The assumption that the network is strongly connected plays a vital role in the aforementioned derivation. If this is not the case, the trajectory length T cannot be sufficiently large. Then, the contribution from the starting points of the random walk may not be negligible in \({\mathbb {E...
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We start with the description of the Hamilton equations that we will use in the most general form we need. Let x1 = x1(s), x2 = x2(s), x3 = x3(s), t = t(s) be the equations of a ray trajectory in parametric form (s is the parameter) in the space x1 –x2 –x3 –t ...
Classical approach leads to a result that the time-independent Schrdinger equation extracts from solutions of the HamiltonJacobi equation only those solutions that satisfy the following two conditions: (I) The orbital angular momentum of a particle moving along a closed trajectory is quantized; (II)...
When the baseball takes a curved trajectory as it passes the plate, in most cases, the batter would not be able to judge the path of the ball and miss the ball. One of the crucial aspects of curveball can be explained using a formula typically used to describe fluid flow. Bernoull’s...
One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to non...
We derive the evolution equations for two-time correlation functions of a generalized non-Markovian open quantum system based on a modified stochastic Schrödinger equation approach. We find that the two-time reduced propagator, an object that used to be characterized by two independent stochastic pr...
The only force acting on the pendulum is the force of gravity. 46.1 Derivation of the Pendulum Equation d2θdt2+gLSin[θ]=0 The force acting on the pendulum can be broken into two components, one in the direction of the rod and the other in the direction of the pendulum's motion. ...
A few common examples of hyperbola include the path followed by the tip of the shadow of a sundial, the scattering trajectory of sub-atomic particles, etc. Here we shall aim at understanding the definition, formula of a hyperbola, derivation of the formula, and standard forms of hyperbola ...
According to the first law of thermodynamics, the system of N particles will move from position A to position B following the less energy trajectory; therefore, the equation of motion can be obtained from the stationary trajectory of the energy function. The Langrange’s function expresses the ...