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Since a particle in one dimension can only move in a straight line. Is it possible to have potential energy? And how would the kinetic energy and potential energy differ in higher dimensions?

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Sure, a particle can have potential energy in one dimension. Just look at Hooke's Law or the gravitational force. Both of those are conservative forces in one dimension ($x$ and $r$, respectively) that have a corresponding potential energy.

In higher dimensions, nothing has to change, but it is possible to have potential energies which depend on the value of more than one of the dimensions (for instance, take $U(x,y)=x^2y^2$).

If your question is about how particular forces would change if the world we lived in were of lower dimension, well, that's more complicated. Suffice it to say that, mathematically, one can write whatever one wants, but physically, it is generally taken to be the case that lower dimensional forms of gravity and the electrostatic force would not be inverse-square laws. In one dimension, it is possible that the inverse-square laws that we all know and love might be constant forces (to maintain a lower dimensional form of Gauss's Law). Constant forces correspond to linear potentials (i.e. $U(x)=kx$); examples of this would be the electric field between a set of parallel plate capacitors or the gravitational potential near the surface of the Earth ($mgh$).

Kinetic Energy is much simpler. In most situations kinetic energy is taken (to a good approximation) to be a quadratic degree of freedom (read: proportional to $v^2$) for all dimensionality.

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Yes, a particle can have potential energy in one dimension as potential energy depends upon the configuration of the body. We can have configuration in one dimension as in hook's law and hence potential energy.

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