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An ODE problem can be expanded with the auxiliary variables which make the power series method trivial for an equivalent, larger system. Expanding the ODE problem with auxiliary variables produces the same coefficients (since the power series for a function is unique) at the cost of also calculating the coefficients of auxiliary equations.
The small divisor problem arises when the difference is small, causing the perturbative correction to "blow up", becoming as large or maybe larger than the zeroth order term. This situation signals a breakdown of perturbation theory: It stops working at this point, and cannot be expanded or summed any further.
Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state x of the system at time t (namely a probability distribution p t ( x ) {\displaystyle p_{t}(x)} ); we want to know the probability distribution of the state at a later time s > t {\displaystyle s>t} .
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
The equations are named after Andrei Kolmogorov since they were highlighted in his 1931 foundational work. [2]William Feller, in 1949, used the names "forward equation" and "backward equation" for his more general version of the Kolmogorov's pair, in both jump and diffusion processes. [1]
The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping.
Courant, R.; Friedrichs, K.; Lewy, H. (September 1956) [1928], On the partial difference equations of mathematical physics, AEC Research and Development Report, vol. NYO-7689, New York: AEC Computing and Applied Mathematics Centre – Courant Institute of Mathematical Sciences, pp. V + 76, archived from the original on October 23, 2008 ...
Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [1] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. [2]