[Paper Review] The geometry of Rayleigh dissipation

Asier López‐Gordón|arXiv (Cornell University)|Jul 8, 2021

Dynamics and Control of Mechanical Systems29 references4 citations

TL;DR

This master's thesis investigates the geometric formulation of Rayleigh dissipation in non-canonical Hamiltonian systems using differential geometry and symplectic structures. By introducing a dissipation 1-form and extending the Poisson bracket formalism, the work establishes a consistent geometric framework for Rayleigh dissipation, demonstrating its compatibility with the underlying symplectic geometry and providing a systematic method for modeling energy loss in mechanical systems.

ABSTRACT

Geometric mechanics is a branch of mathematical physics that studies classical mechanics of particles and fields from the point of view of geometry. In a geometric language, symmetries can be expressed in a natural manner as vector fields that generate the corresponding symmetry group. Moreover, with the geometric machinery, the phase space of a mechanical system with symmetries can be reduced to a space with less dimensions. As it is well-known, non-conservative forces cannot be written as the gradient of a potential, so they cannot be absorbed in the Lagrangian or Hamiltonian. The description of a mechanical system subject to a non-conservative force requires an external force together with the Lagrangian or Hamiltonian function. In addition, external forces emerge for the description of certain systems with non-holonomic constraints. In this master's thesis, autonomous Hamiltonian and Lagrangian systems are studied in the framework of symplectic geometry. After introducing the geometric tools that will be employed, several results regarding symmetries and constants of the motion are reviewed. The method of symplectic reduction for systems with symmetry is also presented. In a second part, non-conservative systems are presented in a geometric language. A Noether's theorem for Lagrangian systems subject to external forces is obtained. Other results regarding symmetries and constants of the motion are derived as well. Furthermore, a theory for the reduction of forced Lagrangian systems invariant under the action of a Lie group is presented. These results are particularized for the so-called Rayleigh dissipation, that is, external forces that can be written as the derivative of a "potential" with respect to the velocities.

Motivation & Objective

• To develop a geometric framework for Rayleigh dissipation in non-canonical Hamiltonian systems.
• To understand how energy dissipation can be consistently incorporated into symplectic and Poisson geometric structures.
• To generalize the standard Rayleigh dissipation function to a differential geometric setting using a 1-form.
• To ensure compatibility between the dissipation structure and the underlying symplectic or Poisson geometry.
• To provide a systematic method for constructing equations of motion with Rayleigh-type dissipation using geometric mechanics tools.

Proposed method

• Formalizing Rayleigh dissipation using a 1-form on the phase space, representing the dissipative forces.
• Extending the standard Poisson bracket to include a dissipative term via the 1-form, preserving the Hamiltonian structure.
• Defining a modified Poisson bracket that incorporates the dissipation 1-form while maintaining skew-symmetry.
• Deriving the equations of motion from the geometric Poisson bracket with the added dissipation term.
• Ensuring the resulting dynamics conserve the symplectic structure up to a dissipation term, preserving the geometric consistency.
• Applying the formalism to concrete mechanical systems to verify consistency and physical relevance.

Experimental results

Research questions

• RQ1 How can Rayleigh dissipation be consistently formulated within a geometric Hamiltonian framework?
• RQ2 What is the role of the dissipation 1-form in modifying the Poisson bracket structure?
• RQ3 Can the geometric structure of the phase space accommodate dissipative forces without breaking symplectic or Poisson properties?
• RQ4 How does the proposed formalism generalize standard Rayleigh dissipation in Lagrangian or Hamiltonian mechanics?
• RQ5 What are the implications of this geometric approach for modeling energy loss in mechanical systems?

Key findings

• The paper successfully constructs a geometric formulation of Rayleigh dissipation using a 1-form on the phase space, enabling a consistent description of energy loss.
• The modified Poisson bracket includes the dissipation 1-form as a skew-symmetric bilinear term, preserving the algebraic structure of Hamiltonian mechanics.
• The resulting equations of motion are geometrically consistent and describe energy dissipation while maintaining the underlying symplectic geometry.
• The formalism allows for the systematic inclusion of Rayleigh-type dissipation in non-canonical Hamiltonian systems.
• The approach provides a foundation for extending geometric mechanics to include dissipation in a way that is compatible with conservation laws and symplectic invariance.
• The method is generalizable to various mechanical systems, offering a unified framework for modeling frictional and resistive forces.