Physics Notes - Herong's Tutorial Notes - v3.25, by Herong Yang
What Is Lagrange Equation
This section describes the Lagrange Equation states that the time derivative of the partial derivative Lagrangian against velocity equals to the partial derivative Lagrangian against position. A proof is provided to show that the Lagrange Equation is equivalent to the Hamilton Principle.
What Is Lagrange Equation? Lagrange Equation states that the time derivative of the partial derivative Lagrangian L against velocity x' equals to the partial derivative Lagrangian L against position x, if x is a true position function of a system.
d(∂L/∂x')/dt = ∂L/∂x (G.5)
# L = L(x,x',t) is the Lagrangian function
# t is time
# x is the position of the object
# x' is the velocity of the object
Lagrange Equation is actually equivalent to the Hamilton's Principle. Here is a simple proof.
According the Hamilton's Principle, if x(t) is a true position function of the system, it must be a stationary point on the Action S[x(t)].
S[x(t)] = ∫ L(x,x',t)dt (G.2)
# Between t1 and t2
Now if we introduce a small variation of x(t) as xs(t) with an error function e(t) that vanishes at t1 and t2. A small parameter s is used to control the scale of the variation.
xs(t) = x(t) + s*e(t) (G.6) # s is small parameter # e(t) is any function with e(t1) = e(t2) = 0 or: xs = x + s*e # (t) omitted
Its velocity function will be xs'(t):
xs'(t) = x'(t) + s*e'(t) (G.7) or: xs' = x' + s*e' # (t) omitted
This will result a new Action value Ss:
Ss = S[xs(t)] or: Ss = ∫ L(xs,xs',t)dt (G.8) or: Ss = ∫ L(x+s*e,x'+s*e',t)dt (G.9) # G.6 and G.7 applied
Now we can expand the Lagrangian function with its first order partial derivatives, because the parameter s is very small.
L(x+s*e,x'+s*e',t) = L(x,x',t) + s*e*∂L/∂x + s*e'*∂L/∂x' (G.10)
Apply G.10 to G.9. We have:
Ss = ∫ (L(x,x',t) + s*e*∂L/∂x + s*e'*∂L/∂x')dt or: Ss = ∫ L(x,x',t)dt + ∫ (s*e*∂L/∂x + s*e'*∂L/∂x')dt (G.11)
According the Hamilton's Principle, x(t) is a stationary point on the Action S[x(t)]. So the small variation x(t) + s*e(t) should result the same Action S value:
S[xs(t)] - S[x(t)] = 0 # Since x(t) is a stationary point and s is small or: Ss - S = 0 or: ∫ L(x,x',t)dt + ∫ (s*e*∂L/∂x + s*e'*∂L/∂x')dt - ∫ L(x,x',t)dt = 0 # G.2 and G.11 applied or: ∫ (s*e*∂L/∂x + s*e'*∂L/∂x')dt = 0 (G.12)
Further reduction of G.12 requires the rule of integration by parts:
s*e*∂L/∂x'| = ∫ (s*e'*∂L/∂x')dt + ∫ (s*e*(∂L/∂x')')dt (G.13) # Between t1 and t2 or: 0 = ∫ (s*e'*∂L/∂x')dt + ∫ (s*e*(∂L/∂x')')dt # Since s*e*∂L/∂x'| = 0 # Remember that e(t1) = e(t2) = 0 or: ∫ (s*e'*∂L/∂x')dt = - ∫ (s*e*(∂L/∂x')')dt (G.14) # Between t1 and t2
Applying G.14 to G.12, we have:
∫ (s*x*∂L/∂x)dt - ∫ (s*x*(∂L/∂x')')dt = 0 # Between t1 and t2 ∫ s*x*(∂L/∂x - (∂L/∂x')')dt = 0 (G.15) # Between t1 and t2
Now here is a big jump. According to many text books, for equation G.15 to be true for any s, the following must be true.
∂L/∂x - (∂L/∂x')' = 0 or: ∂L/∂x - d(∂L/∂x')/dt = 0 # Since ()' = d()/dt or: d(∂L/∂x')/dt = ∂L/∂x (G.16)
Bingo! Equation G.16 is exactly the Lagrange Equation G.5. So we have approved that Hamilton's Principle is equivalent to Lagrange Equation.
By the way, the Lagrange equation is also called the Euler-Lagrange equation.
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