Physics Friday – Hamiltonian Mechanics

Hamiltonian mechanics is an elegant way of formulating problems is classical mechanics.  Also, it provides insight into the world of quantum mechanics, as is evident with the Schrodinger equation.

The basic equations:

$$\dot{q}=\frac{\partial }{\partial p}H$$

$$\dot{p}=-\frac{\partial }{\partial q}H$$

Where $$q=q(t)$$ are the generalized coordinates and $$p=p(t)$$ are the generalized momenta.  H is the Hamiltonian and represents the total energy of the closed system (i.e. conservative) under consideration.  $$H=T+V$$ where T is the kinetic energy and V is the potential energy.  Also note that $$\dot{q}$$ represents the time derivative of the position, or the velocity, and that $$\dot{p}$$ represents the time derivative of the momentum, or the force.

These two equations yield the evolution of the mechanical system.  Aesthetically they are quite pleasing, owing to their nearly perfect symmetry.

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