Physics Notes - Herong's Tutorial Notes - v3.25, by Herong Yang
Lagrangian in Cartesian Coordinates
This section provides a quick introduction to the Lagrangian in Cartesian coordinates, expressed in terms of r, r' and t.
In previous sections, we have discussed Lagrangian with position, velocity, and acceleration as scalar values, which is valid only if the object is moving in a 1-dimensional straight line.
Now, let's re-define Lagrangian for a single object moving a 3-dimensional space using Cartesian coordinates.
First the position of an object as a function of time can be expressed as a vector function, r(t):
r(t) = (x(t), y(t), z(t)) or: r = (x, y, z) r = (rx, ry, rz)
The velocity becomes a vector function of time, r'(t):
r' = dr/dt = (drx/dt, dry/dt, drz/dt)
The acceleration also becomes a vector function of time, r"(t):
r" = dr'/dt = (dr'x/dt, dr'y/dt, dr'z/dt) = (d2x/dt2, d2y/dt2, d2z/dt2)
Lagrangian Function in Cartesian Coordinates
Now let's look at the Lagrangian function in Cartesian coordinates.
H = T - V (G.1)
The kinetic energy part can be expressed in terms of r'
T = 0.5*m*|r'|2 (G.21)
The potential energy part can be expressed in terms of r:
V = V(r) (G.22)
So the Lagrangian function can be expressed in terms of r and r' directly, and t indirectly:
L = T - V (G.1) or: L = 0.5*m*|r'|2 - V(r) (G.23) # G.21 and G.22 applied
In general, the Lagrangian function can be expressed in terms of r, r', and t:
L = L(r,r',t) (G.24) # where r and r' are vectors of 3 coordinates.
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*|r'|2 - H # T = 0.5*m*|r'|2 applied or: L = p∙r' - H # p is the momentum defined as p = m*r' # ∙ is the vector dot product
The last equation is called Legendre transformation, which converts Hamiltonian to Lagrangian, and vice versa.
Table of Contents
Introduction of Frame of Reference
Introduction of Special Relativity
Time Dilation in Special Relativity
Length Contraction in Special Relativity
The Relativity of Simultaneity
Minkowski Spacetime and Diagrams
Action - Integral of Lagrangian
Action - Functional of Position Function x(t)
Hamilton's Principle - Stationary Action
Other Proofs of the Lagrange Equation
Lagrange Equation on Free Fall Motion
Lagrange Equation on Simple Harmonic Motion
►Lagrangian in Cartesian Coordinates
Lagrange Equations in Cartesian Coordinates
Introduction of Generalized Coordinates