Motion of a puck on the frictionless surface of a paraboloïd.

Here is the surface, z = p( (x^2)+(y^2)), (we have set, p=1) :

g=Plot3D[x^2+y^2,{x,-2,2},{y,-2,2},Mesh->None,BoxRatios->Automatic]

Initial conditions for the puck : x[0]=0, x’[0]=-vx0, y[0]=y0, y’[0]=0, z[0]=(y0^2), z’[0]=0. Its motion results from the conjugated action of gravity (g) and of the contact with the surface. Friction is absent.

Clear[sol];sol[p_,g_,y0_,vx0_]:=NDSolve[{- x''[t]+(- g-(2 ((x'[t])^2)+2 ((y'[t])^2)+2 x[t] x''[t]+2 y[t] y''[t]))2p x[t]==0,- y''[t]+(- g- (2 ((x'[t])^2)+2 ((y'[t])^2)+2 x[t] x''[t]+2 y[t] y''[t]))2p y[t]==0,z[t]==p(x[t]^2+y[t]^2),x[0]==0,x'[0]==-vx0,y[0]==y0,y'[0]==0},{x,y,z},{t,0,100}]

A typical trajectory :

a0=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.sol[1,10,1.5,1]],{t,0,10},PlotStyle->Thick]

Show[a0,g]

By suitably adjusting the initial conditions you can obtain a stable circular trajectory : Take  ((x'[0])^2)+((y'[0])^2) =2g z[0].

a1=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.sol[1,10,1,(Sqrt[20])]],{t,0,10},PlotStyle->Thick]

Show[a1,g,PlotRange->All]

What if gravity is absent ? The only force is the upward action due to the contact with the surface : the puck “rises” spiraling in the direction of positive z !

a2=ParametricPlot3D[Evaluate[{x[t],y[t],z[t]}/.sol[1,0,1,1]],{t,0,10},PlotStyle->Thick]

Show[a2,g,PlotRange->All]

Comment : in all cases, the puck minimizes the constraint (Gauss principle).  If gravity is absent this is equivalent to saying that the puck looks for the least curvature or equivalently it follows a geodesic.  If tangential friction is taken into account (f=k N), the minimim constraint principle remains valid so that dissipation is also minimized.  This is a special example of the so-called Minimum entropy principle.

Addendum : Newton equations of motion.
When a particle is forced to stay on the surface, z= f(x,y), and is acted upon by a force F=((Subscript[F, 1]), (Subscript[F, 2]), (Subscript[F, 3])), the equations of motion are :
(Subscript[F, 1])-mx’’+((Subscript[F, 3])-m z’’)(Subscript[f, x])=0  and  (Subscript[F, 2])-my’’+((Subscript[F, 3])-m z’’)(Subscript[f, y])=0
Verify that the (square of the) constraint, ((((Subscript[F, 1])-mx''))^2) +((((Subscript[F, 2])-my''))^2)+((((Subscript[F, 3])-mz''))^2) , is indeed minimized. 
Remember that :

z[t_]:=p(x[t]^2+y[t]^2)

z'[t]

z''[t]