![]() ![]() Noether’s Theorem is an important result in theoretical physics that says, loosely, that if a system’s behavior does not change under a particular infinitesimal transformation then that transformation corresponds to a conserved quantity. The energy associated with the interaction between the two collections of charges is given by the volume integral of ρϕ over V.Īttribution: Hyperphysics Example: Conservation of momentum and translation symmetry The charges that produce ϕ are far away from the sample. Let V be the region occupied by the sample, and assume that this region is very small and includes the origin. Let ϕ represent the known potential and let ρ represent the unknown charge density function for the sample. Such a decomposition is called a multipole expansion. Our strategy will be to represent the unknown distribution as a superposition of elementary charge distributions each of which interacts only with the nᵗʰ-order terms in the Taylor expansion of the potential. We assume that we have complete knowledge of the applied potential because this is our experiment and we get to control all of that. To accomplish this, we have to deduce information about the distribution by observing how it interacts with an electric potential. Suppose that we would like to investigate the distribution of charge inside a sample of material. Whether you agree with this strong interpretation or not, the fact is that a working knowledge of Taylor’s Theorem and its consequences is absolutely essential to physicists and you will not get very far without it. The case has even been made that physics itself is in some sense the study of linearization applied to the natural world. Introductory and intermediate physics courses don’t spend much time on Taylor approximation (or approximation techniques in general) because routine, simple problems with exact answers are more instructive at that educational stage than more open-ended problems that may require some creativity to solve, which may include strategic application of linear approximations. These are situations that you will encounter routinely in both theoretical and experimental physics. By using a linear approximation to express the motion of one of the particles in terms of the motion of the other and interpreting the result, I found expressions for the translation, rotation, and deformation. The information that I had was the empirical fact that the motion of a parcel of an incompressible Newtonian fluid has a translation component, a rotation component, and a component associated with the deformation of the parcel. I had to find the relationship between the velocities of two test particles contained in an infinitesimal parcel of fluid. ![]() That information can be expressed mathematically by associating the that qualitative information with the derivatives that appear in a Taylor expansion of a function that describes the process being studied.Īn example arose in my recent article on the Navier-Stokes Equations. Taylor’s Theorem is also relevant in situations where we have some qualitative information about the relationship between physical processes at nearby points. ![]() The tangent plane to the graph of f at point P approximates nearby points on the graph. ![]()
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