In this paper, we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We show how the dissipative dynamics can be interpreted as a Legendrian submanifold, and also prove a coisotropic reduction theorem similar to the one in symplectic mechanics; as a consequence, we get a method to reduce the dynamics of contact Hamiltonian systems.In this paper, we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We show how the dissipative dynamics can be interpreted as a Legendrian submanifold, and also prove a coisotropic reduction theorem similar to the one in symplectic mechanics; as a consequence, we get a method to reduce the dynamics of contact Hamiltonian systems.

Contact Hamiltonian systems

In this paper, we apply the geometric Hamilton–Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamiltonians including dissipation terms. The interest of a geometric Hamilton–Jacobi equation is the primordial observation that if a hamiltonian vector field XH can be projected into a configuration manifold by means of a 1-form dW , then the integral curves of the projected vector field X H can be transformed into integral curves of XH provided that W is a solution of the Hamilton– Jacobi equation. In this way, we use the geometric Hamilton–Jacobi theory to derive solutions of physical systems with a time-dependent hamiltonian formulation or including dissipative terms. Explicit, new expressions for a geometric Hamilton–Jacobi equation are obtained on a cosymplectic and a contact manifold. These equations are later used to solve physical examples containing explicit time dependence, as it is the case of a unidimensional trigonometric system, and two dimensional nonlinear oscillators as Winternitz–Smorodinsky oscillators. For explicit dissipative behavior, we solve the example of a unidimensional damped oscillator.

https://iopscience.iop.org/article/10.1088/1751-8121/aa711d/pdf

Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems

In this paper we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We prove a coisotropic reduction theorem similar to the one in symplectic mechanics.

Contact Hamiltonian Systems.

We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we rev...

New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetries

In this paper, we apply the geometric Hamilton--Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamiltonians including dissipation terms. 
The interest of a geometric Hamilton--Jacobi equation is the primordial observation that if a hamiltonian vector field $X_{H}$ can be projected into a configuration manifold by means of a 1-form $dW$, then the integral curves of the projected vector field $X_{H}^{dW}$ can be transformed into integral curves of $X_{H}$ provided that $W$ is a solution of the Hamilton--Jacobi equation. In this way, we use the geometric Hamilton--Jacobi theory to derive solutions of physical systems with a time-dependent hamiltonian formulation or including dissipative terms. Explicit, new expressions for a geometric Hamilton--Jacobi equation are obtained on a cosymplectic and a contact manifold. These equations are later used to solve physical examples containing explicit time dependence, as it is the case of a unidimensional trigonometric system, and two dimensional nonlinear oscillators as Winternitz--Smorodinsky oscillators. For explicit dissipative behavior, we solve the example of a unidimensional damped oscillator.

Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems

We give an up-to-date overview of geometric and topological properties of cosymplectic and coKaehler manifolds. We also mention some of their applications to time-dependent mechanics.

A survey on cosymplectic geometry

We give an up-to-date overview of geometric and topological properties of cosymplectic and coKahler manifolds. We also mention some of their applications to time-dependent mechanics.

/pdf/a-survey-on-cosymplectic-geometry-1hdk0c5wj6.pdf

a Survey on Cosymplectic Geometry

We prove that on a compact Sasakian manifold $(M,\eta, g)$ of dimension $2n + 1$, for any $0 \leq p \leq n$ the wedge product with $\eta \wedge (d \eta)^p$ defines an isomorphism between the spaces of harmonic forms $\Omega^{n-p}_{\Delta}(M)$ and $\Omega^{n+p+1}_{\Delta}(M)$. Therefore it induces an isomorphism between the de Rham cohomology spaces $H^{n-p}(M)$ and $H^{n+p+1}(M)$. Such isomorphism is proven to be independent of the choice of a compatible Sasakian metric on a given contact manifold. As a consequence, an obstruction for a contact manifold to admit Sasakian structures is found.

/pdf/hard-lefschetz-theorem-for-sasakian-manifolds-52d5etzzlp.pdf

Hard Lefschetz Theorem for Sasakian manifolds

Sasaki–Einstein and paraSasaki–Einstein metrics from (κ,μ)-structures

Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively.

/pdf/examples-of-compact-k-contact-manifolds-with-no-sasakian-df1mhnvdnk.pdf

Beniamino Cappelletti-Montano

Papers

A survey on cosymplectic geometry

a Survey on Cosymplectic Geometry

Hard Lefschetz Theorem for Sasakian manifolds

Sasaki–Einstein and paraSasaki–Einstein metrics from (κ,μ)-structures

Examples of compact K-contact manifolds with no Sasakian metric