## Summation Convention

We use the Einstein Summation convention here, i.e. twice repeated indices will be summed over. So,

(1)Note that the kinetic energy will be denoted, **not** by $m_{i}\dot{x_{i}}\dot{x_{i}}$ which would be abuse of notation, but rather by $m_{i}\dot{x_{i}}^2$. The bounds on the dummy variable i will be clear from the context. Here, since $x_{i}$ denotes position of the i-th particle of an N particle system interacting with each other, i ranges from 1 to N. The forces will be conservative.

Consider the Lagrangian of such a system,

(2)In general, V may depend explicitly on time, but not on $\dot{x}$. Then, we have

(3)which is a component of the linear momentum. We define canonical or generalized momentum by

(4)in terms of the generalized position $q_{n}$. These need not be momenta in cartesian space, for instance in polar coordinates, we would have $p_{\theta} = \partial L/\partial \dot{\theta}}$, which doesn't even have the dimensions of linear momentum.

Next, consider the time rate of change of the Lagrangian:

(5)From the Euler-Lagrange equations, we have

(6)We define a quantity called the Hamiltonian (H), by

(7)This gives, after some algebra

(8)If L is not an explicit function of time, then $\partial L/\partial t = 0$ and so, the Hamiltonian is constant in time. If the forces are all conservative, we have also

(9)and hence,

(10)So if L and V are both not explicitly dependent on time, then the Hamiltonian is just the total energy of the system and is constant in time (this proves energy conservation in a classical conservative dynamical system).

The definition yields

(11)which simplifies to

(12)This gives

(13)and

(15)which are Hamilton's equations. The set of generalized position and momenta, $(q_{n},p_{n})$ can be replaced by another set $(Q_{n},P_{n})$ where Q and P are in general, functions of q, p and t. If Hamilton's equations remain invariant under such a transformation, then the transformation is said to be a canonical transformation.