Advanced Classical Mechanics/Hamilton's Equations

Hamilton's Equations
Hamilton's equations are an alternative to the Euler-Lagrange equations to find the equations of motion of a system. They are applied to the Hamiltonian function of the system. Hamilton's equations are

$$\dot{q_i} = \frac{\partial H}{\partial p_i}$$

$$\dot{p_i} = -\frac{\partial H}{\partial q_i}$$

$$\frac{\partial L}{\partial t} = -\frac{\partial H}{\partial t}$$

Where $$H$$ is the Hamiltonian, $$q_i$$ are the generalized coordinates, $$p_i$$ are the generalized momenta, and a dot represents the total time derivative. The Hamiltonian can be found by performing a Legendre transformation on the Lagrangian of the system:

$$H = \dot{q_i}p_i - L(q,\dot{q},t)$$

where the Einstein summation notation is used, a dot represents the total time derivative, qi are the generalized coordinates, pi are the generalized momenta.

These momenta are found by differentiating the Lagrangian with respect to the generalized velocities $$\dot{q_i}$$. Mathematically

$$p_i = \frac{\partial L}{\partial \dot{q_i}}$$.

In some cases, $$H = T + V$$ is the total energy of the system. There are two conditions for this. First, the equations defining the generalized coordinates q don't depend on time explicitly. Second, the forces involved in the system are derivable from a scalar potential. The forces must be conservative (such as gravity).

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