Manifolds Pt. 1. PHYS Southern Illinois University. August 24, 2016

Transcription

1 Manifolds Pt. 1 PHYS Southern Illinois University August 24, 2016 PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

2 Motivating Example: Pendulum Example Consider a pendulum of mass m and length l that hangs with angle θ from the vertical. Let M R 2 denote the collection of all possible coordinates for the pendulum. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

3 Motivating Example: Pendulum Example Consider a pendulum of mass m and length l that hangs with angle θ from the vertical. Let M R 2 denote the collection of all possible coordinates for the pendulum. Lesson: The switch to polar coordinates represents a map M R 1 that is locally invertible. The (x, y) coordinates of the pendulum s motion, M, is transformed into a Euclidean space R 1. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

4 Homeomorphisms Definition Two sets S 1 and S 2 are said to be homeomorphic if there exists a continuous and one-to-one map φ : S 1 S 2 whose inverse is also continuous. The map φ is called a homeomorphism. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

5 Homeomorphisms Example The unit circle and unit square are homeomorphic. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

6 Homeomorphisms Solution We have S 1 = {(x 1, x 2 ) : x1 2 + x 2 2 = 1}. Then we can define the homeomorphism (1, x 2 x 1 ); x 2 x 1 1 ( x 1 φ(x 1, x 2 ) = x 2, 1); x 1 x 2 1 ( 1, x 2 x 1 ); x 2 x 1 1 ( x 1 x 2, 1); x 1 x 2 1. (1) This is a continuous map with a continuous inverse. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

7 Topological Manifolds Definition A set of points M is called a manifold if each point of M has an open neighborhood which is homeomorphic to an open, bounded set of R n for some n. The dimension of the manifold is n, the dimension of the associated Euclidean space. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

8 Topological Manifolds Definition For a manifold M, a local coordinate chart on M is a pair (U k, φ k ) where U k is a an open set of M and φ : U k Ũ k is a homeomorphism from U k to an open bounded subset Ũ k = φ k (U k ) R n. For chart (U k, φ k ), the map φ k is called a local coordinate map, and the component functions (x 1,, x n ) of φ k, defined by φ(p) = (x 1 (p),, x n (p)) p U k provide the local coordinates of p on U. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

9 Topological Manifolds Definition For a manifold M, a local coordinate chart on M is a pair (U k, φ k ) where U k is a an open set of M and φ : U k Ũ k is a homeomorphism from U k to an open bounded subset Ũ k = φ k (U k ) R n. For chart (U k, φ k ), the map φ k is called a local coordinate map, and the component functions (x 1,, x n ) of φ k, defined by φ(p) = (x 1 (p),, x n (p)) p U k provide the local coordinates of p on U. Definition A collection of charts that cover the manifold (i.e. each point of M is contained in at least one U k ) is called an atlas. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

10 Topological Manifolds Example (The circle S 1 ) The unit circle S 1 = {(x 1, x 2 ) : x x 2 2 = 1} in R2 is a manifold. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

11 Topological Manifolds Solution We want to provide an atlas for S 1. One choice of atlas consists of four coordinate charts U 1 = {(cos θ, sin θ) : π/2 < θ < π/2}; U 2 = {(cos θ, sin θ) : 0 < θ < π}; U 3 = {(cos θ, sin θ) : π/2 < θ < 3π/2}; U 4 = {(cos θ, sin θ) : π < θ < 2π}; φ 1 (x 1, x 2 ) = arcsin φ 2 (x 1, x 2 ) = arccos φ 3 (x 1, x 2 ) = arcsin φ 4 (x 1, x 2 ) = arccos x 2 x x 2 2 x 1 x x 2 2 x 2 x x 2 2 x 1 x x 2 2. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

12 Topological Manifolds Example (The unit ball B n ) The unit ball B n in R n is defined by the set of points (x 1,, x n ) satisfying x x 2 n < 1. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

13 Topological Manifolds Example (Configuration Space in Classical Mechanics) The coordinates of a classical system of N particles define a manifold called the configuration space of the system. The dimension of this manifold is 3N n where n is the number of constraints placed on the system. Each coordinate on the manifold is a degree of freedom of the system, and the number of degrees of freedom is equal to the dimension of the configuration space 3N n. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

14 Product Manifolds Definition If M and N are manifolds, the product manifold M N is defined as the set of ordered pairs (x, y) where x is a point from M and y is a point from N. If x M has local coordinates (x 1,, x m ) and y N has local coordinates (y 1,, y n ), then (x, y) M N has local coordinates (x 1,, x m, y 1,, y n ). The dimension of M N is m + n. Example (Torus) The torus T is a product manifold of two circles S 1 S 2. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

15 Differentiable Manifolds Motivation Ultimately we will be interested in performing calculus on our manifold (i.e. taking derivatives, integrals, etc.). But how do we define such operations? Definition (Smooth Maps in Euclidean Space) If U and V are open subsets of R m and R n respectively, a function F : U V is said to be smooth if each of its component functions has continuous derivatives of all orders. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

16 Differentiable Manifolds Motivation Ultimately we will be interested in performing calculus on our manifold (i.e. taking derivatives, integrals, etc.). But how do we define such operations? Definition (Smooth Maps in Euclidean Space) If U and V are open subsets of R m and R n respectively, a function F : U V is said to be smooth if each of its component functions has continuous derivatives of all orders. Definition (Diffeomorphisms) A smooth homeomorphism F : U V is called a diffeomorphism. Two sets U and V related by a diffeomorphism are called diffeomorphic. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

17 Differentiable Manifolds Example The unit circle and unit square are homeomorphic but not diffeomorphic. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

18 Clearly this is not a continuous map. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20 Solution Consider again the homeomorphism (1, x 2 x 1 ); x 2 x 1 1 ( x 1 φ(x 1, x 2 ) = x 2, 1); x 1 x 2 1 ( 1/2, x 2 x 1 ); x 2 x 1 1 ( x 1 x 2, 1); x 1 x 2 1. Let φ 1 be the first component function of φ (i.e. φ(x 1, x 2 ) = (φ 1 (x 1, x 2 ), φ 2 (x 1, x 2 )). The x 1 partial derivative of φ 1 is 0; x 2 x φ = x 2 ; x 1 x 2 1 0; x 2 x x 2 ; x 1 x 2 1. (2) (3)

19 Differentiable Manifolds Suppose that (U 1, φ 1 ) and (U 2, φ 2 ) are local charts for M with U 1 U 2. For any p U 1 U 2 we have two sets of local coordinates: φ 1 (p) = (x 1,, x n ), φ 2 (p) = (y 1,, y n ). PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

20 Differentiable Manifolds Suppose that (U 1, φ 1 ) and (U 2, φ 2 ) are local charts for M with U 1 U 2. For any p U 1 U 2 we have two sets of local coordinates: φ 1 (p) = (x 1,, x n ), φ 2 (p) = (y 1,, y n ). This defines a coordinate transformation φ 2 φ 1 1 : R n R 2 : y 1 = y 1 (φ 1 1 (x 1,, x n )) := y 1 (x 1,, x n ) y 2 = y 2 (φ 1 1 (x 1,, x n )) := y 2 (x 1,, x n ) y n = y n (φ 1 1 (x 1,, x n )) := y n (x 1,, x n ). PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

21 Differentiable Manifolds Definition Local charts (U 1, φ 1 ) and (U 2, φ 2 ) are called smoothly compatible if the coordinate transformation φ 2 φ 1 1 is smooth. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20

22 Differntiable Manifolds Definition A manifold M is a differentiable manifold if it has an atlas with all local charts being smoothly compatible. PHYS Southern Illinois University Manifolds Pt. 1 August 24, / 20