I am currently studying non-linear dynamics on my own time. One of the theorems in the material is that systems that can be written as gradient problems cannot have closed orbits i.e. systems like $$\dot{x}=-\nabla V.\tag{1}$$

Isn't this the general form of a gravitational system with $V$ being the gravitational potential (or other conservative systems) and $x$ being the momentum? What am I missing here, knowing that such problems (gravity and like) often have closed orbits?

See this for reference http://www.cds.caltech.edu/archive/help/uploads/wiki/files/224/cds140b-perorb.pdf

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    $\begingroup$ Isn't it usually the force, rather than the momentum, that's proportional to the gradient of potential? If so, then the gravitational system should be of the form $\ddot{x}=-\nabla V$, not $\dot{x}=-\nabla V$. $\endgroup$ – probably_someone Jun 5 '18 at 18:46
  • $\begingroup$ I am taking x=mv to be the momentum. x is not the position. So rate of change of momentum $(\dot{x})$ is the force which is given by the gradient of the potential. Did I miss something? $\endgroup$ – curiousny Jun 5 '18 at 18:49
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    $\begingroup$ In order to be consistent with the definition of gradient system (see e.g. cds.caltech.edu/archive/help/uploads/wiki/files/224/…), your system must be of the form $\dot{x}=-\nabla V(x)$, so your potential is momentum-dependent if $x$ is momentum. This is clearly not equivalent to a gravitational system, where $V$ is dependent on position. This is why omitting the arguments of functions is sometimes dangerous. $\endgroup$ – probably_someone Jun 5 '18 at 18:57
  • $\begingroup$ I knew I was missing something, but couldn't figure our what. Great, thanks $\endgroup$ – curiousny Jun 5 '18 at 19:04
  1. OP's eq. (1) is Aristotelian mechanics $$ m\dot{q}^i~=~-\frac{\partial V(q)}{\partial q^i} \qquad\Rightarrow\qquad V_i-V_f ~=~2 \int_{t_i}^{t_f} \! \mathrm{d}t ~E_{{\rm kin}} \tag{A}$$ This is dissipative. There are no$^1$ closed orbits.

  2. In contrast Newtonian mechanics $$ m\ddot{q}^i~=~-\frac{\partial V(q)}{\partial q^i}\qquad\Rightarrow\qquad V_i+E_{{\rm kin},i} ~=~ V_f+E_{{\rm kin},f} \tag{N}$$ preserves mechanical energy for conservative forces.


$^1$ Indirect one-line proof: A closed orbit would mean that the LHS of the second equality in eq. (A) is zero, but the RHS is clearly positive. Contradiction. $\Box$


What's confusing you is the fact that you have implicitly omitted the argument of $V$. The definition of gradient system (http://www.cds.caltech.edu/archive/help/uploads/wiki/files/224/cds140b-perorb.pdf) is such that

$$\dot{x}=-\nabla V(x)$$

so if you define $x$ to be momentum, then you now have a potential which is dependent on momentum, which is not equivalent to a gravitational system (it has a potential that depends only on position).


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