Contact Hamiltonian and Lagrangian systems with nonholonomic constraints (Q2052470)

This paper aims at using contact and Jacobi geometry to develop the natural geometric framework for studying the dynamics of mechanical systems that are subject to both nonholonomic constraints and Rayleigh dissipation. A \textit{nonholonomic mechanical system} is a mechanical system subject to \textit{nonholonomic constraints}, i.e., constraints (on the position and velocities) that do not derive from constraints only on the positions. Examples include mechanical systems that have rolling contact (like a ball rolling without slipping on a plane) or some kind of sliding contact (like a rigid body sliding on a plane). In the Lagrangian formalism, a mechanical system is described by a \textit{Lagrangian function} \(L\colon TQ\to\mathbb{R},\ (q,\dot q)\mapsto L(q,\dot q)\), where the smooth manifold \(Q\) denotes the \textit{configuration space} of the system. Then a nonholonomic constraint is given by a submanifold \(\mathcal{D}\subset TQ\) such that \(\tau_Q(\mathcal{D})=Q\), where \(\tau_Q\colon TQ\to Q\) denotes the bundle map. In the following, one only considers nonholonomic constraints that are linear in the velocities, i.e., \(\mathcal{D}\subset TQ\) is a vector subbundle. If the mechanical system is conservative, i.e., \(L=K_g-V\), where \(V\in C^\infty(Q)\) and \(K_g\) is the kinetic energy of some pseudo-Riemannian metric \(g\) on \(Q\), then the Lagrangian \(L\) is regular, i.e., the associated Legendre transform \(\mathbb{F}L:TQ\to T^\ast Q\) is a local diffeomorphism. In this case, the natural geometric description of their dynamics is provided in terms of Hamiltonian systems on symplectic manifolds (see, e.g., [\textit{R. Abraham} and \textit{J. E. Marsden}, Foundations of mechanics. 2nd ed., rev., enl., and reset. With the assistance of Tudor Ratiu and Richard Cushman. Reading, Massachusetts: The Benjamin/Cummings Publishing Company, Inc (1978; Zbl 0393.70001)] and references therein). Indeed, the unconstrained dynamics is obtained as the projection on \(Q\) of the flow of the Euler-Lagrange vector field \(\Gamma_L\), i.e., the Hamiltonian vector field of the system \((TQ,\omega_L,E_L)\), where \(E_L=\Delta(L)-L\) is the energy, with \(\Delta\) the Euler vector field on \(TQ\), and \(\omega_L=(\mathbb{F}L)^\ast\omega_{\text{can}}\) is the pull-back along \(\mathbb{F}L\) of the canonical symplectic form on \(T^\ast Q\). This \(\Gamma_L\) is a SODE (second-order differential equation) on \(TQ\) and its flow is obtained integrating the standard Euler-Lagrange equations. This description of the dynamics is consistent with the one arising from D'Alembert principle. If the conservative mechanical system is additionally subject to nonholonomic constraints, then its dynamics can be still described in terms of Hamiltonian systems on symplectic manifolds (see, e.g., [\textit{C.-M. Marle}, Rep. Math. Phys. 42, No. 1--2, 211--229 (1998; Zbl 0931.37023)] and references therein). Indeed, its dynamics is the projection on \(Q\) of the flow of a nonholonomic Euler-Lagrange vector field \(\Gamma_L^\mathcal{D}\). The latter is still a SODE on \(TQ\) and is obtained from \(\Gamma_L\) by projection with respect to a certain decomposition of \((TTQ)|_\mathcal{D}\). This description of the nonholonomic dynamics is consistent with the one arising from Chetaev version of D'Alembert principle. Moreover, this nonholonomic dynamics is almost-Poisson but not Poisson. Indeed, there is a bracket \(\{-,-\}\) on \(C^\infty(\mathcal{D})\) that satisfies the Leibniz rule in each entry and, together with the energy \(E_L\) on \(TQ\), controls the time evolution of the observables, but in generally it fails to satisfy the Jacobi identity. The authors start from the observation that there are other kinds of nonholonomic mechanical systems that do not fit in the previous framework. As a first example, one can consider a nonholonomic mechanical system that is also subject to Rayleigh dissipation and so non-conservative. Additional examples come from thermodynamics. These mechanical systems can be described by a Lagrangian function \(L\colon TQ\times\mathbb{R}\to\mathbb{R},\ (q,\dot q,z)\mapsto L(q,\dot q,z):=L_z(q,\dot q),\) where the smooth manifold \(Q\) is the configuration space and the parameter \(z\) on \(\mathbb{R}\) denotes friction (or a thermal variable in thermodynamics). If the Lagrangian \(L\) is regular, in the sense that, for any \(z\in\mathbb{R}\), the associated Legendre transform \(\mathbb{F}L_z:TQ\to T^\ast Q\) is a local diffeomorphism, the natural geometric framework of their dynamics is provided by the theory of Hamiltonian systems on contact manifolds (cf., e.g., [\textit{M. de León} and \textit{M. Lainz Valcázar}, J. Math. Phys. 60, No. 10, 102902, 18 p. (2019; Zbl 1427.70039)] and references therein). Indeed, the unconstrained dynamics is obtained as the projection on \(Q\) of the flow of the Euler-Lagrange vector field \(\Gamma_L\), i.e., the Hamiltonian vector field of the system \((TQ\times\mathbb{R},\eta_L,E_L)\), where \(E_L=\Delta(L)-L\) is the energy, with \(\Delta\) the Euler vector field on \(TQ\), and \(\eta_L=(\mathbb{F}L\times\operatorname{id}_\mathbb{R})^\ast\eta_{\text{can}}\) is the pull-back along \(\mathbb{F}L\times\operatorname{id}_\mathbb{R}\) of the canonical contact form on \(T^\ast Q\times\mathbb{R}\). In the current setting, this \(\Gamma_L\) is still an SODE on \(TQ\times\mathbb{R}\) (in the sense recalled in Definition~5) and its flow is obtained integrating the so-called Herglotz equations. Indeed, this description of the dynamics is consistent with the one arising from the Herglotz variational principle (as recalled in Section~4). In this paper the authors show that the theory of Hamiltonian systems on contact manifolds can be adapted to provide a geometric interpretation of the dynamics of mechanical systems that are subject to both dissipation and nonholonomic constraints. Section~5 defines a version of the Herglotz principle in presence of nonholonomic constraints: essentially, one restricts the variations so that they satisfy the constraints. Then the dynamics is described by the extremals of this Herglotz principle with constraints and they are given by the solutions of the so-called constrained Herglotz equations (see Theorem~5). Actually, this description admits a geometric description similar to the one obtained when there are no constraints. Indeed, Theorem 6 shows that, if the Lagrangian \(L\colon TQ\times\mathbb{R}\to\mathbb{R}\) is regular (i.e., \(F_z\colon TQ\to\mathbb{R}\) is regular, for any \(z\in\mathbb{R}\), as recalled above), the solutions of the constrained Herglotz equations are the projections on \(Q\) of the integral curves of the nonholonomic Euler-Lagrange vector field \(\Gamma_L^\mathcal{D}\). The latter is still a SODE on \(TQ\times\mathbb{R}\) (see Definition~5) and is obtained from \(\Gamma_L\) by projection with respect to a certain decomposition of \(T(TQ\times\mathbb{R})\) along \(\mathcal{D}\). The authors also prove that the time evolution of these mechanical systems subject to both dissipation and nonholonomic constraints is governed by an almost-Jacobi bracket (see Definition 7). Indeed, in Section 6, they first construct a nonholonomic bracket from functions on \(TQ\times\mathbb{R}\) to functions on \(\mathcal{D}\times\mathbb{R}\) (see Equation 100), then they prove that this nonholonomic bracket (together with the Energy \(E_L\)) controls the time evolution of the observables (see Theorem 12) and it is an almost Jacobi bracket (see Proposition 6). Further, it turns out that this nonholonomic bracket is actually a Jacobi structure (i.e., it satisfies the Jacobi identity) if and only if the constraint \(\mathcal{D}\subset TQ\) is an involutive vector subbundle (see Theorem 13). Finally, in Example 2, the authors illustrate their results applying them to a particular example given by a model of the Chaplygin's sleight subject to Rayleigh dissipation.