Why did nature decide to make conjugate of position to be momentum? Since energy and position do not commute, why not energy? What determines the pairing of time with energy and momentum with position?
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1$\begingroup$ Related: physics.stackexchange.com/q/18280/2451 $\endgroup$– Qmechanic ♦Oct 28, 2013 at 22:41
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The derivatives of action are conjugate variables to the quantity with respect to which one is differentiating. Action is an attribute of the dynamics of a physical system
The energy of a particle at a certain event is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the time of the event.
The linear momentum of a particle is the derivative of its action with respect to its position.
The angular momentum of a particle is the derivative of its action with respect to its orientation (angular position).
The electric potential (φ, voltage) at an event is the negative of the derivative of the action of the electromagnetic field with respect to the density of (free) electric charge at that event.
The magnetic potential (A) at an event is the derivative of the action of the electromagnetic field with respect to the density of (free) electric current at that event.
The electric field (E) at an event is the derivative of the action of the electromagnetic field with respect to the electric polarization density at that event..
The magnetic induction (B) at an event is the derivative of the action of the electromagnetic field with respect to the magnetization at that event.
The Newtonian gravitational potential at an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to the mass density at that event.
answered Oct 28, 2013 at 22:28