Consider a particle of mass m moving on the inner surface of a smooth torus placed in a uniform gravitational field:
r=xE₁+yE₂+zE₃=aeᵣ+beϕ,
Where
eᵣ=cosθE₁+SinθE₂,eϕ=cosϕ+sinϕE₃
and a and b are positive constants and a>b; also,V=mgz.
Use θ and ϕ as generalized coordinates.
(a) Calculate the velocity and the kinetic energy of the particle. (b) Use the expression for velocity to find the covariant basis vectors for the torus. (c) Calculate the metric coefficients and the expression for the square of the line element of the torus. Observe that a₁₁ varies with ϕ.
(d) Calculate the unit normal vectora₃
(e) Argue that the covariant components of the constraint force are zero.
(f) Apply Lagrange's equations to obtain second-order differential equations for the generalized coordinates.
(g) What is the physical meaning of the conservation equation you just found?
(h) Argue that energy is conserved in this problem.
(i) Obtain an uncoupled differential equation for ϕ.