By Dr. Walter Thirring, Dr. Evans Harrell (auth.)

ISBN-10: 3709185262

ISBN-13: 9783709185261

ISBN-10: 3709185289

ISBN-13: 9783709185285

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**Extra info for A Course in Mathematical Physics 1: Classical Dynamical Systems**

**Example text**

8]. 5) 1. M = ~n, X: (Xl> •.. •. , xn; V, 0, ... ,0). V = ~n, '1 = 00, u(t, x(O»: (t, Xi(O» --+ (Xl (0) + vt, Xz{O), ... , xn(O». A constant vector field induces a linear field of motion. 2. M = ~n\{(o, 0, ... , O)}, X: (Xl, ... , Xn) --+ (Xl, ... , Xn; V, 0, ... ,0), V arbitrary, but '1 is the smallest value of t for which V + (vt, 0, ... ,0) contains the origin. u(t, x(O» is again as in Example 1. The constant field of motion may leave M, in a length of time that depends on V. ~ 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ x 0 ~ ~ ~ ~ ~ ~ ~ ~n 3.

XiXjh{O) + .... From (i) and (ii) it follows that L(f(O» = L(constant) = 0, and then from (ii), L(XJ'i(O» = L(Xi)UO) and L(XiXj[,i;)/2 = XiL(X)f,JO). Therefore L(f)(O) = j;iL(X} The L(x;) are the components of the vector field X. It is obviously necessary that LU) E COO for the components to be COO. 3 Flows A vector field X defines a motion in the direction of X at all points of a manifold. feomorphisms of M. A vector field X is to be regarded as a field of direction indicators: to every point of M it assigns a vector in the tangent space at that point.

Gik = gki> and all eigenvalues are positive). Then Ilvll := (VIV)1/2 can be interpreted as the length of the vector v. If all the eigenvalues of gik = gki are different from zero but not necessarily positive, then we can still make the weaker statement (vlw)=O 'r/vEYq(M)~w=O. 4 Tensors 47 If this holds for all q E M, g is said to be nondegenerate. By the equation (vlw) = (v*lw) Vw E Yq(M), to every v is associated a v* = e*igikvk. 14) If a manifold M is given a nondegenerate, symmetric tensor field g E ffg(M), it is called a pseudo-Riemannian space.

### A Course in Mathematical Physics 1: Classical Dynamical Systems by Dr. Walter Thirring, Dr. Evans Harrell (auth.)

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