and τ is the timestep. Weights w i = exp where σ is another parameter. y x τ y 0 x 0 τ 2, ( 1 ( δ((x0 2σ 2, y 0 ), (x i, y i )) 2 + δ((x 0, y 0 ), (y i, z i )) 2) ), Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

27 The Simple Pendulum The Lagrangian L d (x, y) = τ ( 1 2 ( y x τ ) 2 ( ( x + y 1 cos 2 )) ). Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

28 The Simple Pendulum The Lagrangian L d (x, y) = τ ( 1 2 ( y x True Values of Lagrangian Parameters τ ) 2 ( ( x + y 1 cos 2 )) ). Using a Taylor approximation to the Lagrangian, we find that B A = 4 + τ 2 ( cos x0 +y 0 ) 2 4 τ 2 cos ( D x 0 +y 0 ), A = 4τ 2 y 0 x 0 2 sin ( x 0 +y 0 2 ) 4 τ 2 cos ( x 0 +y 0 2 ). Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

29 The Simple Pendulum The Lagrangian L d (x, y) = τ ( 1 2 ( y x True Values of Lagrangian Parameters τ ) 2 ( ( x + y 1 cos 2 )) ). Using a Taylor approximation to the Lagrangian, we find that B A = 4 + τ 2 ( cos x0 +y 0 ) 2 4 τ 2 cos ( D x 0 +y 0 ), A = 4τ 2 y 0 x 0 Parameters Computed From Trajectories 2 sin ( x 0 +y 0 2 ) 4 τ 2 cos ( x 0 +y 0 2 I computed the parameters from the trajectories with Matlab. The graphs of B A + 1 and D A y 0 x 0 are on the following slides. ). Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

30 x Pendulum simulation at high energy B/A+1 0 D y 0 x 0 / A x Pendulum Position Time Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

31 x Pendulum simulation at high energy with noise B/A+1 0 D y 0 x 0 / A x Pendulum Position Time Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

32 x Pendulum simulation at high energy with noise B/A+1 0 D y 0 x 0 / A x Pendulum Position Time Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

33 x Pendulum simulation at high energy with noise B/A+1 0 D y 0 x 0 / A x Pendulum Position Time Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

34 What Next? Future Directions Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

35 What Next? Future Directions Recover the Lagrangian from data of several trajectories, and then use it to predict new trajectories. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

36 What Next? Future Directions Recover the Lagrangian from data of several trajectories, and then use it to predict new trajectories. Investigate the best choices for τ s and σ. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

37 What Next? Future Directions Recover the Lagrangian from data of several trajectories, and then use it to predict new trajectories. Investigate the best choices for τ s and σ. Try adding other kinds of noise to the system. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

38 What Next? Future Directions Recover the Lagrangian from data of several trajectories, and then use it to predict new trajectories. Investigate the best choices for τ s and σ. Try adding other kinds of noise to the system. Try the method with real data. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

39 What Next? Future Directions Recover the Lagrangian from data of several trajectories, and then use it to predict new trajectories. Investigate the best choices for τ s and σ. Try adding other kinds of noise to the system. Try the method with real data. Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

40 What Next? Future Directions Recover the Lagrangian from data of several trajectories, and then use it to predict new trajectories. Investigate the best choices for τ s and σ. Try adding other kinds of noise to the system. Try the method with real data. Evan S. Gawlik, Patrick Mullen, Dmitry Pavlov, Jerrold E. Marsden, and Mathieu Desbrun, Geometric, variational discretization of continuum theories, Ari Stern and Mathieu Desbrun, Discrete geometric mechanics for variational time integrators, Discrete Differential Geometry: An Applied Introduction, 2006, pp Yakov Berchenko-Kogan (MIT) Uncovering the Lagrangian 7 January, / 13

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