Physics Notes - Herong's Tutorial Notes - v3.25, by Herong Yang
Lagrange Equations in Cartesian Coordinates
This section provides a quick introduction to the Lagrange Equations in Cartesian coordinates, expressed in terms of r, r' and t.
If the Lagrangian function is expressed in Cartesian Coordinates as shown in the last section, the Action, Hamilton's Principle and Lagrange Equations can also be expressed in Cartesian Coordinates.
Action in Cartesian Coordinates
If the Lagrangian function is expressed in Cartesian Coordinates, the Action can also be expressed in Cartesian Coordinates.
S = S[r(t)]
or:
S[r(t)] = ∫ L(r,r',t)dt (G.25)
Hamilton's Principle in Cartesian Coordinates
In Cartesian Coordinates, the Hamilton's Principle stay the same: the true position function r(t) of a system is a stationary point of the Action S[r(t)] functional.
Hamilton's Principle is also called Stationary Action.
Lagrange Equations in Cartesian Coordinates
Now the Lagrange Equation becomes the Lagrange Equations in Cartesian Coordinates:
d(∂L/∂r')/dt = ∂L/∂r (G.26) or: d(∂L/∂r'x)/dt = ∂L/∂rx d(∂L/∂r'y)/dt = ∂L/∂ry d(∂L/∂r'z)/dt = ∂L/∂rz
We can prove that Lagrange Equations is equivalent to Hamilton's Principle in Cartesian Coordinates in the same way as we did earlier in the 1 dimension case.
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