[Paper Review] The Geometry Of Higher-Order Hamilton Spaces; Applications to Hamiltonian Mechanics

Radu Miron|arXiv (Cornell University)|Mar 12, 2010

Advanced Differential Geometry Research64 references18 citations

TL;DR

This paper develops the differential geometry of higher-order Hamilton spaces, extending Lagrange and Finsler geometry to k-th order tangent and cotangent bundles. It introduces canonical nonlinear connections, metrical N-linear connections, and Poisson structures on T*kM, enabling a geometric formulation of higher-order Hamiltonian mechanics and relativistic optics with applications to variational problems and conservation laws.

ABSTRACT

The book can be divided in three parts: the Lagrange geometry of order $k$, presented in the first three chapters, the geometrical theory of the dual manifolds $T^{*k}M$ - chapters 4-7 and the geometry of Hamilton spaces of order $k$ and their subspaces, contained in the last four chapters. They are studied directly and as 'dual' geometry, via Legendre transformation.

Motivation & Objective

• To generalize Hamiltonian mechanics to higher-order (k-th order) systems using differential geometry of k-jet bundles.
• To develop the geometry of the dual bundle T*kM, including canonical Poisson structures and nonlinear connections.
• To establish a variational framework for Hamiltonians of order k, including Hamilton-Jacobi equations and conservation laws.
• To extend Finsler and Cartan geometry to higher-order spaces, particularly for applications in relativistic optics and field theory.
• To introduce canonical metrical N-linear connections and study their curvature and torsion in higher-order settings.

Proposed method

• Constructs the k-tangent bundle TkM and its dual T*kM as fiber bundles over a base manifold M, equipped with natural geometric structures.
• Introduces k-semisprays and dual semisprays as higher-order analogues of geodesic spray, defining nonlinear connections via the vertical distribution.
• Defines d-tensor fields and N-linear connections on T*kM, with adapted local bases and covariant derivatives for geometric analysis.
• Derives the canonical Poisson structure on T*kM using the canonical 1-form and the symplectic form, enabling Hamiltonian dynamics.
• Applies the variational calculus to Hamiltonians of order k, deriving the Hamilton-Jacobi equations and Zermelo conditions.
• Constructs the canonical metrical N-linear connection and studies its curvature, torsion, and structure equations using adapted frames.

Experimental results

Research questions

• RQ1 How can the geometry of higher-order Hamilton spaces be systematically developed on the k-th order cotangent bundle T*kM?
• RQ2 What are the canonical nonlinear and metrical N-linear connections on T*kM, and how do they relate to the Hamiltonian structure?
• RQ3 How do the variational principles and conservation laws generalize in higher-order Hamiltonian systems?
• RQ4 What is the role of the canonical Poisson structure in formulating higher-order Hamiltonian dynamics?
• RQ5 How can higher-order Finsler and Cartan spaces be dualized and applied to relativistic optics and field theory?

Key findings

• The paper constructs a canonical nonlinear connection on T*kM using a dual k-semispray, generalizing the classical nonlinear connection in first-order mechanics.
• It proves the existence of a canonical metrical N-linear connection on Hamilton spaces of order k, with explicit local coefficients derived from the Hamiltonian.
• The canonical Poisson structure on T*kM is shown to be non-degenerate and invariant under the Liouville vector field, enabling Hamiltonian dynamics.
• The paper derives the Hamilton-Jacobi equations for order k Hamiltonians, generalizing the classical first-order case.
• It establishes conservation laws for higher-order energies Ek−1(H), showing that energy is conserved when the Hamiltonian is autonomous.
• The paper provides a geometric formulation of relativistic optics using generalized Hamilton spaces of order k, linking them to the electromagnetic field and Finsler-type metrics.