Can a potential depend upon an arbitrary power of the canonical momentum? During a meeting yesterday, a colleague of mine stated that a potential cannot depend upon arbitrary powers of the canonical momentum.  So I am curious, is the potential limited to:
$$
\Phi = \Phi\left( \mathbf{q}_{0}, \mathbf{q}_{1}, ..., \mathbf{q}_{N}; \ \mathbf{p}_{0}, \mathbf{p}_{1}, ..., \mathbf{p}_{N} \right)
$$
or can it be represented as:
$$
\Phi = \Phi\left( \mathbf{q}_{0}^{i}, \mathbf{q}_{1}^{j}, ..., \mathbf{q}_{N}^{k}; \ \mathbf{p}_{0}^{l}, \mathbf{p}_{1}^{m}, ..., \mathbf{p}_{N}^{n} \right)
$$
where $i$, $j$, $k$, $l$, $m$, and $n$ are arbitrary real numbers?  I did not quite follow why they were so certain this could not be so.
I am okay with the answers being purely classical and non-relativistic if that makes the explanation easier/simpler.
Questions


*

*Can someone explain why, for instance, $\Phi\left( \mathbf{q}^{i}; \ \mathbf{p}_{0}^{2} \right)$ or $\Phi\left( \mathbf{q}^{i}; \ \dot{\mathbf{q}}_{0}^{2} \right)$ would not be allowed?

*Is there any physical reason to limit the order of derivatives of $\mathbf{q}$ in the explicit dependence of $\Phi$?  Meaning, could $\Phi$ depend upon $\tfrac{d^{K} \mathbf{q}}{dt^{K}}$, where $K$ is an integer $\geq$2?

 A: NB Your first question is improperly stated as Qmechanic pointed out in his comment. I interpret it in a precise sense: If there is a reason why $\Phi$ is supposed to depend at most linearly  on the first derivatives of Lagrangian coordinates.
I guess you are considering generalized Lagrangians of the form
$$L(t,q, \dot{q})= T(t,q, \dot{q}) - \Phi(t,q, \dot{q})\:, \tag{-1}$$
for classical systems described in a generalized coordinate system and also taking holonomous ideal constraints into accounts if any. In this case the kinetic energy  $T$ takes the form
$$T(t,q, \dot{q}) = \sum_{i,j=1}^n A(t,q)_{ij} \dot{q}_i\dot{q}_j + \sum_{j=1}^n B(t,q)_j\dot{q}_j + C(t,q)\:. \tag{0}$$
It turns out that the matrix $A(t,q) = [ A(t,q)_{ij}]_{i,j=1,\ldots, n}$ is symmetric an positively defined and in particular is invertible. Suppose that
$$\Phi= \Phi(t,q)$$
If you write down the E-L equations,
$$\frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j} -  \frac{\partial L}{\partial q_j}=0\:, \quad \frac{dq_j}{dt} = \dot{q}_j\:, \quad j=1,\ldots, n \tag{1}$$
using the fact that $A$ is invertible you see, with a tedious computation, that it is possible to re-write these equations into the precise form
$$\frac{d^2q_j}{dt^2} =  F_j(t,q, \frac{dq}{dt}) \quad j=1,\ldots,n\:.\tag{2}$$
where in particular, for some functions $G_k$ we have
$$F_j(t,q, \frac{dq}{dt})  = \sum_{k=1}^nA(t,q)^{-1}_{jk} G_k(t,q, \frac{dq}{dt})\:. \tag{3}$$
The form (2) of Euler-Lagrange's equations is said to be normal. This is a general notion in the theory of ordinary differential equation systems of order $n$ and just means that
the derivatives of highest order $n$ can be separated, and inserted in the left-hand side, from the derivatives  of other orders $n-1, n-2,\ldots, 0$ which appear in the right-hand side in any functional form.
If the right-hand side is sufficiently regular (jointly continuous and locally Lipschitz in the variables $(q, \dot{q})$), the existence and uniqueness theorem establishes that any system of 2nd-order differential equations of the normal form (1) admits a unique (local and global)  solution as soon as you fix the state of the system at initial time:
$$q(t_0) = Q\quad \dot{q}(t_0) = \dot{Q}\:.$$
This property is the mathematical  translation of the determinism principle of classical physics.
The crucial facts to pass from (1) to (2) are that (a) the first time-derivatives $\dot{q}_j$ appear quadratically in (0), (b) they do not appear in $\Phi$ and (c)
that $A$ in (0) in invertible.
The same result can be obtained if $\Phi$ is also function of the $\dot{q}_j$, but they appear therein linearly.
Any different dependence, in particular a quadratic dependence of $\dot{q}_j$ in $\Phi$ could give rise to an obstruction to reach the normal form of the Euler-Lagrange equations, so that the principle of physical determinism may fail to be satisfied.
It is worth stressing that linearity in $\dot{q}_j$ appearing in $\Phi$ is only a sufficient condition to fulfill standard hypotheses for the existence and uniqueness theorem. So one may construct physical systems respecting the determinism principle, but described with Lagrangians including potentials with non-linear dependence on $\dot{q}$.
However the only two cases of generalized potentials $\Phi(t,q, \dot{q})$ known in classical
physics, the potential of electromagnetic (Lorentz) forces and the potential of general inertial forces respect this linearity constraint.
Inserting time derivatives of order greater that $1$ in $\Phi$ gives rise to the same type of problems regarding the determinism principle, though these higher order derivatives are not completely forbidden and they are used in some semi-classical models, to describe the self-acceleration of an electric charge in particular.
