Preliminaries to Studying Non-Euclidean Geometry
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== Metric and Metric Tensor == | == Metric and Metric Tensor == | ||
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A metric specifies how we compute distances between points in space | A metric specifies how we compute distances between points in space | ||
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(ds)^2 = d\mathbf{x}^T\mathbf{g}d\mathbf{x} | (ds)^2 = d\mathbf{x}^T\mathbf{g}d\mathbf{x} | ||
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+ | === Examples === | ||
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+ | ==== Cartesian Coordinates ==== | ||
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+ | ==== Skewed Cartesian Coordinates ==== | ||
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+ | ==== Spherical Coordinates ==== | ||
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+ | ==== Poincaré Disk ==== | ||
== Geometric Connections and Co-variant Derivative == | == Geometric Connections and Co-variant Derivative == |
Revision as of 13:06, 9 May 2015
Contents |
Introduction
Continuum mechanics applies the tools of differential geometry to study stress and strain fields within a material by treating that material as an idealized continuum. In most cases, we assume that we assume Euclidean geometry for the continuum, but there are several situations when the assumptions of Euclidean geometry no longer apply.
One example is using the tools of continuum mechanics at astronomical length-scale where the effects of gravity, i.e. the curvature of space, become non-negligible, such as when studying neutron stars. ^{[1]} ^{[2]}
Another example is when applying continuum mechanics at the mesoscale to study material defects such as dislocations, voids and disclinations. ^{[3]} ^{[4]} ^{[5]} Normally, the continuum idealization would have meant that we cannot model explicitly discontinuities in the material such as due to dislocations and disclinations, but as the aforementioned references show, this problem can be overcome by applying non-Euclidean geometry to the discontinuous material.
Here is an intuitive way to see how non-Euclidean geometry helps: Consider a wall map of the Earth and notice that any such map would have a one or more discontinuities, e.g. the line where east-west hemispheres appear separated on the projection. A globe, however, does not manifest such discontinuity, because the globe's geometry is no longer Euclidean (flat) but spherical.
There are other potential future applications for why it maybe worth pursuing the generalization of continuum mechanics to non-Euclidean geometry. For example. non-Euclidean computational models could be more accurate as they will be able to naturally handle discontinuities and incompatibilities.
Nevertheless, understanding how the geometry of space should reflect in the equations of continuum mechanics is a challenging exercise, which is why perhaps it is not a part of the typical continuum mechanics curriculum. Non-Euclidean geometry is often veiled in highly abstracted mathematical language (tangent bundles, vector fields, Lie groups and Lie algebra, differential forms, ... ). While some of these abstractedness is necessary to ensure that we are not overly limiting or overly extending the applicability of the math, it is often more helpful to first grasp the basic concepts intuitively and then grok the generalization.
Here we will focus on an attempt to intuitively present the most basic abstractions in a way that would hopefully lend itself to better intuitive and not just syntactic understanding. The following sections attempt to introduce and motivate some of the key vocabulary of differential geometry. Specifically, we are interested in Riemannian geometry which is a particular generalization of Euclidean geometry. The reader is encourage to lookup the references for further reading.
Here is a brief overview of the basic concepts described in later sections:
- Contravariant and covariant representations are different representations of a given vector or point in space.
- The metric defines how we measure distances between points in space. In general the metric can vary from one point in space to another. In Riemannian geometry, the metric is specified by the metric tensor.
- Geometric connection defines how vectors at different points in space can be compared to each other.
Contravariant and Covariant Representation of Points, Reciprocal Basis
Choice of Basis Vectors
Fundamental to a physical space is the ability to refer to points in space. In order to refer to points in space we need a set of basis vectors and a reference point (the origin). For example let and be basis vectors in 2D space. Think of these as two non-overlapping meter sticks extending from the origin. Each meter stick represents both a direction as well as a unit of length in that direction. Since and are not parallel to each other, we can describe any point in space by stating how many times one needs to count off the length of in the direction followed by how many times one should count off the length of in the direction. Thus,
is a way to describe a point in terms of the given measuring sticks.
In summary:
- The choice of basis vectors and is arbitrary with the only limitatoin being that these two should not be parallel.
- The point exist independent of and does not change with the choice of basis vectors. However, the indices and are only meaningful in the context of the given basis vectors and would change with different choice of basis vectors.
Contravariant Representation of a Point
The coefficients and form the contravariant representation of point . The term "contravariant" signifies that these coefficients vary inversely with respect to the length of their respective meter sticks. In other words, if we were to choose longer meter sticks then the same point would be represented by smaller coefficients and vice versa.
To see this more intuitively, think of each basis vector as analogous to a unit of length, e.g. m, cm, etc. Then, notice for example that, is an illustration of how making the unit smaller requires a larger numerical value to represent the same length.
In short:
- A point in space is independent of the chosen basis, but its representation depends on the choice of basis.
- The term contravariant characterizes the point's representaiton and not the point itself and signifies how the representation changes with a change of basis.
Covariant Representation of a Point
One might ask how can one determine the contravariant coefficients and given the point and the basis vectors and . If the given basis vectors formed a Cartesian orthonormal basis, then the coefficients would be simply the orthogonal projections of the point onto the respective basis vectors, which would be the same as computing the dot product (a.k.a inner product) between and the respective basis vector. However, for a general set of basis vectors that are not orthonormal this is not the case.
Nevertheless, it turns out that we can choose a reciprocal set of basis vectors and such that
With the help of these reciprocal basis vectors we can compute each contravariant coefficient as the dot product between the given point and the respective resiprocal basis vector. For example,
and likewise for .
Since and themselves form another basis, the given point has a representation in that basis as follows,
The coefficients and form the covariant representation of point . The term "covariant" signifies that these coefficients would grow or shrink when the original set of basis vectors and grow or shrink respectively. That is, the covariant coefficients vary in the same way as the original basis vectors.
Metric and Metric Tensor
Definition
A metric specifies how we compute distances between points in space given a set of basis vectors.
For exmaple, let and be the basis vectors and let and be two points in space. If the line through and happened to be along one of the basis vectors, then we could use that basis vector as the measuring stick to count off the distance between the two points. However, for general positions of points and we have no apriori mechanism for determining distance between them. That is because we only know apriori how to use each basis vector as a measuring stick in the direction of that basis vector. For an arbitrary direction, we would have to apply a combination of all measuring sticks and how exactly to do that is something that needs to be specified in addition to the basis vectors. The metric tensor provides this additional information.
Let be the vector between points and so that
and
We will define the length of the segment connecting points and as follows:
In the above equation, the coefficients are called the metric. It is possible to show that these coefficients obey tensor transformation rules under change of basis and thus show that they are the components of a tensor known as the metric tensor. Being a tensor means that represents a quantity that is independent of the choice of basis and that only the particular representation of depends on the basis.
Indeed, we can write the equation above in a basis-independent way as follows:
Therefore it is the tensor alone, independent from the choice of basis, that is responsible for defining distnaces between points in space.
In general, may vary from point to point in space, i.e. is a tensor field. Therefore, the metric equation above is only valid within a small region in space for which we can treat as approximately constant. To signify that, we typically write the metric equation in terms of the infinitesimal distance like so:
Examples
Cartesian Coordinates
Skewed Cartesian Coordinates
Spherical Coordinates
Poincaré Disk
Geometric Connections and Co-variant Derivative
TODO
Materials for Further Study
TODO
References
- ↑ Gerard A Maugin, Magnetized deformable media in general relativity, Annales de l'I.H.P., section A, tome 15, #4(1971), p.275-302
- ↑ G.A.Maugin, On the covariant equaitons of the relativistic electrodynamics of continua. I. General equations, Universite de Paris VI, Laboratoire de Mecanique Theorique associe au C.N.R.S., tour 66, 75230 Paris, Cedex 05, France
- ↑ Kazuo Kondo, ON THE ANALYTICAL AND PHYSICAL FOUNDATIONS OF THE THEORY OF DISLOCATIONS AND YIELDING BY THE DIFFERENTIAL GEOMETRY OF CONTINUA, International Journal of Engineering Science, Vol. 2, pp. 219-251, Pergamon Press 1964
- ↑ John D. Clayton, Douglas J. Bammann, and David L. McDowell, A Geometric Framework for the Kinematics of Crystals With Defects, Philosophical Magazine, vol 85, nos. 33-35, pp. 3983-4010, February 2005
- ↑ J.D. Clayton, D.L. McDowell, D.J. Bammann, Modeling dislocations and disclinations with finite micropolar elastoplasticity, International Journal of Plasticity, 22 (2006) 210-256
- ↑ http://commons.wikimedia.org/wiki/File%3AGoode_homolosine_projection_SW.jpg, By Strebe (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
- ↑ http://commons.wikimedia.org/wiki/File%3ABasis.svg, By Original diagram (File:Basis.gif) due to Hernlund, redrawn by Maschen. (Own work) [Public domain], via Wikimedia Commons