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Coordinate Transformations

Introduction​

Coordinate transformations are mathematical operations that convert the description of a point or a physical quantity from one coordinate system to another. They are essential in physics and mathematics for solving problems, simplifying equations, and understanding how physical laws appear in different reference frames.

Types of Coordinate Transformations​

1. Between Different Coordinate Systems​

a. Cartesian to Polar (2D)​

x = r cos(θ) y = r sin(θ)

r = √(x² + y²) θ = tan⁻¹(y/x)

b. Cartesian to Spherical (3D)​

x = r sin(θ) cos(φ) y = r sin(θ) sin(φ) z = r cos(θ)

r = √(x² + y² + z²) θ = cos⁻¹(z/r) φ = tan⁻¹(y/x)

c. Cartesian to Cylindrical (3D)​

x = r cos(φ) y = r sin(φ) z = z

r = √(x² + y²) φ = tan⁻¹(y/x) z = z

2. Within the Same Coordinate System​

a. Translation​

x' = x + a y' = y + b z' = z + c

Where (a, b, c) is the translation vector.

b. Rotation​

For a rotation by angle α around the z-axis in 2D: x' = x cos(α) - y sin(α) y' = x sin(α) + y cos(α)

c. Scaling​

x' = kx y' = ky z' = kz

Where k is the scaling factor.

Matrix Representation of Transformations​

Many transformations can be represented as matrix operations:

1. 2D Rotation Matrix​

[x'] = [cos(α) -sin(α)] [x] [y'] [sin(α) cos(α)] [y]

2. 3D Rotation Matrices​

Around x-axis: [1 0 0 ] [0 cos(α) -sin(α)] [0 sin(α) cos(α)]

Around y-axis: [ cos(α) 0 sin(α)] [ 0 1 0 ] [-sin(α) 0 cos(α)]

Around z-axis: [cos(α) -sin(α) 0] [sin(α) cos(α) 0] [ 0 0 1]

Jacobian Matrix and Determinant​

The Jacobian matrix represents the best linear approximation to a differentiable function near a given point. It's crucial in coordinate transformations, especially for transforming integrals.

For a transformation from (x, y) to (u, v):

J = [∂u/∂x ∂u/∂y] [∂v/∂x ∂v/∂y]

The Jacobian determinant |J| gives the factor by which the transformation changes small volumes.

Applications in Physics​

  1. Relativity: Lorentz transformations between inertial reference frames
  2. Quantum Mechanics: Transformations between position and momentum representations
  3. Electromagnetism: Transforming fields between different coordinate systems
  4. Classical Mechanics: Describing motion in rotating reference frames
  5. Fluid Dynamics: Transforming between Eulerian and Lagrangian descriptions

Important Concepts​

1. Invariance​

Some quantities remain unchanged under certain transformations. For example, the distance between two points is invariant under rotations and translations.

2. Covariance​

The form of physical laws should be the same in all coordinate systems. This principle leads to tensor formulations of physical laws.

3. Active vs. Passive Transformations​

  • Active: Actually moving the object
  • Passive: Changing the coordinate system while the object remains fixed

Examples​

1. Transforming a Vector Field​

Consider the vector field F = (y, -x) in Cartesian coordinates. In polar coordinates:

Fr = y cos(θ) - x sin(θ) = -r sin²(θ) - r cos²(θ) = -r Fθ = -y sin(θ) - x cos(θ) = r sin(θ) cos(θ) - r sin(θ) cos(θ) = 0

So, F = (-r, 0) in polar coordinates.

2. Lorentz Transformation (Special Relativity)​

For a boost in the x-direction with velocity v:

t' = γ(t - vx/c²) x' = γ(x - vt) y' = y z' = z

Where γ = 1/√(1 - v²/c²) is the Lorentz factor.

Conclusion​

Coordinate transformations are a fundamental tool in physics and mathematics. They allow us to choose the most convenient coordinate system for a given problem, relate observations made in different reference frames, and reveal the underlying symmetries of physical laws.

Mastering coordinate transformations involves not only learning the mathematical techniques but also developing the intuition to choose the most appropriate transformation for a given problem. As you progress in your studies, you'll find that the ability to move fluidly between different coordinate representations is crucial for solving complex problems in physics, from classical mechanics to quantum field theory.