# Lagrangians in an isotropic plane from complex coordinates

TL;DR: In this paper, a point having rectangular coordinates (x, y) and polar coordinates (r, theta) has complex coordinates (z, z) where z =x+iy=r exp(i theta ) and z=x-iy =r exp(- i theta ).

Abstract: A point having rectangular coordinates (x, y) and polar coordinates (r, theta ) has complex coordinates (z, z) where z=x+iy=r exp(i theta ) and z=x-iy=r exp(-i theta ). The versatility of complex coordinates leads to computational advantages in writing and solving equations of motion in two dimensions. Potential energies, lagrangians, and equations of motion are given in three examples to illustrate the method.

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TL;DR: In this paper, an interpretation of the B-spline in probability is presented, and it is shown how this result could be useful in teaching, and how this interpretation could be used in teaching.

Abstract: An interpretation of the B‐spline in probability is presented, and it is shown how this result could be useful in teaching.

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TL;DR: In this paper, the Lagrangian functions for one-dimensional lattices and the two-dimensional rhombic crystal are modeled as systems of arbitrarily many point masses interconnected by ideal springs, and the irreducible matrix representations of the symmetry groups for these lattices generate unitary transformations into symmetry coordinates.

Abstract: Computational aspects of the representation theory of finite Abelian groups provide means for exploiting the symmetries of classical systems of harmonic oscillators. Coupled one-dimensional lattices and the two-dimensional rhombic crystal are modelled as systems of arbitrarily many point masses interconnected by ideal springs. After application of the Born cyclic condition, the Lagrangian functions for these systems are written in matrix notation. The irreducible matrix representations of the symmetry groups for these lattices generate unitary transformations into symmetry coordinates. Under these transformations, the Lagrangian matrices are either diagonalized or reduced to block diagonal form, thereby separating the equations of motion to the maximum extent made possible by system-wide considerations of a purely geometrical nature. Natural lattice frequencies are computed once the transformations to symmetry coordinates have been made.

3 citations