PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Geometry → Differential Geometry ↓


Geodesics


Differential geometry is a branch of mathematics that uses techniques from calculus and linear algebra to study problems related to geometry. A central concept within this field is the idea of the geodesic.

In simple terms, a geodesic is the shortest path between two points on the surface. Imagine you are walking along the Earth – wherever you travel, you are following a path along its curved surface. If you want to take the shortest path between two cities, you are essentially walking along a geodesic. However, finding these paths on different surfaces can be mathematically challenging and reveals a deep understanding of the geometry involved.

The concept of geodesics

To understand the concept of a geodesic, start by thinking about the simplest surface, the Euclidean plane. Here, the geodesic path between two points is simply a straight line. However, when moving to curved surfaces such as the surface of a sphere or torus, the idea of a "straight line" needs to be transformed into something more complex.

Mathematically, geodesics generalize the concept of a straight line and can be thought of as the "straightest possible paths" in a given space. They respect the intrinsic curvature of the surface or manifold on which they lie.

Geodesy on a sphere

One of the most common examples of geodesy is a great circle on a sphere. For example, consider the Earth. The shortest path between two points on the globe is along the surface of the Earth, and this path traces out a great circle.

A great circle is any circle on the surface of a sphere whose center coincides with the center of the sphere. The equator is an example of a great circle, as well as any longitudinal line. Note that these paths are not "straight lines" in the Euclidean sense, but represent the shortest travel distance on the curved surface of the sphere.

Imagine the Earth as a perfect sphere. To find the geodesic line between New York City and London, one would have to travel along the arc of a great circle crossing the Atlantic.
Great Circle

Mathematical definition

The precise mathematical definition of a geodesic is based on the calculus of variations. It is a curve that shows the constant value of the arc length under specific boundary conditions. In simple terms, it is a curve that minimizes (or sometimes only approximately minimizes) the distance between points.

For a surface embedded in space, the geodesic can be found by solving the geodesic equation. The general form of such an equation is:

[ frac{d^2 x^i}{dt^2} + Gamma^i_{jk}frac{dx^j}{dt}frac{dx^k}{dt} = 0 ]

Here, ( Gamma^i_{jk} ) represents the Christoffel symbols, which denote the coordinate curve of the space, and ( x^i(t) ) represents the parametric form of the geodesic.

Examples of geodesics

To understand how geodesics appear in more complex contexts, consider the surface of a torus (a doughnut-shaped surface). On a torus, geodesics can wrap around a central hole, move along a spherical body, or move in complex spirals. Each individual path can represent a geodesic, depending on the start and end positions.

Geodesic path

Applying geodesics within physics, they capture the motion of particles and light under the influence of gravity, which was accurately predicted by Einstein's general theory of relativity. Thus, geodesics are of paramount importance in describing natural phenomena such as planetary orbits, the trajectories of space probes, and light paths in gravitational lenses.

Properties of geodesics

Geodesics have many interesting properties, among which the following are noteworthy:

  • Local minimizers: Geodesics minimize distances locally. However, they may not always represent the globally shortest path on tubes such as spheres.
  • Parallel transport: vectors that parallel transport along a geodesic maintain constant magnitude and direction relative to the manifold.
  • Conjugate Points: Two distinct points on a manifold where a geodesic connecting them experiences a sort of concentrated mapping (like a lens), potentially recombining paths.

Due to these properties, geodesics serve as the cornerstone in optimization, global positioning system (GPS), map making, etc.

Calculating geodesics

Finding geodesics on complex surfaces involves solving differential equations numerically. Techniques include using numerical solvers to approximate the solution to the geodesic equations while ensuring boundary conditions are respected. General steps include:

  1. Define the metric tensor describing the geometry of the surface.
  2. Derive the Christoffel symbols based on this metric.
  3. Form the geodetic equation using these symbols.
  4. Implement numerical methods, such as the Runge-Kutta method, to solve the equations.

By following these algorithms, scientists can determine the shortest paths on unconventional surfaces.

Visualizing geodesics in the real world

Understanding geodesics conceptually and visually helps us recognize them in the world around us. For example, airplane routes and ship paths around the world are aligned with geodesics on the Earth's spherical surface.

In urban planning, geodesics play a subtle role in determining optimal traffic flow in urban landscapes, taking into account distances between streets rather than between buildings.

Conclusion

Geodesics are fundamental curves within differential geometry that extend our understanding of straight paths into new dimensions. They reveal the shortest or most direct possible routes on curved surfaces, impacting myriad scientific, engineering, and practical applications. From celestial mechanics to terrestrial navigation, recognizing and taking advantage of geodesics leads to both theoretical insights and concrete solutions.


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