Spray (mathematics) - Semisprays in Lagrangian Mechanics

Semisprays in Lagrangian Mechanics

A physical system is modeled in Lagrangian mechanics by a Lagrangian function L:TM→R on the tangent bundle of some configuration space M. The dynamical law is obtained from the Hamiltonian principle, which states that the time evolution γ:→M of the state of the system is stationary for the action integral

.

In the associated coordinates on TM the first variation of the action integral reads as

\frac{d}{ds}\Big|_{s=0}\mathcal S(\gamma_s) = \Big|_a^b \frac{\partial L}{\partial\xi^i}X^i - \int_a^b \Big(\frac{\partial^2 L}{\partial \xi^j\partial \xi^i} \ddot\gamma^j + \frac{\partial^2 L}{\partial x^j\partial\xi^i} \dot\gamma^j - \frac{\partial L}{\partial x^i} \Big) X^i dt,

where X:→R is the variation vector field associated with the variation γs:→M around γ(t) = γ0(t). This first variation formula can be recast in a more informative form by introducing the following concepts:

  • The covector with is the conjugate momentum of .
  • The corresponding one-form with is the Hilbert-form associated with the Lagrangian.
  • The bilinear form with is the fundamental tensor of the Lagrangian at .
  • The Lagrangian satisfies the Legendre condition if the fundamental tensor is non-degenerate at every . Then the inverse matrix of is denoted by .
  • The Energy associated with the Lagrangian is .

If the Legendre condition is satisfied, then dα∈Ω2(TM) is a symplectic form, and there exists a unique Hamiltonian vector field H on TM corresponding to the Hamiltonian function E such that

.

Let (Xi,Yi) be the components of the Hamiltonian vector field H in the associated coordinates on TM. Then

and

dE = \Big(\frac{\partial^2 L}{\partial x^i \partial \xi^j}\xi^j - \frac{\partial L}{\partial x^i}\Big)dx^i + \xi^j \frac{\partial^2 L}{\partial\xi^i\partial x^j} d\xi^i

so we see that the Hamiltonian vector field H is a semispray on the configuration space M with the spray coefficients

Now the first variational formula can be rewritten as

\frac{d}{ds}\Big|_{s=0}\mathcal S(\gamma_s) = \Big|_a^b \alpha_i X^i - \int_a^b g_{ik}(\ddot\gamma^k+2G^k)X^i dt,

and we see γ→M is stationary for the action integral with fixed end points if and only if its tangent curve γ':→TM is an integral curve for the Hamiltonian vector field H. Hence the dynamics of mechanical systems are described by semisprays arising from action integrals.

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