Analytical Mechanics - Properties of The Lagrangian and Hamiltonian Functions

Properties of The Lagrangian and Hamiltonian Functions

Following are overlapping properties between the Lagrangian and Hamiltonian functions.

• All the individual generalized coordinates q_i(t), velocities q̇_i(t) and momenta p_i(t) for every degree of freedom are mutually independent. Explicit time-dependence of a function means the function actually includes time t as a variable in addition to the q(t), p(t), not simply as a parameter through q(t) and p(t), which would mean explicit time-independence.

• The Lagrangian is invariant under addition of the total time derivative of any function of q and t, that is:

so each Lagrangian L and L' describe exactly the same motion.

• Analogously, the Hamiltonian is invariant under addition of the partial time derivative of any function of q, p and t, that is:

(K is a frequently used letter in this case). This property is used in canonical transformations (see below).

• If the Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are constants of the motion, i.e. are conserved, this immediately follows from Lagrange's equations:

Such coordinates are "cyclic" or "ignorable". It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates.

• If the Lagrangian is time-independent the Hamiltonian is also time-independent (i..e both are constant in time).

• If the kinetic energy is a homogeneous function (of degree 2 - quadratic) of the generalized velocities and the Lagrangian is explicitly time-independent:

where λ is a constant, then the Hamiltonian will be the total conserved energy, equal to the total the kinetic and potential energies of the system:

This is the basis for the Schrödinger equation, inserting quantum operators directly obtains it.