346 APPENDIX B. CURVILINEAR COORDINATES

in which n = 2, l is fixed and y1 = l sinθ ,y2 = l− l cosθ . Thus, in this simple exam-ple, m = 1. If l were changing in a known way with respect to t, then this would beof the form y = y (x,t) . In general, the kinetic energy is defined as

T ≡ 12

mẏ · ẏ (∗)

where the dot on the top signifies differentiation with respect to t. Show

∂T∂ ẋk = m ẏ· ∂y

∂xk .

Hint: First show

ẏ =∂y

∂x j ẋ j +∂y

∂ t(∗∗)

and so∂ ẏ

∂ ẋ j =∂y

∂x j .

3. ↑ Showddt

(∂T∂ ẋk

)= m ÿ· ∂y

∂xk +m ẏ· ∂ 2y

∂xk∂xr ẋr +mẏ · ∂ 2y

∂ t∂xk .

4. ↑ Show∂T∂xk = m ẏ·

(∂ 2y

∂xr∂xk ẋr +∂ 2y

∂ t∂xk

).

Hint: Use ∗ and ∗∗ .

5. ↑ Now show from Newton’s second law ( mass times acceleration equals force ) thatfor F the force,

ddt

(∂T∂ ẋk

)− ∂T

∂xk = m ÿ· ∂y

∂xk = F · ∂y∂xk . (∗∗∗)

6. ↑ In the example of the simple pendulum above,

y =

(l sinθ

l− l cosθ

)= l sinθ i+ (l− l cosθ)j.

Use ∗∗∗ to find a differential equation which describes the vibrations of the pendu-lum in terms of θ . First write the kinetic energy and then consider the force actingon the mass which is −mgj.

7. Of course, the idea is to write equations of motion in terms of the variables xk, insteadof the rectangular variables yk. Suppose y = y (x) and x is a function of t. Letting Gdenote the metric tensor, show that the kinetic energy is of the form 1

2 mẋT Gx wherem is a point mass with m its mass.

8. The pendulum problem is fairly easy to do without the formalism developed. Nowconsider the case where x = (ρ,θ ,φ) , spherical coordinates, and write differentialequations for ρ,θ , and φ to describe the motion of an object in terms of these coor-dinates given a force, F.