Notes Dirichlet's condition Dirichlet's condition is a mathematical condition used in the study of Fourier series. Specifically, it is a sufficient condition for the convergence of a Fourier series to the function it ...
Notes Fourier series In mathematics, a Fourier series is a way to represent a periodic function as the sum of simple sine and cosine functions. It is named after Joseph Fourier, who introduced ...
Notes Grand Cannonical Ensemble Each system in the microcannonical ensemble has the same fixed energy and the same amount of particles (N, V, E). This ensemble's system is a closed, isolated thermodynamic system (because ...
Notes Euler's angles The orientation of a rigid body must be specified by three independent factors in order for the associated orthogonal transformation matrix to have the determinant +1 and to be used ...
Notes Force free motion of a symmetrical rigid body Let us choose the symmetry axis as the principle z axis, so that I1 = I2 and Euler's equations simplifies towhere A is some constant and we have chosen ...
Notes Lagrangian for a simple pendulum and equation describing its motion The angle θ between rest position and deflected position is chosen as generalized coordinate. If the string is of length l, then kinetic energy iswhere m is the ...
Notes D'Alembert's principle Consider a system in equilibrium, i.e. total force Fi on every particle is zero. Then work done by this force in a small virtual displacement δr will also vanish i.e., ...
Notes Euler's theorem A rigid body is capable of two independent motions: rotation and translation. The position and orientation of such a body in motion can therefore be fully described. Three of these ...
