What Is Lagrangian

∟What Is Lagrangian

This section provides a quick introduction to the Lagrangian, which is a derived property of a mechanical system defined as the difference of kinetic energy and potential energy of the system.

What Is Lagrangian? Lagrangian, L, is a derived property of a a mechanical system defined as the difference of kinetic energy and potential energy of the system:

L = T - V                          (G.1)
  # T is the kinetic energy
  # V is the potential energy
  # L the difference of kinetic energy and potential energy

When a mechanical system changes over time, t, the value of Lagrangian may change too. The kinetic energy, the potential energy and Lagrangian of the system can be viewed as functions of t. So Lagrangian can also be expressed as:

L(t) = T(t) - V(t)                 (G.1)

Now let's compare it with the Hamiltonian of the system introduced earlier in the book:

H(t) = T(t) + V(t)                 (H.1)
  # H is the total of kinetic energy and potential energy

So Lagrangian L is related to Hamiltonian H through the following equation:

H = 2*T - L

or:
  H(t) = 2*T(t) - L(t)

The physical meaning of Hamiltonian is much easier to understand than the Lagrangian function. Hamiltonian function can be directly related to the Law of conservation of Energy For an isolated conservative system, the derivative of Hamiltonian against time is zero:

T(t) + V(t) = constant
H(t) = constant                    (H.2)

or:
  dH/dt = 0                        (H.3)

Lagrangian was introduced by Joseph-Louis Lagrange (1736-1813), an Italian mathematician and astronomer.

Lagrangian for a Single Object

For a single object, the kinetic energy, T, can be expressed as:

T = 0.5*m*v*v
  # m is the mass of the object
  # v is the velocity of the object

So the Lagrangian for a single object can be expressed as:

L = T - V

or:
  L = 0.5*m*v*v - V
    # T = 0.5*m*v*v applied

If we apply the Hamiltonian definition to Lagrangian, we can have:

L = T - V

or:
  L = 2*T - H
    # H = T + V applied

or:
  L = m*v*v - H
    # T = 0.5*m*v*v applied

or:
  L = p*v - H
    # p is the momentum defined as p = m*v