Not that it's a favourite of mine or particularly difficult, just something i noticed earlier the other day which I kind of found interesting.. The aim of the problem is to gain insight into the role played by vector potential fields in generalising the relationship ##U(q_1,q_2,\cdots,q_n) = -V(q_1,q_2,\cdots,q_n)## between a velocity independent work function and the potential energy function, to the situation of a velocity dependent work function of a charge interacting with the E-M field.
Consider a charge ##+e## traveling at some moment in time with non-relativistic velocity ##\textbf{v}_0## in a uniform static magnetic field orthogonal to ##\textbf{v}_0## and uniform time independent background scalar potential ##\phi##. In such a situation, the charge undergoes uniform circular motion with energy $$E=\frac{1}{2}m\textbf{v}_0\cdot\textbf{v}_0 +e\phi,$$ which remains constant during the motion and is independent of the strength of the uniform magnetic field. Thus, the effect of the rotational vector field ##\textbf{A}## is simply to rotate ##\textbf{v}_0## at a constant rate in time without changing its magnitude.
1) Show that the Hamiltonian function is a constant of the motion which can be interpreted as the sum of kinetic energy ##T## and potential energy ##V##, provided the potential energy is defined as ##V=\sum_i A_iv_i - U(\textbf{x},\textbf{v})##.
2) What form should ##V## take generally (in terms of ##U##) for such an interpretation to be valid?
3) Thus, find the relationship between ##A_i## and the work function ##U##. Compare this expression to the definition of canonical momentum and comment on the nature of the mathematical relationship between the potential and work functions and of the variable ##A_i##.
Bonus (purely analytical mechanics really):
Given the invariant differential form of the work function ##dU=\sum_i F_i dq_i##, find an expression for the generalised forces ##F_i## in terms of a velocity dependent work function ##U(q_1,q_2,\cdots,q_n;\dot{q}_1,\dot{q}_2,\cdots , \dot{q}_n)##.
