Question

How are Hamilton's generalized coordinates twice the Lagrange coordinates in classical mechanics?

Answer 1

In classical mechanics, Hamilton's generalized coordinates and Lagrange coordinates are related, but they are not generally equal to each other. The relationship between them can be more accurately described as the velocity being the time derivative of the position coordinates in both cases.

Let's consider a system with $n$ degrees of freedom. In Lagrangian mechanics, we typically describe the system using $n$ generalized coordinates, denoted as $q_i$ (where $i=1,2,\dots ,n$). The Lagrange coordinates describe the configuration of the system at a given time.

The Lagrange coordinates ($q_i$) are related to the Cartesian coordinates ($x_i$) of the system. The relationship between them is determined by the specific configuration of the system and can be quite complex. The Lagrange equations of motion are then derived using the Lagrangian function, $L$, which is a function of the generalized coordinates and their time derivatives ($\dot{q_i}$).

On the other hand, Hamiltonian mechanics introduces a different set of generalized coordinates, denoted as $p_i$ (where $i=1,2,\dots ,n$), known as the canonical momenta. The Hamiltonian function, $H$, is defined as a function of the canonical coordinates ($q_i$