Talk:Spray (mathematics)

Comments on Introduction

Semisprays should be included in this article.
Are "normal coordinates" traditionally associated with non-metric sprays? (I don't have the reference at hand.) Even if they are, regularity around the origin is not so crucial. For example, "normal coordinates" are definitely used in Finsler geometry, even though there this property fails.
Perhaps the fact that semisprays are invariant versions of general second order ODE systems on smooth manifolds should be stressed and Riemann/Finsler sprays given as examples of full sprays.
There is material on sprays on double tangent bundle and Finsler manifold that should be moved into this page.

Lapasotka (talk) 20:03, 26 August 2009 (UTC)[reply]

I second the motion that the fact that semisprays are invariant versions of general second order ODE systems on smooth manifolds should be stressed

Jrolland (talk • contribs) 04:29, 19 May 2016 (UTC)[reply]

Comments on Semisprays in Lagrangian mechanics

I think maybe the canonical symplectic form ω on TM (ω = -dΘ, Θ = Σ p_idq_i, where (q_i, p_i) are coordinates on TM) might be a better way to define the Hamiltonian vector field associated to a Lagrangian function on TM.

I'm not an expert on classical mechanics, but in Foundations of Mechanics by Abraham and Marsden, there is a nice discussion on the canonical symplectic form on TM in Chapter III, Section 14 (Theorem 14.14 in particular defines the canonical symplectic form) and on global Hamiltonian vector fields in Chapter III, Section 16 (Definition 16.14 in particular defines global Hamiltonian functions and vector fields).

Jrolland (talk • contribs) 01:27, 19 May 2016 (UTC)[reply]

The canonical symplectic form ω is defined on the cotangent bundle T^*M, not on the tangent bundle TM. This form can be pulled back to TM by the Legendre map of a Lagrangian, and then result will be symplectic if (and only if) the Lagrangian is regular. 81.78.200.135 (talk) 18:14, 15 December 2024 (UTC)[reply]