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July  2021, 8(3): 241-271. doi: 10.3934/jcd.2021011

Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics

Luis C. García-Naranjo 1, and  Mats Vermeeren 2,,

1. 

Departamento de Matemáticas y Mecánica, IIMAS, UNAM, Apdo. Postal 20-126, 01000 Mexico City, Mexico, Current address: Dipartimento di Matematica "Tullio Levi-Civita", Università di Padova, Via Trieste 63, 35121 Padova, Italy
2. 

School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

* Corresponding author: Mats Vermeeren

Received  September 2020 Revised  February 2021 Published  July 2021 Early access  May 2021

We propose a discretization of vector fields that are Hamiltonian up to multiplication by a positive function on the phase space that may be interpreted as a time reparametrization. We construct a family of maps, labeled by an arbitrary $ \ell \in \mathbb{N} $ indicating the desired order of accuracy, and prove that our method is structure preserving in the sense that the discrete flow is interpolated to order $ \ell $ by the flow of a continuous system possessing the same structure as the vector field that is being discretized. In particular, our discretization preserves a smooth measure on the phase space to the arbitrary order $ \ell $. We present applications to a remarkable class of nonholonomic mechanical systems that allow Hamiltonization. To our best knowledge, these results provide the first instance of a measure preserving discretization (to arbitrary order) of measure preserving nonholonomic systems.

Keywords: Geometric integration, nonholonomic mechanics, measure preservation, conformally Hamiltonian systems.
Mathematics Subject Classification: Primary: 37M15; Secondary: 37C40, 37J60, 70G45.
Citation: Luis C. García-Naranjo, Mats Vermeeren. Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics. Journal of Computational Dynamics, 2021, 8 (3) : 241-271. doi: 10.3934/jcd.2021011
References:
[1]

A. Anahory Simoes, J. C. Marrero and D. Martín de Diego, Exact discrete Lagrangian mechanics for nonholonomic mechanics, preprint, 2020, arXiv: 2003.11362.

[2]

L. Bates and J. Śniatycki, Nonholonomic reduction, Reports on Mathematical Physics, 32 (1993), 99-115.  doi: 10.1016/0034-4877(93)90073-N.

[3]

G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms, Journal of Statistical Physics, 74 (1994), 1117-1143.  doi: 10.1007/BF02188219.

[4]

A. M. Bloch, P. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Archive for Rational Mechanics and Analysis, 136 (1996), 21-99.  doi: 10.1007/BF02199365.

[5]

A. V. Borisov and I. S. Mamaev, Isomorphism and Hamilton representation of some nonholonomic systems, Siberian Mathematical Journal, 48 (2007), 26-36.  doi: 10.1007/s11202-007-0004-6.

[6]

A. V. Borisov and I. S. Mamaev, Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Regular and Chaotic Dynamics, 13 (2008), 443-490.  doi: 10.1134/S1560354708050079.

[7]

A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Regular and Chaotic Dynamics, 18 (2013), 277-328.  doi: 10.1134/S1560354713030064.

[8]

R. C. Calleja, A. Celletti and R. de la Llave, A KAM theory for conformally symplectic systems: Efficient algorithms and their validation, Journal of Differential Equations, 255 (2013), 978-1049.  doi: 10.1016/j.jde.2013.05.001.

[9]

F. Cantrijn, J. Cortés, M. De León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Mathematical Proceedings of the Cambridge Philosophical Society, 132 (2002), 323-351.  doi: 10.1017/S0305004101005679.

[10]

S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, Regular and Chaotic Dynamics, 13 (2008), 369-376.  doi: 10.1134/S1560354708040102.

[11]

J. Cortés and S. Martínez, Non-holonomic integrators, Nonlinearity, 14 (2001), 1365-1392.  doi: 10.1088/0951-7715/14/5/322.

[12]

K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The Breadth of Symplectic and Poisson Geometry Birkhäuser Boston, Boston, MA, (2005), 75–120. doi: 10.1007/0-8176-4419-9_4.

[13]

Y. N. Fedorov and B. Jovanovic, Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces, Journal of Nonlinear Science, 14 (2004), 341-381.  doi: 10.1007/s00332-004-0603-3.

[14]

Y. N. Fedorov and B. Jovanović, Hamiltonization of the generalized Veselova LR system, Regular and Chaotic Dynamics, 14 (2009), 495-505.  doi: 10.1134/S1560354709040066.

[15]

Y. N. Fedorov, L. C. García-Naranjo and J. C. Marrero, Unimodularity and preservation of volumes in nonholonomic mechanics, Journal of Nonlinear Science, 25 (2015), 203-246.  doi: 10.1007/s00332-014-9227-4.

[16]

O. E. Fernandez, T. Mestdag and A. M. Bloch, A generalization of Chaplygin's reducibility theorem, Regular and Chaotic Dynamics, 14 (2009), 635-655.  doi: 10.1134/S1560354709060033.

[17]

O. E. Fernandez, A. M. Bloch and P. J. Olver, Variational integrators for Hamiltonizable nonholonomic systems, Journal of Geometric Mechanics, 4 (2012), 137-163.  doi: 10.3934/jgm.2012.4.137.

[18]

O. E. Fernandez, Poincaré transformations in nonholonomic mechanics, Applied Mathematics Letters, 43 (2015), 96-100.  doi: 10.1016/j.aml.2014.12.004.

[19]

S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928.  doi: 10.1088/0951-7715/21/8/009.

[20]

S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator, Nonlinearity, 28 (2015), 871-900.  doi: 10.1088/0951-7715/28/4/871.

[21]

L. C. García-Naranjo, Generalisation of Chaplygin's reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics, Journal of Physics A: Mathematical and Theoretical, 52 (2019), 205203, 16 pp. doi: 10.1088/1751-8121/ab15f8.

[22]

L. C. García-Naranjo and J. C. Marrero, The geometry of nonholonomic Chaplygin systems revisited, Nonlinearity, 33 (2020), 1297-1341.  doi: 10.1088/1361-6544/ab5c0a.

[23]

E. Hairer, Variable time step integration with symplectic methods, Applied Numerical Mathematics, 25 (1997), 219-227.  doi: 10.1016/S0168-9274(97)00061-5.

[24]

E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators, Numerische Mathematik, 76 (1997), 441-462.  doi: 10.1007/s002110050271.

[25]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.

[26]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic lagrangian systems on lie groupoids, Journal of Nonlinear Science, 18 (2008), 221-276.  doi: 10.1007/s00332-007-9012-8.

[27]

B. Jovanović, Note on a ball rolling over a sphere: Integrable Chaplygin system with an invariant measure without Chaplygin Hamiltonization, Theoretical and Applied Mechanics, 46 (2019), 97-108.  doi: 10.2298/TAM190322003J.

[28]

M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 61-84.  doi: 10.3934/dcdss.2010.3.61.

[29]

J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Archive for Rational Mechanics and Analysis, 118 (1992), 113-148.  doi: 10.1007/BF00375092.

[30]

B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511614118.

[31]

T. Levi-Civita, Sur la résolution qualitative du probleme restreint des trois corps, Acta Mathematica, 30 (1906), 305-327.  doi: 10.1007/BF02418577.

[32]

C.-M. Marle, A property of conformally Hamiltonian vector fields; Application to the Kepler problem, Journal of Geometric Mechanics, 4 (2012), 181-206.  doi: 10.3934/jgm.2012.4.181.

[33]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.

[34]

R. McLachlan and M. Perlmutter, Conformal Hamiltonian systems, Journal of Geometry and Physics, 39 (2001), 276-300.  doi: 10.1016/S0393-0440(01)00020-1.

[35]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.  doi: 10.1007/s00332-005-0698-1.

[36]

K. Modin and O. Verdier, What makes nonholonomic integrators work?, Numerische Mathematik, 145 (2020), 405-435.  doi: 10.1007/s00211-020-01126-y.

[37]

S. Reich, Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36 (1999), 1549-1570.  doi: 10.1137/S0036142997329797.

[38]

S. Stanchenko, Non-holonomic Chaplygin systems, Journal of Applied Mathematics and Mechanics, 53 (1989), 11-17.  doi: 10.1016/0021-8928(89)90126-3.

[39]

M. Vermeeren, Modified equations for variational integrators, Numerische Mathematik, 137 (2017), 1001-1037.  doi: 10.1007/s00211-017-0896-4.

[40]

M. Vermeeren, Support code for "Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics", https://github.com/mvermeeren/conf-ham-sys-2020. doi: 10.5281/zenodo.3988087.

[41]

A. P. Veselov and L. Veselova, Integrable nonholonomic systems on Lie groups, Mathematical Notes of the Academy of Sciences of the USSR, 44 (1988), 810-819.  doi: 10.1007/BF01158420.

[42]

P. Woronetz, Über die Bewegung eines starren Körpers, der ohne Gleitung auf einer beliebigen Fläche rollt, Mathematische Annalen, 70 (1911), 410-453.  doi: 10.1007/BF01564505.

show all references

References:
[1]

A. Anahory Simoes, J. C. Marrero and D. Martín de Diego, Exact discrete Lagrangian mechanics for nonholonomic mechanics, preprint, 2020, arXiv: 2003.11362.

[2]

L. Bates and J. Śniatycki, Nonholonomic reduction, Reports on Mathematical Physics, 32 (1993), 99-115.  doi: 10.1016/0034-4877(93)90073-N.

[3]

G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms, Journal of Statistical Physics, 74 (1994), 1117-1143.  doi: 10.1007/BF02188219.

[4]

A. M. Bloch, P. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Archive for Rational Mechanics and Analysis, 136 (1996), 21-99.  doi: 10.1007/BF02199365.

[5]

A. V. Borisov and I. S. Mamaev, Isomorphism and Hamilton representation of some nonholonomic systems, Siberian Mathematical Journal, 48 (2007), 26-36.  doi: 10.1007/s11202-007-0004-6.

[6]

A. V. Borisov and I. S. Mamaev, Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Regular and Chaotic Dynamics, 13 (2008), 443-490.  doi: 10.1134/S1560354708050079.

[7]

A. V. Borisov, I. S. Mamaev and I. A. Bizyaev, The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Regular and Chaotic Dynamics, 18 (2013), 277-328.  doi: 10.1134/S1560354713030064.

[8]

R. C. Calleja, A. Celletti and R. de la Llave, A KAM theory for conformally symplectic systems: Efficient algorithms and their validation, Journal of Differential Equations, 255 (2013), 978-1049.  doi: 10.1016/j.jde.2013.05.001.

[9]

F. Cantrijn, J. Cortés, M. De León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Mathematical Proceedings of the Cambridge Philosophical Society, 132 (2002), 323-351.  doi: 10.1017/S0305004101005679.

[10]

S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, Regular and Chaotic Dynamics, 13 (2008), 369-376.  doi: 10.1134/S1560354708040102.

[11]

J. Cortés and S. Martínez, Non-holonomic integrators, Nonlinearity, 14 (2001), 1365-1392.  doi: 10.1088/0951-7715/14/5/322.

[12]

K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The Breadth of Symplectic and Poisson Geometry Birkhäuser Boston, Boston, MA, (2005), 75–120. doi: 10.1007/0-8176-4419-9_4.

[13]

Y. N. Fedorov and B. Jovanovic, Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces, Journal of Nonlinear Science, 14 (2004), 341-381.  doi: 10.1007/s00332-004-0603-3.

[14]

Y. N. Fedorov and B. Jovanović, Hamiltonization of the generalized Veselova LR system, Regular and Chaotic Dynamics, 14 (2009), 495-505.  doi: 10.1134/S1560354709040066.

[15]

Y. N. Fedorov, L. C. García-Naranjo and J. C. Marrero, Unimodularity and preservation of volumes in nonholonomic mechanics, Journal of Nonlinear Science, 25 (2015), 203-246.  doi: 10.1007/s00332-014-9227-4.

[16]

O. E. Fernandez, T. Mestdag and A. M. Bloch, A generalization of Chaplygin's reducibility theorem, Regular and Chaotic Dynamics, 14 (2009), 635-655.  doi: 10.1134/S1560354709060033.

[17]

O. E. Fernandez, A. M. Bloch and P. J. Olver, Variational integrators for Hamiltonizable nonholonomic systems, Journal of Geometric Mechanics, 4 (2012), 137-163.  doi: 10.3934/jgm.2012.4.137.

[18]

O. E. Fernandez, Poincaré transformations in nonholonomic mechanics, Applied Mathematics Letters, 43 (2015), 96-100.  doi: 10.1016/j.aml.2014.12.004.

[19]

S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928.  doi: 10.1088/0951-7715/21/8/009.

[20]

S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator, Nonlinearity, 28 (2015), 871-900.  doi: 10.1088/0951-7715/28/4/871.

[21]

L. C. García-Naranjo, Generalisation of Chaplygin's reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics, Journal of Physics A: Mathematical and Theoretical, 52 (2019), 205203, 16 pp. doi: 10.1088/1751-8121/ab15f8.

[22]

L. C. García-Naranjo and J. C. Marrero, The geometry of nonholonomic Chaplygin systems revisited, Nonlinearity, 33 (2020), 1297-1341.  doi: 10.1088/1361-6544/ab5c0a.

[23]

E. Hairer, Variable time step integration with symplectic methods, Applied Numerical Mathematics, 25 (1997), 219-227.  doi: 10.1016/S0168-9274(97)00061-5.

[24]

E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators, Numerische Mathematik, 76 (1997), 441-462.  doi: 10.1007/s002110050271.

[25]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.

[26]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic lagrangian systems on lie groupoids, Journal of Nonlinear Science, 18 (2008), 221-276.  doi: 10.1007/s00332-007-9012-8.

[27]

B. Jovanović, Note on a ball rolling over a sphere: Integrable Chaplygin system with an invariant measure without Chaplygin Hamiltonization, Theoretical and Applied Mechanics, 46 (2019), 97-108.  doi: 10.2298/TAM190322003J.

[28]

M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 61-84.  doi: 10.3934/dcdss.2010.3.61.

[29]

J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Archive for Rational Mechanics and Analysis, 118 (1992), 113-148.  doi: 10.1007/BF00375092.

[30]

B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511614118.

[31]

T. Levi-Civita, Sur la résolution qualitative du probleme restreint des trois corps, Acta Mathematica, 30 (1906), 305-327.  doi: 10.1007/BF02418577.

[32]

C.-M. Marle, A property of conformally Hamiltonian vector fields; Application to the Kepler problem, Journal of Geometric Mechanics, 4 (2012), 181-206.  doi: 10.3934/jgm.2012.4.181.

[33]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.

[34]

R. McLachlan and M. Perlmutter, Conformal Hamiltonian systems, Journal of Geometry and Physics, 39 (2001), 276-300.  doi: 10.1016/S0393-0440(01)00020-1.

[35]

R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.  doi: 10.1007/s00332-005-0698-1.

[36]

K. Modin and O. Verdier, What makes nonholonomic integrators work?, Numerische Mathematik, 145 (2020), 405-435.  doi: 10.1007/s00211-020-01126-y.

[37]

S. Reich, Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36 (1999), 1549-1570.  doi: 10.1137/S0036142997329797.

[38]

S. Stanchenko, Non-holonomic Chaplygin systems, Journal of Applied Mathematics and Mechanics, 53 (1989), 11-17.  doi: 10.1016/0021-8928(89)90126-3.

[39]

M. Vermeeren, Modified equations for variational integrators, Numerische Mathematik, 137 (2017), 1001-1037.  doi: 10.1007/s00211-017-0896-4.

[40]

M. Vermeeren, Support code for "Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics", https://github.com/mvermeeren/conf-ham-sys-2020. doi: 10.5281/zenodo.3988087.

[41]

A. P. Veselov and L. Veselova, Integrable nonholonomic systems on Lie groups, Mathematical Notes of the Academy of Sciences of the USSR, 44 (1988), 810-819.  doi: 10.1007/BF01158420.

[42]

P. Woronetz, Über die Bewegung eines starren Körpers, der ohne Gleitung auf einer beliebigen Fläche rollt, Mathematische Annalen, 70 (1911), 410-453.  doi: 10.1007/BF01564505.

Figure 1.  Evolution of a spherical cloud (black) of $ 5000 $ points with radius $ 0.3 $ at times $ 0 $, $ 2.75 $, $ \ldots $, $ 13.75 $, projected to the $ (x,p_x) $ and $ (y,p_y) $ planes, using a high-accuracy method. The longer the time, the lighter the color
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Figure 2.  Evolution of the volume of a spherical cloud of $ 120 $ points, arranged in a $ 600 $-cell, with radius $ 0.01 $ and centered around $ (x,y,p_x,p_y) = (0,0,1,1) $, using our proposed method $ \Phi_h^{(4)} $ with time step $ h = 0.25 $ for the five variational integrators in (24) and a high-accuracy reference solution. The volume is computed with respect to the measure indicated at the top of each graph
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Figure 3.  Application of $ \Phi_h^{(4)} $ to the nonholonomic particle in a harmonic potential for each discretization in (24): overview of the numerical values of the different energy functions involved in the algorithm. The initial condition is $ (x,y,p_x),p_y) = (0,0,1,1) $ and the time step $ h = 0.25 $
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Figure 4.  Evolution of the volume of a spherical cloud of $ 120 $ points, arranged in a $ 600 $-cell, with radius $ 0.01 $ and centered around $ (x,y,p_x,p_y) = (0,0,1,1) $, using the method $ \Phi_h^{(0)} $ proposed in [23,37,17] with time step $ h = 0.25 $ for the five variational integrators in (24) and a high-accuracy reference solution. The volume is computed with respect to the measure indicated at the top of each graph
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Figure 5.  Application of $ \Phi_h^{(4)} $ to the free nonholonomic particle for each discretization in (24). Overview of the numerical values of the different energy functions involved in the algorithm and error norm of the solutions. The initial condition is $ (x,y,p_x,p_y) = (0,0,1,1) $ and the step size $ h = 0.25 $
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Figure 6.  Application of $ \Phi_h^{(0)} $ to the free nonholonomic particle for each discretization in (24): overview of the numerical values of the different energy functions involved in the algorithm and error norm of the solutions. The initial condition and the time step coincide with those of Figure 5
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Table 1.  Overview of all relevant Hamiltonians
Eqn. Notation Name Equations of motion
Original system
(3) $ H $ Conformal Hamiltonian $ {\bf{i}}_{ X} \Omega = \mathcal{N} \mathsf{d} H $
(5) $ K_E = \mathcal{N}(H-E) $ Altered Hamiltonian $ {\bf{i}}_{ X} \Omega = \mathsf{d} K_E $
on $ \{ K_E = 0 \} $
Modified system, interpolating numerical solutions
(16) $ K_{mod} $ Modified altered Hamiltonian $ {\bf{i}}_{X_{mod}} \Omega = \mathsf{d} K_{mod} $
on $ \{ K_{mod} = 0 \} $
$ K_{mod}^{(\ell)} $ Truncated modified altered Hamiltonian
(18) $ \mathcal{E} $ Modified conformal Hamiltonian $ {\bf{i}}_{X_{mod}} \Omega = \mathcal{N}_{mod} \mathsf{d} \mathcal{E} $
$ \mathcal{E}^{(\ell)} $ Truncated modified conformal Hamiltonian
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Eqn. Notation Name Equations of motion
Original system
(3) $ H $ Conformal Hamiltonian $ {\bf{i}}_{ X} \Omega = \mathcal{N} \mathsf{d} H $
(5) $ K_E = \mathcal{N}(H-E) $ Altered Hamiltonian $ {\bf{i}}_{ X} \Omega = \mathsf{d} K_E $
on $ \{ K_E = 0 \} $
Modified system, interpolating numerical solutions
(16) $ K_{mod} $ Modified altered Hamiltonian $ {\bf{i}}_{X_{mod}} \Omega = \mathsf{d} K_{mod} $
on $ \{ K_{mod} = 0 \} $
$ K_{mod}^{(\ell)} $ Truncated modified altered Hamiltonian
(18) $ \mathcal{E} $ Modified conformal Hamiltonian $ {\bf{i}}_{X_{mod}} \Omega = \mathcal{N}_{mod} \mathsf{d} \mathcal{E} $
$ \mathcal{E}^{(\ell)} $ Truncated modified conformal Hamiltonian
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