Recent content by eztum

The output you got from Mathematica means: 'Sorry, I did not find a symbolic solution for the problem'. As the previous answer suggests, you should probably use NDSove to find a numerical solution (for which graphical representations can easily be created by Mathematica). For this to work, you...

You can considerably increase the power of such a numerical method by following Verlet (actually the method is much older and is often associated with Störmer, sometimes it is called explicit midpoint method) Treat r (position) and v (velocity) as vectors (in 3-space, or 2-space). (1) determine...

The Wikipedia article 'Kepler orbit' has this and much more. Notice that solving Kepler's equation solves the problem where on his orbit the planet is at a given instant of time. Solving this equation by some iteration scheme is easy and fast and the computational burden is independent of how...

Who is more deeply interested in the problem should study the exact solution based on Kepler's famous transcendental equation. Compared with this, the present approach is a 'toy solution'. Comparing the poor physics textbook method with Kepler's method allows one to appreciate what genius in...

I had in mind the view held in most mathematics textbooks, where linear spaces are discussed as associated with some 'field K of scalars' and all matrices under considerations have elements from K. Then, when it comes to eigenvalues, these are defined to belong to K (since otherwise...

When I had to deal with non-seperable Hilbert spaces, decades ago, all interested students knew the (in a sense trivial) example cited here by dextercioby and the non-trivial example of 'almost-periodic functions'. Also everybody knew that the Fock space over a seperable Hilbert space (acting...

What is your thinking with respect to the type of the matrix elements: real, complex, Galois, ...? Characterizations in terms of eigenvalues depend on this. There are real 2x2 matrices, the rotations of the euclidean plane, that have no eigenvalues at all (iff they differ from the unit...

An natural question, particularly for a mathematically interested programmer: Wheras defining arrays (i.e. tuples) of any length is commonplace in all sensible programming languages, sets are absent from wide spread languages such as C. Actually one can build the foundations of mathematics in a...

Dear \aleph_0, also I was not interested in whatever kind of competition about "best" methods. My only desire was to get help in locating the proposed method within the spectrum of published or of common-place methods. Your localization in a family of predictor-corrector methods looks not very...

You already have d2f/dxdy(i,j,k) = [f(i+1,j+1,k) - f(i+1,j,k) - f(i,j+1,k) + f(i-1,j-1,k)] / 4 so you have to do the first order derivation for z, which means for indexes in short-hand notation [(k->k+1) - (k->k-1)]/2 By the way, congratulations for having chosen the symetrical representation...

What I proposed in my previous post as an computer experiment can be done by mere thinking: I assume that 'Cantor steps' means indicator function of Cantor's ternary set. In particular U(x) \in \{0,1\}. Two cases: (a) \quad \dot{x}(0) < \sqrt{2} (i.e. E_\text{kin} < U_\text{max}) (b)...

By energy conservation you see that a step in U makes an instantaneous change in velocity of the particle. You represent the Cantor steps as a limit of step functions of increasing detail. For each such approximating 'pre Cantor' potential the motion is thus well defined. My further proceeding...

Thanks for your post. Could you please make your statement more explicit? The method under consideration is an explicit method that needs just one evaluation of the ODE's right-hand side per step like Euler) and is second order (unlike Euler, which is first order only). From step to step...

Consider an ODE of general form: \frac{d}{dt}y(t) = f(y(t),t) with an initial condition y(t_0) = y_0 . We create an approximate solution by an intitialization step v := f(y_0,t_0) t:= t_0 y:= y_0 and iteration of the following modified leapfrog step \tau := h/2 t += \tau y += \tau v...