- Email: [email protected]

ScienceDirect

IFAC PapersOnLine 51-3 (2018) 1–6

Homogeneous Homogeneous Hamiltonian Hamiltonian Control Control HomogeneousSystems Hamiltonian Control HomogeneousSystems Hamiltonian Control Systems Part I: Geometric Formulation Part I: Geometric Formulation Systems Part I: Geometric Formulation ∗ ∗∗ Part I: Geometric Formulation Arjan van der Schaft Bernhard ∗∗ Arjan van der Schaft ∗∗ Bernhard Maschke Maschke ∗∗

Arjan van der Schaft ∗∗ Bernhard Maschke ∗∗ ∗∗ Arjan van der Schaft Bernhard Maschke ∗ ∗∗ Science, Johann Bernoulli Institute for Mathematics Computer Arjan van der Schaft Bernhardand Maschke Johann Institute for Mathematics and Computer Science, Johann Bernoulli Bernoulli Institute for Mathematics and Computer Science, University of Groningen, the Netherlands Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, the Netherlands University of Groningen, the Netherlands ∗ University of Groningen, the Netherlands (e-mail: [email protected]) Johann Bernoulli Institute for Mathematics and Computer Science, (e-mail: [email protected]) (e-mail: [email protected]) ∗∗ (e-mail:Bernard [email protected]) ee Claude Lyon 1, LAGEP, France ∗∗ University of Groningen, theCNRS, Netherlands ∗∗ Universit´ Universit´ Claude Bernard CNRS, LAGEP, e Claude Bernard Lyon Lyon 1, 1, CNRS, LAGEP, France France ∗∗ Universit´ ∗∗ (e-mail: [email protected]) Universit´ e Claude Bernard Lyon 1, CNRS, LAGEP, France (e-mail: [email protected]) (e-mail: (e-mail: [email protected]) [email protected]) ∗∗ (e-mail: [email protected]) Universit´ e Claude Bernard Lyon 1, CNRS, LAGEP, France (e-mail: [email protected]) Abstract: Contact geometry has been successfully employed for the geometric formulation Abstract: Contact Contact geometry geometry has has been been successfully successfully employed employed for for the the geometric geometric formulation formulation Abstract: and control of systems containing thermodynamic components. In this paper we elaborate on Abstract: Contact geometry has been successfully employed for the geometric formulation and control control of of systems systems containing containing thermodynamic thermodynamic components. components. In In this this paper paper we we elaborate elaborate on on and and control of systems containing thermodynamic components. In this paper we elaborate on the geometric theory of symplectization of contact manifolds in order to lift contact control Abstract: Contact geometry has been successfully employed for the geometric formulation the geometric geometric theory theory of of symplectization symplectization of of contact contact manifolds manifolds in in order order to to lift lift contact contact control control the systems to Hamiltonian control systems with a Hamiltonian that is homogeneous in the co-state the geometric symplectization of contact manifolds to lift we contact control and control of theory systemsofcontaining thermodynamic components. Inorder this paper on systems to Hamiltonian Hamiltonian control systems with Hamiltonian thatinis is homogeneous inelaborate the co-state co-state systems to control systems with aa Hamiltonian that homogeneous in the variables. This provides a new view on contact control systems as used in thermodynamics, and systems to Hamiltonian control systems with a Hamiltonian that is homogeneous in the co-state the geometric theory of symplectization of contact manifolds in order to lift contact control variables. This provides a new view on contact control systems as used in thermodynamics, and variables. This provides a new the view on contact control systems as usedHamiltonian in thermodynamics, and offers possibilities for unifying theories of contact control systems, input-output variables. provides a new the view on contact control systems asisused in thermodynamics, and systems toThis Hamiltonian control systems with a Hamiltonian that homogeneous ininput-output the co-state offers possibilities possibilities for unifying unifying theories of contact contact control systems, Hamiltonian input-output offers for the theories of control systems, Hamiltonian offers possibilities for unifying the theories of contact control systems, Hamiltonian input-output systems and port-Hamiltonian systems. variables. This provides a new view on contact control systems as used in thermodynamics, and systems and port-Hamiltonian systems. systems and port-Hamiltonian systems. systems and port-Hamiltonian systems. offers possibilities for unifying the theories of contact control systems, Hamiltonian input-output © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. systems andHamiltonian port-Hamiltonian systems. Keywords: systems, nonlinear control, thermodynamics, contact geometry, Keywords: Hamiltonian Hamiltonian systems, systems, nonlinear nonlinear control, control, thermodynamics, thermodynamics, contact contact geometry, geometry, Keywords: Keywords: Hamiltonian systems, nonlinear control, thermodynamics, contact geometry, homogeneous functions, invariant Lagrangian manifolds, liftings homogeneous functions, invariant Lagrangian manifolds, liftings homogeneous functions, invariant Lagrangian manifolds, liftings homogeneous functions, systems, invariantnonlinear Lagrangian manifolds, liftings Keywords: Hamiltonian control, thermodynamics, contact geometry, homogeneous functions, invariant Lagrangian manifolds, liftings 1. On 1. INTRODUCTION INTRODUCTION On the the other other hand, hand, it it is is well-known well-known that that contact contact manifolds manifolds 1. On the other it that manifolds 1. INTRODUCTION INTRODUCTION On the other hand, hand, it is is well-known well-known that contact contact manifolds can be naturally symplectized to symplectic manifolds with can be naturally symplectized to symplectic manifolds with can be naturally symplectized to symplectic manifolds with an additional structure of homogeneity; see Arnold (1989), The geometric formulation of Hamiltonian dynamics, incan be naturally symplectized to symplectic manifolds with 1. INTRODUCTION On the other hand, it is well-known that contact manifolds an additional structure of homogeneity; see Arnold (1989), The geometric formulation of Hamiltonian dynamics, inan additional structure of homogeneity; see Arnold (1989), The geometric formulation of Hamiltonian dynamics, inLibermann and Marle (1987) for textbook expositions. cluding geometric mechanics, through symplectic geometry an additional structure of homogeneity; see Arnold (1989), can be naturally symplectized to symplectic manifolds with The geometric formulation of Hamiltonian dynamics, inand Marle for expositions. cluding geometric geometric mechanics, mechanics, through through symplectic symplectic geometry geometry Libermann Libermann andapplications Marle (1987) (1987) for textbook textbook expositions. cluding However, the of, and motivations for, this has been a flourishing area of research for quite some time; Libermann and Marle (1987) for textbook expositions. an additional structure of homogeneity; see Arnold (1989), cluding geometric mechanics, through symplectic geometry The geometric formulation of Hamiltonian dynamics, inthe applications of, and motivations for, this has been been aa flourishing flourishing area area of of research research for for quite quite some some time; time; However, However, the applications of, and motivations for, has However, the applications of,confined and motivations for, this this theory, seem to have been to time-dependent see e.g. the classical textbooks Arnold (1989), Abraham Libermann and Marle (1987) for textbook expositions. has been a flourishing area of research for quite some time; cluding geometric mechanics, through symplectic geometry theory, seem to have been confined to time-dependent see e.g. e.g. the the classical classical textbooks textbooks Arnold Arnold (1989), (1989), Abraham Abraham theory, seem to have been confined to time-dependent see Hamiltonian mechanics (see e.g. Libermann and Marle see e.g. the classical textbooks Arnold (1989), Abraham and Marsden (1978), Libermann and Marle (1987). This theory, seem to have been confined to time-dependent However, the applications of, and motivations for, this has a flourishing of research quite(1987). some time; mechanics (see Libermann and Marle and been Marsden (1978), area Libermann and for Marle (1987). This Hamiltonian Hamiltonian mechanics (see e.g. e.g. Libermann and(1989). Marle and Marsden (1978), Libermann and Marle This (1987)), or partial differential equations Arnold theory was also underlying the formulation of Hamiltonian Hamiltonian mechanics (see e.g. Libermann and Marle theory, seem to have been confined to time-dependent and Marsden (1978), Libermann and Marle (1987). This see e.g. the classical textbooks Arnold (1989), Abraham or partial differential equations Arnold (1989). theory was was also also underlying underlying the the formulation formulation of of Hamiltonian Hamiltonian (1987)), (1987)), or differential equations Arnold (1989). theory or partial partial differential equations Arnold Only in Balian and Valentin (2001) it was argued that input-output systems, starting with the groundbreaking Hamiltonian mechanics (see e.g. Libermann and(1989). Marle theory was also underlying the formulation of Hamiltonian and Marsden (1978), Libermann andthe Marle (1987). This (1987)), Only in Balian and Valentin (2001) it was argued that input-output systems, starting with the groundbreaking Only in Balian and Valentin (2001) it was argued that input-output systems, starting with groundbreaking Only in Balian anddifferential Valentin (2001) it was argued that the symplectization of contact manifolds provides a new input-output systems, starting with the groundbreaking paper Brockett (1977) and continued in e.g. Van der Schaft (1987)), or partial equations Arnold (1989). theory was also underlying the formulation of Hamiltonian the symplectization of contact manifolds provides a paper Brockett Brockett (1977) (1977) and and continued continued in in e.g. e.g. Van Van der der Schaft Schaft the symplectization of contact manifolds provides a new new paper and insightful view point to thermodynamics as well. paper Brockett (1977) and continued in e.g. Van Schaft (1982); Van der Schaft and Crouch (1987); Van der the symplectization of contact manifolds provides a new Only in Balian and Valentin (2001) it was argued that input-output systems, with the groundbreaking (1982); Van Van der der Schaft starting and Crouch Crouch (1987); Van der der Schaft Schaft and and insightful insightful view view point point to to thermodynamics thermodynamics as as well. well. (1982); Schaft and (1987); Van (1989). By generalizing symplectic and Poisson structures and insightful view point to thermodynamics as well. the symplectization of contact manifolds provides a new (1982); Van der Schaft and Crouch (1987); Van der paper Brockett (1977) and continued in e.g. Van Schaft (1989). By By generalizing generalizing symplectic symplectic and and Poisson Poisson structures structures In this paper, inspired by Balian and Valentin (2001) and (1989). In this paper, inspired by Balian and Valentin (2001) and to Dirac structures, and by emphasizing port-based modIn this paper, inspired by Balian and Valentin (2001) and and insightful view point to thermodynamics as well. (1989). By generalizing symplectic and Poisson structures (1982); Van der Schaft and Crouch (1987); Van der Schaft to Dirac Dirac structures, structures, and and by by emphasizing emphasizing port-based port-based modmod- In motivated by control problems in multi-physics systems this paper, inspired by Balian and Valentin (2001) and to motivated by control problems in multi-physics systems to Dirac structures, and by emphasizing port-based modeling of multi-physics systems it also led to the theory of motivated by control problems in multi-physics systems (1989). By generalizing symplectic and Poisson structures eling of of multi-physics multi-physics systems systems it it also also led led to to the the theory theory of of In including thermodynamic components, we will further exmotivated by control problems in multi-physics systems eling this paper, inspired by Balian and Valentin (2001) and thermodynamic components, we will further exeling of multi-physics systems ite.g. also led port-based to the of including port-Hamiltonian systems; see Maschke and van der including thermodynamic components, we will further exto Dirac structures, and by see emphasizing port-Hamiltonian systems; see e.g. Maschke andtheory vanmodder including thermodynamic components, we will further expand the symplectization point of view. This will result port-Hamiltonian systems; e.g. Maschke and van der motivated by control problems in multi-physics systems pand the symplectization point of view. This will result Schaft (1992); Van der Schaft and Maschke (1995), and the pand the symplectization point of view. This will result port-Hamiltonian systems; see e.g. Maschke and van der eling of multi-physics systems it also led to the theory of Schaft (1992); Van der Schaft and Maschke (1995), and the pand the symplectization point of view. This will result in the definition of homogeneous Hamiltonian control sysSchaft (1992); Van der Schaft and Maschke (1995),(2014). and the including thermodynamic components, we willcontrol furthersysexthe definition of Hamiltonian introductory der Schaft and Jeltsema in the by definition of homogeneous homogeneous Hamiltonian control sysSchaft (1992);surveyVan Vansystems; der Schaft and Maschke (1995), and der the in port-Hamiltonian e.g. Maschke and (2014). van introductory surveyVan dersee Schaft and Jeltsema (2014). tems, symplectization of the notion of contact control in the the definition of homogeneous Hamiltonian control sysintroductory surveyVan der Schaft and Jeltsema pand symplectization point of view. This will result tems, by symplectization of the notion of contact control introductory surveyVan der Schaft and Jeltsema (2014). tems, bydeveloped symplectization of the notion of contact control Schaft (1992); Van der Schaft of andthermodynamics Maschke (1995), and the systems Eberard et al. Favache et al. The geometric formulation has retems, symplectization of the of contact control the by definition of in homogeneous Hamiltonian control systems developed in Eberard et notion al. (2007); (2007); Favache etsysal. The geometric geometric formulation of thermodynamics thermodynamics has rere- in systems developed in Eberard et al. (2007); Favache et al. The formulation of has introductory surveyVan der Schaft and Jeltsema (2014). systems developed in Eberard et al. (2007); Favache et al. (2009, 2010); Ramirez et al. (2013a, 2017), Merker and mained more elusive. Starting from Gibbs fundamental tems, by symplectization of the notion of contact control The geometric formulation of thermodynamics has re(2009, 2010); 2010); Ramirez Ramirez et et al. al. (2013a, (2013a, 2017), 2017), Merker Merker and and mained more more elusive. elusive. Starting Starting from from Gibbs Gibbs fundamental fundamental (2009, mained Kr¨ u ger (2013). Because of space limitations, this paper mained more elusive. Starting from Gibbs fundamental relation, contact geometry was recognized as an essential (2009, 2010); Ramirez et al. (2013a, 2017), Merker and developed in Eberard et al.limitations, (2007); Favache et al. The geometric of recognized thermodynamics has re- systems Kr¨ u (2013). Because of space this paper relation, contactformulation geometry was was recognized as an an essential essential Kr¨ uger ger (2013). Because of theory space limitations, thisHamilpaper relation, contact geometry as will focus on the geometric of homogeneous element; see e.g. Hermann (1973), Mruga lla (1978, 2000b); Kr¨ uger (2013). Because ofal. space limitations, thisHamilpaper (2009, 2010); Ramirez et (2013a, 2017), Merker and relation, contact geometry was recognized as an essential mained more elusive. Starting from Gibbs fundamental will focus on the geometric theory of homogeneous element; see e.g. Hermann (1973), Mruga a (1978, 2000b); will focus on the geometric theory of homogeneous Hamilelement; see e.g. Hermann (1973),and Mruga la (1978, 2000b); tonian control systems, only providing the formulation of Mruga l a et al. (1991); Balian Valentin (2001) for will focus on the geometric theory of homogeneous HamilKr¨ u ger (2013). Because of space limitations, this paper element; see e.g. Hermann (1973), Mruga l a (1978, 2000b); relation, contact geometry was recognized as an essential control systems, only providing the formulation of Mrugallaa et et al. al. (1991); (1991); Balian Balian and and Valentin Valentin (2001) (2001) for for tonian tonian control systems, only providing the formulation of Mruga tonian control systems, only providing the formulation of simple thermodynamic systems as an example. Further important contributions. Recently, the interest in contactwill focus on the geometric theory of homogeneous HamilMruga l a et al. (1991); Balian and Valentin (2001) for element; see e.g. Hermann (1973), Mruga l a (1978, 2000b); simple thermodynamic systems as an example. Further important contributions. contributions. Recently, Recently, the the interest interest in in contactcontact- simple thermodynamic systems as an example. Further important examples of multi-physics systems in this framework will important contributions. Recently, the interest in contactgeometric descriptions of thermodynamics has been intensimple thermodynamic systems as an example. Further control systems, only providing the framework formulationwill of Mruga la et al. (1991);of and Valentin (2001) for tonian examples of systems in geometric descriptions of Balian thermodynamics has been been intenexamples ofinmulti-physics multi-physics systems in this this framework will geometric descriptions thermodynamics has intenbe treated the companion paper and van der sified; see e.g. Eberard et al. (2007), Favache al. (2009, examples ofinmulti-physics systems framework will thermodynamic systems asinMaschke anthis example. Further geometric descriptions of thermodynamics haset been inten- simple important contributions. Recently, the interest in contactbe treated the companion paper Maschke and van der sified; see e.g. Eberard et al. (2007), Favache et al. (2009, be treated in the companion paper Maschke and van der sified; see e.g. Eberard et al. (2007), Favache et al. (2009, Schaft (2018). 2010), Ramirez et al. (2013a, 2017), Merker and u ger be treated the companion paperinMaschke and van will der ofinmulti-physics systems this framework sified; e.g. Eberard et thermodynamics al. (2007), al.Kr¨ (2009, geometric descriptions of haset been intenSchaft (2018). 2010), see Ramirez et al. al. (2013a, (2013a, 2017),Favache Merker and Kr¨ uger ger examples Schaft (2018). 2010), Ramirez et 2017), Merker and Kr¨ u Schaft (2018). (2013), Bravetti al. (2015). In particular, this has led to be treated in the companion paper Maschke and van der 2010), Ramirez et (2013a, 2017), Merker and Kr¨ u ger sified; see e.g. Eberard et al. (2007), Favache et al. (2009, (2013), Bravetti Bravetti et et al. (2015). (2015). In In particular, particular, this this has has led led to to (2013), (2013), Bravetti et al. al.control (2015). In2017), particular, this hasKr¨ led to Schaft the theory of contact systems, see e.g. Favache et al. (2018). 2. THE SYMPLECTIZATION OF CONTACT 2010), Ramirez (2013a, Merker and u ger the theory of contact control systems, see e.g. Favache et al. 2. THE SYMPLECTIZATION OF CONTACT the theory of contact control systems, see e.g. Favache et al. 2. OF (2009, 2010), Ramirez et al. (2013a, 2017), Merker 2. THE THE SYMPLECTIZATION SYMPLECTIZATION OF CONTACT CONTACT the theory of contact see e.g.this Favache etand al. GEOMETRY; CONTACT AND HOMOGENEOUS (2013), Bravetti et al.control (2015). In particular, has led to (2009, 2010), Ramirez et al. al.systems, (2013a, 2017), Merker and GEOMETRY; CONTACT AND HOMOGENEOUS (2009, 2010), Ramirez et (2013a, 2017), Merker and GEOMETRY; CONTACT AND HOMOGENEOUS Kr¨ u ger (2013). (2009, 2010), Ramirez et al. (2013a, 2017), Merker and HAMILTONIAN VECTOR FIELDS the theory of contact control systems, see e.g. Favache et al. GEOMETRY; CONTACT AND HOMOGENEOUS 2. THE SYMPLECTIZATION OF CONTACT Kr¨ uger (2013). (2013). HAMILTONIAN Kr¨ u HAMILTONIAN VECTOR VECTOR FIELDS FIELDS Kr¨ uger ger 2010), (2013).Ramirez et al. (2013a, 2017), Merker and (2009, HAMILTONIAN VECTOR FIELDS GEOMETRY; CONTACT AND HOMOGENEOUS paper was written as the result of the stay in 2017 of the This This paper was written as result of the in of This paper written as the the result the stay stay in 2017 2017 of the the Kr¨ uger (2013). First, notion manifold will VECTOR FIELDS first author as was invited professor with the of research group DYCOP of First, the the HAMILTONIAN notion of of aaa contact contact manifold will be be recalled. recalled. This paper was written as the result of the stay in 2017 of the First, the notion of contact manifold will be recalled. first author as invited professor with the research group DYCOP of first author as invited professor with the research group DYCOP of Then, following Arnold (1989); Libermann and Marle First, the notion of a contact manifold will be recalled. theThis Laboratoire d’Automatique et de G´ e nie des Proc´ e d´ e s (LAGEP), first author as invited professor with the research group DYCOP of Then, following Arnold (1989); Libermann and Marle paper was written as the result of the stay in 2017 of the the Laboratoire d’Automatique et de G´ e nie des Proc´ e d´ e s (LAGEP), Then, following Arnold (1989); Libermann and Marle the Laboratoire d’Automatique G´ enie des Proc´ ed´ es (LAGEP), Then, following Arnold (1989); Libermann andrecalled. Marle (1987), we will discuss the natural symplectization of Universit´ e Claude Bernard Lyonet 1.de First, the notion of a contact manifold will be the Laboratoire d’Automatique et de the G´ enie des Proc´ ed´ es DYCOP (LAGEP), first author as invited professor with research group of (1987), we will discuss the natural symplectization Universit´ e Claude Bernard Lyon 1. (1987), we will discuss the natural symplectization of of Universit´ e Claude Bernard Lyon 1. (1987), we will discuss the natural symplectization of Universit´ e Claude Bernard Lyon 1. Then, following Arnold (1989); Libermann and Marle the Laboratoire d’Automatique et de G´ enie des Proc´ ed´ es (LAGEP), (1987), we will discuss the natural symplectization of Universit´ e Claude Bernard Lyon 1. Copyright 1 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) ∗ ∗ ∗ ∗ ∗

Copyright © 2018 1 Copyright © under 2018 IFAC IFAC 1 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 1 10.1016/j.ifacol.2018.06.001 Copyright © 2018 IFAC 1

IFAC LHMNC 2018 2 Valparaíso, Chile, May 1-4, 2018

Arjan van der Schaft et al. / IFAC PapersOnLine 51-3 (2018) 1–6

see Arnold (1989). In case of M = P(T ∗ Q) this is very clear: the symplectization is simply given by the cotangent bundle T ∗ Q without its zero-section; denoted by T0∗ Q. The projection from T0∗ Q to P(T ∗ Q), taking non-zero cotangent vectors to the corresponding equivalence classes (rays) in the cotangent space will be denoted by π : T0∗ Q → P(T ∗ Q) (4) The cotangent bundle T ∗ Q, as well as T0∗ Q, is endowed with its canonical 1-form α (called Liouville form), in natural coordinates (q, p) = (q 0 , q 1 , · · · , q n , p0 , p1 , · · · , pn ) (5) ∗ for T Q given by n pi dq i , (6) α=

contact manifolds, and describe how contact vector fields on a contact manifold are naturally lifted to ordinary Hamiltonian vector fields on the symplectization, with respect to a Hamiltonian function that is homogeneous of degree 1 in the co-state variables. In the same vein Legendre submanifolds of a contact manifold are lifted to Lagrangian submanifolds of the symplectization, with a generating function that is again homogeneous of degree 1 in the co-state variables. As main example we will discuss simple thermodynamic systems (Section 3). Finally in Section 4 this will culminate in the definition of a homogeneous Hamiltonian control system. 2.1 Contact manifolds Let us start by recalling the definition of a contact manifold; see Arnold (1989); Libermann and Marle (1987). A contact manifold is a (2n + 1)-dimensional manifold M equipped with a maximally non-integrable field of hyperplanes ξ. This means that ξ = ker θ ⊂ T M for a, possibly only locally defined, 1-form θ on M satisfying θ ∧ (dθ)n = 0 (1) By Darboux’s theorem there exist local coordinates (2) q 0 , q 1 , · · · , q n , γ1 , · · · , γn for M such that n γi dq i (3) θ = dq 0 −

i=0

as well as its canonical symplectic form ω := dα expressed as n ω = dα = dpi ∧ dq i (7) i=0

Note that the contact form θ on P(T ∗ Q), as well as local Darboux coordinates as in (2),(3) for θ, are obtained from α and the natural coordinates (5) for T0∗ Q as follows. Consider local coordinates q 0 , q 1 , · · · , q n , p0 , p1 , · · · , pn as in (5), and consider a neighborhood where e.g. p0 = 0. Then define pi γi := − , i = 1, · · · , n (8) p0 It follows that n α = p0 (dq 0 − γi dq i ) = p0 θ (9)

i=1

The canonical example of a contact manifold is as follows; see e.g. Arnold (1989). Consider an (n + 1)-dimensional manifold Q, and consider at any point q ∈ Q the set of n-dimensional subspaces of the (n + 1)-dimensional tangent space Tq Q. This defines an (2n + 1)-dimensional manifold M , which is a vector bundle over the base space manifold Q with projection denoted by Π. A field of hyperplanes ξ on M is defined by considering at each point (q, S) ∈ M , with q ∈ Q and S an n-dimensional subspace of Tq Q, the subspace of tangent vectors to M at (q, S) which are such that the projection under Π to Tq Q is contained in S. It can be readily verified that the thus defined field of hyperplanes ξ is indeed maximally non-integrable. Obviously, any n-dimensional subspace of the tangent space Tq Q can be identified with all non-zero multiples of some cotangent vector in Tq∗ Q, whose kernel equals this subspace. Hence it follows that the thus defined canonical contact manifold is equal to the projectivization P(T ∗ Q) of the cotangent bundle T ∗ Q, i.e., the fiber bundle over Q with fiber at any point q ∈ Q given by the projective space P(Tq∗ Q). (Recall that elements of P(Tq∗ Q) are identified with rays in Tq∗ Q, i.e. non-zero multiples of non-zero cotangent vectors.) Furthermore, q 0 , · · · , q n in (2) can be taken to be coordinates for Q.

i=1

Performing the same construction for any coordinate pj = 0 (instead of p0 ) this yields different definitions for γ1 , · · · , γn . As we will see later on this corresponds in the case of thermodynamical systems to the choice of different representations; e.g. the energy representation of a thermodynamical systems instead of its entropy representation. 2.3 Homogeneous Hamiltonians

In order to elucidate the exact connection between the contact manifold P(T ∗ Q) with contact form θ and the symplectic manifold T0∗ Q with canonical 1-form α, and in particular the connection between (contact) Hamiltonian dynamics on both of them, one needs the notion of homogeneous functions on T0∗ Q. Definition 1. Let k = 0, 1, · · · . A function h : T0∗ Q → R is called homogeneous of degree k (in the variables p0 , p1 , · · · , pn ) if h(q 0 , q 1 , · · · , q n , λp0 , λp1 , · · · , λpn ) =

From Darboux’s theorem it follows that any (2n + 1)dimensional manifold M is locally contactomorphic to a canonical contact manifold P(T ∗ Q) with dim Q = n+1. In the rest of the paper, for clarity of exposition, we will only consider the canonical contact manifolds given as P(T ∗ Q) for some (n + 1)-dimensional manifold Q.

λk h(q 0 , q 1 , · · · , q n , p0 , p1 , · · · , pn ),

∀λ = 0

(10)

Note that this definition is independent of the choice of natural coordinates (5) for T0∗ Q. This will also become clear from the geometric characterization of homogeneous functions given below. First let us recall Euler’s theorem on homogeneous functions. Theorem 2. A differentiable function h : T0∗ Q → R is homogeneous of degree k (in the variables p0 , p1 , · · · , pn ) if and only if

2.2 Symplectization of contact manifolds Any (2n+1)-dimensional contact manifold M can be symplectized to an (2n + 2)-dimensional symplectic manifold; 2

IFAC LHMNC 2018 Valparaíso, Chile, May 1-4, 2018

n i=0

pi

∂h (q, p) = kh(q, p), ∂pi

Arjan van der Schaft et al. / IFAC PapersOnLine 51-3 (2018) 1–6

for all (q, p) ∈ T0∗ Q

3

homogeneous of degree 1. This can be verified by the following computation, cf. Arnold (1989). Take natural coordinates q 0 , q 1 , · · · , q n , p0 , p1 , · · · , pn for T0∗ Q, and Darboux coordinates q 0 , q 1 , · · · , q n , γ1 , · · · , γn for P(T ∗ Q) as in (8),(9); i.e., γi = − pp0i , i = 1, · · · , n. Now consider any differentiable function K : P(T ∗ Q) → R (17) expressed in Darboux coordinates (2). Then define the corresponding function h : T0∗ Q → R by setting

(11)

Geometrically, Euler’s theorem can be equivalently stated as follows. Recall that the Hamiltonian vector field Xh on T0∗ Q with symplectic form ω = dα corresponding to a Hamiltonian h : T0∗ Q → R is defined by (12) iXh ω = −dh In natural coordinates (q, p) for T0∗ Q this amounts to the standard Hamiltonian equations ∂h q˙i = (q, p), i = 0, 1, · · · , n ∂pi (13) ∂h p˙ i = − (q, p), i = 0, 1, · · · , n ∂q i Then the following formulation of Euler’s theorem is immediate. Proposition 3. h : T0∗ Q → R is homogeneous of degree k iff α(Xh ) = kh (14)

h(q 0 , q 1 , · · · , q n , −1, γ1 , ·, γn ) := K(q 0 , q 1 , · · · , q n , γ1 , ·, γn ) (18) By requiring h to be homogeneous of degree 1 we extend its definition to T0∗ Q as h(q 0 , q 1 , · · · , q n , p0 , p1 , · · · , pn ) = p0 p1 pn −p0 h(q 0 , q 1 , · · · , q n , − , − , · · · , − ) = p0 p0 p0 −p0 h(q 0 , q 1 , · · · , q n , −1, γ1 , · · · , γn ) =

(19)

−p0 K(q 0 , q 1 , · · · , q n , γ1 , · · · , γn )

with γi := − pp0i , i = 1, · · · , n. This implies that the derivatives of h with respect to q 0 , q 1 , · · · , q n and p1 , · · · , pn at any point (q 0 , · · · , q n , −1, p1 , · · · , pn ) = (q 0 , · · · , q n , −1, γ1 , · · · , γn ) are simply given by ∂K ∂h = , i = 0, · · · , n i ∂q ∂q i (20) ∂h ∂K = , i = 1, · · · , n ∂pi ∂γi while on the other hand n pi ∂h ∂K d = −K − p0 (− ) = ∂p0 ∂γ dp p i 0 0 i=1 (21) n n pi ∂K ∂K = −K + γi −K − p ∂γi ∂γi i=1 0 i=1

In the sequel we will only use the notion of homogeneity for k = 0 and especially for k = 1. Indeed, it is clear that physical variables defined on the contact manifold P(T ∗ Q) correspond to functions on T0∗ Q which are homogeneous of degree 0. On the other hand, it will turn out that contact dynamics on the contact manifold P(T ∗ Q) can be lifted to Hamiltonian dynamics on T0∗ Q with respect to a Hamiltonian that is homogeneous of degree 1. 2.4 Contact vector fields and homogeneous Hamiltonian vector fields The basic relation between, on the one hand, contact vector fields on the contact manifold P(T ∗ Q) , and, on the other hand, ordinary Hamiltonian vector fields on T0∗ Q, with respect to a Hamiltonian h that is homogeneous of degree 1 is provided by the following observation. First recall that a vector field X on a contact manifold is called a contact vector field if (15) LX θ = ρθ for some function ρ. Furthermore, the function K := θ(X) is called the contact Hamiltonian of the contact vector field X. Conversely, for any differentiable function K it can be shown that there exists a unique contact vector field X such that K = θ(X), and we denote this contact vector field by XK . In Darboux coordinates q 0 , q 1 , · · · , q n , γ1 , · · · , γn as above (see (2), (3)) it follows that the contact vector field XK with contact Hamiltonian K is given as n n ∂ ∂K ∂K ∂ XK = K − γi − + 0 ∂γ ∂q ∂pi ∂q i i i=1 i=1 (16) n ∂K ∂ ∂K + γi 0 ∂q i ∂q ∂pi i=1

Comparing this with the local Darboux coordinate expression of XK in (16) it indeed follows that the contact vector field XK on P(T ∗ Q) is the projection of the ordinary Hamiltonian vector field Xh on T0∗ Q, with h the Hamiltonian (homogeneous of degree 1) corresponding to the contact Hamiltonian K. This can be summarized by the equation (22) π∗ Xh = XK , where π : T0∗ Q → P(T ∗ Q) denotes the projection (4). The above computation also yields the following expression of Poisson brackets of homogeneous degree 1 Hamiltonians. Consider two differentiable functions K 1 , K 2 on the contact manifold P(T ∗ Q), and define their homogeneous degree 1 extensions h1 , h2 : T0∗ Q → R as above. Then the Poisson bracket {h1 , h2 } at any point (q 0 , q 1 , ·, q n , −1, p1 , ·, pn ) = (q 0 , q 1 , ·, q n , −1, γ1 , ·, γn ) satisfies n ∂K 2 ∂K 1 ∂K 1 ∂K 2 {h1 , h2 } = − i ∂q ∂γi ∂q i ∂γi i=1 n n ∂K 2 ∂K 1 ∂K 1 ∂K 2 2 1 − 0 K − γi γi + K − ∂q ∂γi ∂q 0 ∂γi i=1 i=1 (23)

Any ordinary Hamiltonian vector field Xh on T0∗ Q with h homogeneous of degree 1 satisfies by Proposition 3 LXh α = 0. It follows that Xh projects to a vector field X on P(T ∗ Q) satisfying LX θ = ρθ for some function ρ, i.e., a contact vector field. Conversely, any contact vector field XK on P(T ∗ Q) gives rise to an ordinary Hamiltonian vector field on T0∗ Q with a Hamiltonian that is

3

IFAC LHMNC 2018 4 Valparaíso, Chile, May 1-4, 2018

Arjan van der Schaft et al. / IFAC PapersOnLine 51-3 (2018) 1–6

G(q 0 , · · · , q n , p0 , · · · , pn ) = −p0 F (q 1 , · · · , q n ) (27) which obviously is homogeneous of degree 1. More generally, the Lagrangian submanifold Ls corresponding to the Legendre submanifold L given by (25) is given by the generating function (28) G(q 0 , · · · , q n , p0 , · · · , pn ) = −p0 F (q I , γJ ) where in multi-index notation γJ = − ppJ0 .

The expression on the right-hand side of (23) is known as the Jacobi bracket of the two contact Hamiltonians K 1 , K 2 ; cf. Libermann and Marle (1987); ?); Eberard et al. (2007). 2.5 Legendre and Lagrangian submanifolds The relation between the contact geometry of the contact manifold P(T ∗ Q), and the symplectic geometry of T0∗ Q translates in the following relation between Legendre submanifolds of P(T ∗ Q) and Lagrangian submanifolds of T0∗ Q defined by a generating function which is homogeneous of degree 1.

An interesting feature, both of Legendre and Lagrangian submanifolds, is the characterization of invariance with respect to contact vector fields, respectively Hamiltonian vector fields 1 . We have the following two propositions. Proposition 5. (Mrugala et al. (1991)). A Legendre submanifold L is invariant with respect to the contact vector field XK if and only if the contact Hamiltonian K restricted to L is zero. Proposition 6. (Abraham and Marsden (1978)). The Lagrangian submanifold Ls is invariant with respect to the Hamiltonian vector field Xh if and only if the Hamiltonian h restricted to Ls is constant.

Recall that a Legendre submanifold L of the contact manifold P(T ∗ Q) is an integral manifold of θ of maximal dimension. It follows that for an (n+ 1)-dimensional Q the dimension of a Legendre submanifold of P(T ∗ Q) is n. The standard example of a Legendre submanifold of P(T ∗ Q) is a submanifold of the form L = {(q 0 , q 1 , · · · , q n , γ1 , · · · , γn ) | q 0 = F (q 1 , · · · , q n ), γi =

∂F 1 (q , · · · , q n ), i = 1, · · · , n} ∂q i

3. SIMPLE THERMODYNAMIC SYSTEMS

(24) for some function F (q 1 , · · · , q n ), which is called the generating function of the Legendre submanifold L. In general, an arbitrary Legendre submanifold can be locally represented as ∂F L = {(q 0 , q 1 , · · · , q n , γ1 , · · · , γn ) | q 0 = F − γJ , ∂γJ ∂F ∂F , γI = I } qJ = − ∂γJ ∂q (25) for some partitioning I ∪ J = {1, · · · , n} of the index set, and some generating function F (q I , γJ ).

As initiated in Balian and Valentin (2001), while contact geometry has long been recognized as an appropriate geometric framework for thermodynamics, the symplectization of contact geometry leads to new view points, which in particular look promising for the system-theoretic description and the control of thermodynamic systems. Briefly, in the set-up given in Balian and Valentin (2001) the (n + 1)-dimensional Q is described by coordinates q 0 , q 1 , · · · , q n which comprise the extensive variables of the thermodynamic system, including the entropy. So in the energy representation the coordinates q 0 , q 1 , · · · , q n are given as (29) (q 0 , q 1 , · · · , q n ) = (U, S, V, N 1 , · · · , N m ) with U the internal energy, S the entropy, V the volume and N k the mole numbers of the k-th chemical species with k = 1, · · · , m := n − 2. The corresponding co-state variables γ1 , · · · , γn in this case correspond to the intensive variables (T, −P, µ1 , · · · , µm ) (30) (with T the temperature, P the pressure and µk the chemical potential of the k-th chemical species) once a state equation is specified. Such a state equation (describing the thermodynamic properties of the system at hand) is given by a Legendre submanifold L of the contact manifold with coordinates q 0 , q 1 , · · · , q n and γ1 , · · · , γn . In particular, let L be defined as in (24) by a generating function F , then the intensive variables of the thermodynamical system are defined as ∂F ∂F ∂F , −P = , µk = , (31) T = ∂S ∂V ∂N k with k = 1, · · · , m. Hence the generating function F of L equals the internal energy U of the system expressed as a function of S, V, N 1 , · · · , N m , corresponding to Gibbs law m µk dN k (32) dU = T dS − P dV +

On the other hand, a Lagrangian submanifold Ls of the symplectic space T0∗ Q is a manifold of maximal dimension restricted to which the symplectic form ω = dα is zero. For Q being (n+1)-dimensional the dimension of a Lagrangian submanifold Ls ⊂ T0∗ Q is n + 1. A general Lagrangian submanifold Ls of T0∗ Q is locally represented as ∂G ∂G Ls = {(q 0 , · · · , q n , p0 , · · · , pn ) | q J = − , pI = I } ∂pJ ∂q (26) for some partitioning I ∪J of the (n+1)-dimensional index set {0, · · · , n}, and a generating function G(q I , pJ ).

The following tight connection between Legendre submanifolds of the contact manifold P(T ∗ Q) and Lagrangian submanifolds of the symplectic space T0∗ Q enjoying a homogeneity property can be found in Libermann and Marle (1987), Proposition 10.16. Proposition 4. Consider the projection π : T0∗ Q → P(T ∗ Q) as in (4). An integral submanifold L of θ is a Legendre submanifold of P(T ∗ Q) if and only if Ls := π −1 L is a Lagrangian submanifold of T0∗ Q. Note that the corresponding Lagrangian submanifold Ls = π −1 L is homogeneous in the sense that (q, p) ∈ Ls implies that (q, λp) ∈ Ls for any λ = 0. Equivalently, Ls has a homogeneous generating function. For example, Ls corresponding to L in (24) is defined by the generating function

k=1

1

Recall that a submanifold is invariant for a vector field if the vector field is everywhere tangent to it; and thus solutions remain on it.

4

IFAC LHMNC 2018 Valparaíso, Chile, May 1-4, 2018

Arjan van der Schaft et al. / IFAC PapersOnLine 51-3 (2018) 1–6

On the other hand, it is well-known that different choices of the intensive variables can be made. In particular, in the entropy representation the coordinates q 0 , q 1 , · · · , q n are given as (q 0 , q 1 , · · · , q n ) = (S, U, V, N 1 , · · · , N m ), (33) i.e., U and S are swapped with respect (29). In this case, the generating function F equals the entropy S expressed as a function of U, V, N 1 , · · · , N m , corresponding to the differently expressed Gibbs law m 1 P µk dS = dU − dV + dN k (34) T T T

5

ing equations can be normalized by taking the constant pS (called a gauge variable in Balian and Valentin (2001)) to be equal to e.g. −1. This leads restricted to the Lagrangian ∂S submanifold Ls to pU = ∂U (U ) = T1 . In the next section we will see how the lifting procedure employed in this example can be generalized to the definition canonical lifting, and how the Hamiltonian dynamics (37) is an example of what we will define as a homogeneous Hamiltonian control system. 4. HOMOGENEOUS HAMILTONIAN CONTROL SYSTEMS

k=1

In the symplectized point of view this takes the following appealing form. Take, for example, the identification of the extensive variables as in (29), leading to the intensive variables (31). In particular q 0 corresponds to internal energy, while q 1 corresponds to entropy. Now consider the symplectization T0∗ Q with Q = Rn+1 with n + 1 co-state variables p0 , p1 , · · · , pn . Considering p0 = 0 and defining γi = − pp0i , i = 1, · · · , n, leads to a contact manifold, and Legendre submanifold corresponding to intensive variables as in (31). With this same ordering of extensive variables one can however also define γi = − pp1i , i = 1, · · · , n (assuming p1 = 0); leading to the intensive variables as in (34). A nice feature of the symplectization is that the corresponding Lagrangian submanifold Ls is the same in both cases.

Let us start from the symplectized space T0∗ Q with canonical Liouville 1-form α and symplectic form ω = dα. Motivated both by thermodynamics as sketched above, and by the theory of input-output Hamiltonian and portHamiltonian systems, we will define the following class of control systems on T0∗ Q. First ingredient is a Lagrangian submanifold Ls ⊂ T0∗ Q, with generating function G which is homogeneous of degree 1 (and thus of the form π −1 L, with L a Legendre submanifold of P(T ∗ Q)). This Lagrangian submanifold describes the state properties of the system. Second ingredient is a control Hamiltonian (38) h : T0∗ Q × U → R with U an m-dimensional input space (e.g., U = Rm ), which is homogeneous of degree 1 in the co-state variables p0 , · · · , pn . Furthermore, in analogy with thermodynamics (at least for quasi-static processes) we will additionally require that the control Hamiltonian h is zero on Ls for every u ∈ U , i.e., (39) h(q, p, u) = 0, for all (q, p) ∈ Ls , u ∈ U This is summarized in the following definition. Definition 7. Consider the symplectized space T0∗ Q, with canonical Liouville 1-form α and symplectic form ω = dα. Furthermore, consider a Lagrangian submanifold Ls ⊂ T0∗ Q, with generating function homogeneous of degree 1, and a control Hamiltonian h : T0∗ Q × U → R as in (38) which is homogeneous of degree 1, and moreover satisfying (39). The triple (T0∗ Q, Ls , h) is called a homogeneous Hamiltonian control system.

Dynamics is defined by considering a Hamiltonian h : T0∗ Q → R which is homogeneous of degree 1. Furthermore, at least for quasi-reversible dynamics, h is required to be zero restricted to the Lagrangian submanifold Ls , implying invariance of this submanifold. Note however that for a simple thermodynamic system dynamics can only arise from interaction with its environment. For k instance, assume that the system is closed ( dN = 0) dt dV and isochore ( dt = 0), and only exchanges a heat flux Φ with its environment, corresponding to the energy balance dU 2 0 1 dt = Φ. Take Q = R with coordinates (q , q ) = (S, U ). Let Ls be the Lagrangian submanifold of T0∗ Q given as ∂S {(S, U, pS , pU ) | S = S(U ), pU = −pS (U )} (35) ∂U (i.e., we denote pS = p0 , pU = p1 ). Note the use of S both as an extensive variable, and as a function of U . A corresponding homogeneous Hamiltonian is defined as ∂S (U ))Φ, (36) h = (pU + pS ∂U which obviously is zero on Ls . The resulting Hamiltonian dynamics on T0∗ Q is given by ∂h ∂S dS = (U )Φ = dt ∂pS ∂U ∂h dU = =Φ dt ∂pU (37) dpS = 0 dt ∂h dpU ∂2S = − = −pS (U )Φ dt ∂U ∂U 2 Here the second equation replicates the energy balance, while the first equation defines the entropy balance. Note ∂S (U ) equals T1 , with T the temperature that the quantity ∂U of the thermodynamic system. Also notice that the result-

Note that this definition bears similarity to the definition of input-output Hamiltonian systems Brockett (1977); Van der Schaft (1982); Van der Schaft and Crouch (1987); Van der Schaft (1989). This also leads to the construction of the canonical lifting of a control system x˙ = f (x, u), x ∈ X, u ∈ U (40) to a homogenous Hamiltonian control system. This is done by first specifying a generating function S : X → R, and considering the corresponding Legendre submanifold ∂S (x)} (41) L = {(q 0 , x, γ) | q 0 = S(x), γ = ∂x with ∂S ∂x (x) denoting the n-dimensional column vector of partial derivatives of S with respect to x1 , · · · , xn . Equivalently (see the previous section), one defines the 5

IFAC LHMNC 2018 6 Valparaíso, Chile, May 1-4, 2018

Arjan van der Schaft et al. / IFAC PapersOnLine 51-3 (2018) 1–6

Lagrangian submanifold Ls of T0∗ (R × X) with generating function (homogeneous of degree 1) −p0 S(x), (42) 0 where p0 is the co-state variable conjugated with q .

D. Eberard, B.M. Maschke, A.J. van der Schaft. An extension of pseudo-Hamiltonian systems to the thermodynamic space: towards a geometry of non-equilibrium thermodynamics. Reports on Mathematical Physics, 60(2):175–198, 2007. A. Favache, B.M. Maschke, V. Dos Santos, D. Dochain. Some properties of conservative control systems. IEEE trans. on Automatic Control, 54(10):2341–2351, 2009. A. Favache, D. Dochain, B.M. Maschke. An entropy-based formulation of irreversible processes based on contact structures,. Chemical Engineering Science, 65:5204– 5216, 2010. R. Hermann Geometry, physics and systems. Marcel Dekker, New York, 1973. P. Libermann, C.-M. Marle. Symplectic geometry and analytical mechanics. D. Reidel Publishing Company, Dordrecht, Holland, 1987. B.M. Maschke, A.J. van der Schaft. Port controlled Hamiltonian systems: modeling origins and system theoretic properties. In Proc. 3rd Int. IFAC Conf. on Nonlinear Systems Theory and Control,, NOLCOS’92, 282–288, Bordeaux, 1992. B.M. Maschke, A.J. van der Schaft. Homogeneous Hamiltonian control systems, Part II: Application to thermodynamic systems. Submitted to 6th IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Control 2018. J. Merker, M. Kr¨ uger. On a variational principle in thermodynamics. Continuum Mechanics and Thermodynamics, 25(6):779–793, 2013. R. Mruga¨la. Geometric formulation of equilibrium phenomenological thermodynamics. Reports in Mathematical Physics, 14:419, 1978. R. Mruga¨la. On contact and metric structures on thermodynamic spaces. RIMS, Kokyuroku, 1142:167–181, 2000. R. Mruga¨la, J.D. Nulton, J.C. Sch¨on, P. Salamon. Contact structures in thermodynamic theory. Reports in Mathematical Physics, 29(1):109–121, 1991. H. Ramirez, B. Maschke, and D. Sbarbaro. Feedback equivalence of input-output contact systems. Systems and Control Letters, 62(6):475 – 481, 2013. H. Ramirez, B. Maschke, and D. Sbarbaro. Partial stabilization of input-output contact systems on a Legendre submanifold. IEEE Transactions on Automatic Control, 62(3):1431–1437, 2017. A.J. van der Schaft. Hamiltonian dynamics with external forces and observations, Mathematical Systems Theory, 15:145–168, 1982. A.J. van der Schaft. Three Decades of Mathematical System Theory, volume 135 of Lect. Notes Contr. Inf. Sci., Chapter System Theory and Mechanics, 426–452. Springer, Berlin, 1989. A.J. van der Schaft, P.E. Crouch. Hamiltonian and selfadjoint control systems. Systems & Control Letters, 8:289–295, 1987. A.J. van der Schaft, B.M. Maschke. The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv f¨ ur Elektronik und ¨ Ubertragungstechnik, 49: 362–371, 1995. A.J. van der Schaft, D. Jeltsema, ”Port-Hamiltonian Systems Theory: An Introductory Overview,” Foundations and Trends in Systems and Control, 1, 173–378, 2014.

The dynamics (40) can now be lifted to T0∗ (R × X), while leaving Ls invariant, by defining the control Hamiltonian ∂S h(q 0 , x, p0 , p¯, u) = (¯ (x))T f (x, u) p + p0 (43) ∂x Clearly, this control Hamiltonian is zero on Ls and homogeneous of degree 1 in p = (p0 , p¯), while the resulting Hamiltonian dynamics on T0∗ (R×X) replicates the original plant dynamics (40). This is summarized in Definition 8. Consider the control system (40) on a state space manifold X, together with a generating function S : X → R defining the Legendre submanifold (41). The canonical lifting of (40) with respect to S is defined as the homogeneous Hamiltonian control system (T0∗ (R × X), Ls , h) with Lagrangian submanifold defined by the generating function (42) and Hamiltonian h given by (43). The canonical lifting was already used in the example in the previous Section 3, cf. (35), (36), (37); see also Favache et al. (2010); Ramirez et al. (2013a). It will be extensively used in the companion paper Maschke and van der Schaft (2018). 5. CONCLUSIONS Inspired by Balian and Valentin (2001) we have elaborated on the symplectization of contact manifolds and on the lifting of contact vector fields to Hamiltonian vector fields with respect to a Hamiltonian that is homogeneous of degree 1 in the co-state variables. Based on this we have established the definition of homogeneous Hamiltonian control systems, linking to the theory of input-output Hamiltonian, as well as port-Hamiltonian, systems. This provides a new view on contact control systems as used in thermodynamics. In particular, it unifies the description of thermodynamic systems in energy or entropy representation. In the companion paper Maschke and van der Schaft (2018) the established theory will be applied to a number of physical examples, including the heat exchanger, the mass-spring-damper system, the gas-piston system, and the Continuous Stirred Tank Reactor (CSTR). REFERENCES R.A. Abraham, J.E. Marsden. Foundations of Mechanics, 2nd ed. Benjamin/Cummings, Reading, MA, 1978. V.I. Arnold. Mathematical Methods of Classical Mechanics. Springer, 2nd edition, 1989. R. Balian, P. Valentin. Hamiltonian structure of thermodynamics with gauge. Eur. J. Phys. B, 21:269–282, 2001. A. Bravetti, C.S. Lopez-Monsalvo, and F. Nettel. Contact symmetries and Hamiltonian thermodynamics. Annals of Physics, 361:377 – 400, 2015. R.W. Brockett. Geometric Control Theory, vol. 7 of Lie groups: History, Frontiers and Applications, ’Control theory and analytical mechanics’, 1–46. MathSciPress,, Brookline, 1977. C. Martin and R. Hermann eds. 6

Copyright © 2022 COEK.INFO. All rights reserved.