Mechanical Systems with Random Perturbations

Part of the Theoretical and Mathematical Physics book series (TMP)


It is a well-known fact that a second order differential equation \(\ddot{x}(t)=\bar{\alpha}(t,x(t),\dot{x}(t))\) expressing Newton’s law in ℝ n may be represented as a first order system on the space of dimension 2n:
$$ \left\{\begin{array}{rcl}\dot{x}(t) & = & v(t) \\\dot{v}(t) & = & \bar{\alpha}(t,x(t),v(t)).\end{array}\right.$$
We call the first equation of the above system horizontal and the second one vertical. This is consistent with the terminology in the general case of a mechanical system on a non-linear configuration space M, where Newton’s law is presented by means of covariant derivatives in the form (11.2), equivalent to equation (11.3) with a special vector field (second order differential equation) on the phase space TM. Recall that the field (11.3) is the sum of the Levi-Civitá geodesic spray \(\mathcal{Z}\) (which is horizontal, i.e. belongs to the connection) and the vertical lift of the vector force field (which belongs to the vertical subspace).


Weak Solution Mechanical System Sample Path Wiener Process Langevin Equation 
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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentVoronezh State UniversityVoronezhRussia

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