Stochastic Mechanics of Particles and Fields

The configuration space of a physical system is a differentiable manifold M. The state of a system is given by a point x in M and a tangent vector v at x, the velocity of the configuration. Call the tangent bundle T M of M the state space of the system (though the phrase " velocity phase space " is often used). A dynamical variable is a possibly time-dependent function on state space. Newtonian mechanics. The kinetic energy T of a system is given, using tensor notation and the summation convention, by (1) T = 1 2 m ij v i v j where m ij is a Riemannian metric, called the mass tensor. Using the Riemannian connection we can define acceleration. If x(t) is the configuration of the system at time t, then in local coordinates the acceleration a is given by a i = ˙ v i + Γ