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F=m(dv/dt) F=m(dv/dx)*(dx/dt) F=m(dv/dx) *v F*dx=mdv *v w =∑F*dx=∑mdv *v=1/2 mv^2-1/2mv0^2 then we define 1/2mv^2 as the kinetic energy

I can't understand the formula U(P)=K*∫∫U(Q)*F(θ0，θ0)*exp{ikr}/r*d∑. Can you express it more clearly?

The second question may be easier. Newton's second law should be expressed as F = dp/dt rather than F = ma,because F=dp/dt may be more universal than F =ma. When the velocity is far smaller than the speed of light,the mass can be seen as a constant , F=ma and F=dp/dt can be all right...

According to Rigid-body rotation Law ，you can get a formula mglsinθ=-ml^2*(d2θ/dt2) m is the mass of the ball ,l is the length of the rope and θ is the angle between the rope and vertical. According to Talyor series ,when θ is very small,sinθ=θ. Then it turns to (d2θ/dt2)+g/l *θ=0,and...