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Using standard but slightly laborious methods (involving switching to polar coordinates), the following nonlinear dynamical system can be shown to exhibit an unstable spiral at the origin and a stable limit cycle at : A student contacted me for help after unsuccessfully attempting to plot the phase portrait of this system, showing both the unstableContinue reading “Using wxdrawdf in MAXIMA to plot a phase portrait with a stable limit cycle”

The usual model of two masses connected by a spring has one of its eigenvalues equal to zero, meaning that the system could be free to move in space rather than being fixed in position. The corresponding eigenvector has components which are equal to each other, meaning that the system would move through space asContinue reading “A note on an unusual system of two masses connected by a spring”

I was having a discussion about different approaches for quadratic spline interpolation and I introduced an example involving interpolation of points on a modulus function. This led to a deeper exploration of how altering the knot sequence of a B-Spline interpolation can substantially improve the results. I want to record this here. The question thatContinue reading “Quadratic B-Spline interpolation of points on a modulus function”

I record here an experience of manually calculating a Fast Fourier Transform (FFT). Carrying out the computations by hand turned out to be relatively straightforward and quite instructive. A number of different FFT algorithms have been developed and the underlying equations are provided in numerous journal articles and books. I chose to calculate an eight-pointContinue reading “Fast Fourier Transform by hand”

The following is a useful coding technique using Boolean array indexing in Python for quickly finding many zeros of a function at once. It avoids having to search for the multiple roots using root-finding functions of various kinds, with inconveniences such as having to identify suitable starting values for the numerical iterations, etc. I willContinue reading “Boolean array indexing and finding multiple zeros of a function at once”

Let denote an data matrix, e.g., observations on variables for individuals. In what follows, we assume has been centered at zero, i.e., each element of the original uncentered data matrix has had the mean of its column subtracted from it to form the corresponding element of . If denotes the transpose of , then theContinue reading “A note on Singular Value Decomposition and Principal Components”

Quantum mechanical models in three dimensions and involving many particles, etc., use a formalism which is largely based on some key mathematical modelling ideas pertaining to single particle systems. These mathematical modelling ideas are principally designed to tell us what sets of values we are `allowed’ to observe when we measure some aspect of aContinue reading “Overview of mathematical modelling ideas in single particle quantum mechanics”

The Legendre transform is a mechanism for converting a relationship like into a relationship like In other words, by going from a function to its Legendre transform and vice versa, the roles of and in (1) and (2) can be reversed. In moving from (1) to (2), the required Legendre transform is because differentiating bothContinue reading “Legendre-transforming a Lagrangian mechanics problem into a Hamiltonian one”

A heat flow problem led to the following general series solution: An initial condition at required , so it was necessary to find the coefficients in the following sine series: The sine functions here involve coefficients of of the form rather than the usual . They have period and by looking at graphs it isContinue reading “A note on a Fourier sine series solution to a heat flow problem”

Bessel’s equation arises in countless physics applications and has the form where the constant is known as the order of the Bessel function which solves (1). The method of Frobenius can be used to find series solutions for of the form where is a number to be found by substituting (2) and its relevant derivativesContinue reading “Equivalent integral formulae for Bessel functions of the first kind, order zero”