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```Dynamic Simulation:
Lagrange’s Equation
Objective

The objective of this module is to derive Lagrange’s equation, which
along with constraint equations provide a systematic method for
solving multi-body dynamics problems.
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Calculus of Variations
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 2
 Problems in dynamics can be formulated in such a way that it is
necessary to find the stationary value of a definite integral.
 Lagrange (1736-1813) created the Calculus of Variations as a
method for finding the stationary value of a definite integral. He
was a self taught mathematician who did this when he was
nineteen.
 Euler (1707-1783) used a less rigorous but completely
independent method to do the same thing at about the same
time.
 They were both trying to solve a problem with constraints in the
field of dynamics.
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Section 4 – Dynamic Simulation
Euler and Lagrange
Module 6 – Lagrange’s Equation
Page 3
Leonhard Euler
Joseph-Louis Lagrange
1707-1783
1736-1813
http://en.wikipedia.org/wiki/Leonhard_Euler
http://en.wikipedia.org/wiki/Lagrange
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Section 4 – Dynamic Simulation
Hamilton’s Principle
Module 6 – Lagrange’s Equation
Page 4
Hamilton’s Principle states that the path followed by a mechanical
system during some time interval is the path that makes the
integral of the difference between the kinetic and the potential
energy stationary.
t2
A

Ldt
t1
L=T-V is the Lagrangian of the system.
T and V are respectively the kinetic and potential energy of the
system.
The integral, A, is called the action of the system.
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Section 4 – Dynamic Simulation
Principle of Least Action
Module 6 – Lagrange’s Equation
Page 5
Hamilton’s Principle is also called the “Principle of Least Action”
since the paths taken by components in a mechanical system are
those that make the Action stationary.
t2
A
Ldt

Action
t1
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Section 4 – Dynamic Simulation
Stationary Value of an Integral
Module 6 – Lagrange’s Equation
Page 6
 The application of Hamilton’s Principle requires that
we be able to find the stationary value of a definite
integral.
 We will see that finding the stationary value of an
integral requires finding the solution to a differential
equation known as the Lagrange equation.
 We will begin our derivation by looking at the
stationary value of a function, and then extend these
concepts to finding the stationary value of an integral.
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Section 4 – Dynamic Simulation
Stationary Value of a Function
Module 6 – Lagrange’s Equation
Page 7
 A function is said to have
a “stationary value” at a
certain point if the rate of
change of the function in
every possible direction
from that point vanishes.
y
y=f(x)
 In this example, the
function has a stationary y1
point at x=x1. At this
point, its first derivative is
equal to zero.
x1
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3D Stationary Points
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 8
In 3D the rate of change of the function in any direction is
zero at a stationary point. Note that the stationary point is
not necessarily a maximum or a minimum.
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Section 4 – Dynamic Simulation
Variation of a Function
Module 6 – Lagrange’s Equation
Page 9
g  x   f  x     x 
 (x) is an arbitrary
function that satisfies the
boundary conditions at a
and b.
y
y  g x 
Candidate
Path
infinitely close to f(x) by
making the parameter 
infinitesimally small.
dy
y=f(x)
dx
Actual
Path
a
dy
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Meaning of dy
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 10
 The Calculus of Variations
considers a virtual
infinitesimal change of
function y = f(x).
y
y  g x 
dy
dy
 The variation dy refers to
an arbitrary infinitesimal
change of the value of y at
the point x.
 The independent variable
x does not participate in
the process of variation.
y=f(x)
dx
a
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Section 4 – Dynamic Simulation
Variation of a Derivative
Module 6 – Lagrange’s Equation
Page 11
In the calculus of variations, the derivative of the variation and the
variation of the derivative are equal.
Derivative of the Variation
d d y 
dx

d
dx

d
dx
d
 g  x   f  x 
  x   
dy

dx
d x 

dx
dx
d
dy
dx
dg  x 
d   x 
dx
d d y 
Variation of the Derivative
dx


df  x 
dx
d x 
dx
The order of operation
is interchangeable.
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Section 4 – Dynamic Simulation
Variation of a Definite Integral
Module 6 – Lagrange’s Equation
Page 12
In the calculus of variations, the variation of a definite integral is equal
to the integral of the variation.
Integral of a Variation
Variation of an Integral
d
b
b
b
a
a
 f  x dx   g  x dx   f  x dx
b
b
 d f  x dx    g  x   f  x dx
a
a
a
b

b
  g  x   f  x dx     x dx
a

   x dx
a
d
b
a
b
b
a
a
 f  x dx   d f  x dx
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The order of operation
is interchangeable.
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Section 4 – Dynamic Simulation
Specific Definite Integral
Module 6 – Lagrange’s Equation
Page 13
The specific definite integral that we want to find the stationary
value of is the Action from Hamilton’s Principle. It can be
written in functional form as
t2
A

t1
L  q i , q i , t 
n
L
 T  q t   V  q t 
i
i
i 1
qi are the generalized coordinates used to define the position
and orientation of each component in the system.
The actual path that the system will follow will be the one
that makes the definite integral stationary.
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Section 4 – Dynamic Simulation
Euler-Lagrange Equation Derivation
Module 6 – Lagrange’s Equation
Page 14
The stationary value of an integral is found by setting its
variation equal to zero.
 L


L
d L  q i , q i , t   L q i   i , q i   i , t  L  q i , q i , t    
i 
i 
 q i 
 qi


A first order Taylor’s Series was used in the last step.
t2
dA  d

t1
t2
t2
 L
L  
Ldt   d Ldt    
i 
 i dt  0
 q i 
 qi
t1
t1
t2
For an arbitrary value of ,
 L
L  
   q  i   q  i dt  0
i
i

t1 
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Euler-Lagrange Equation Derivation
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 15
The second integral is integrated by parts.
Integration by Parts
Substitutions
t2
 L
L  
   q  i   q  i dt  0
i
i

t1 
t2

t1
t2
 L 
L 
 i dt  
i  
 q i
  q i  t1
t2

t1
d  L

dt   q i
d uv   udv  vdu

 i dt


 is equal to zero at t1 and t2.
 L
d  L



   q dt   q
i
 i
t1 
t2
udv  d uv   vdu
u 
L
 q i
v

  i dt  0


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Euler-Lagrange Equation Derivation
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 16
 L
d  L



   q dt   q
i
 i
t1 
t2
d  L


 q i dt   q i
L

  i dt  0


The only way that this definite integral
can be zero for arbitrary values of i is for
the partial differential equation in
parentheses to be zero at all values of x in
the interval t1 to t2.

0


or
d  L

dt   q i
 L


0
 q
i

Lagrange’s equation
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Section 4 – Dynamic Simulation
Euler-Lagrange Summary
Module 6 – Lagrange’s Equation
Page 17
Finding the stationary value of the Action, A, for a mechanical
system involves solving the set of differential equations known as
Lagrange’s equation.
Solving these equations
d  L

dt   q i
 L

 q  0
i

n
L
 T q t   V q t 
i
i
i 1
Makes this integral stationary
t2
A
 L q , q , t 
i
i
t1
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Section 4 – Dynamic Simulation
Examples
Module 6 – Lagrange’s Equation
Page 18
 Although the derivation of Lagrange’s equation that
provides a solution to Hamilton’s Principle of Least Action,
seems abstract, its application is straight forward.
 Using Lagrange’s equation to derive the equations of
motion for a couple of problems that you are familiar with
will help to introduce their application.
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Section 4 – Dynamic Simulation
Vibrating Spring Mass Example
Module 6 – Lagrange’s Equation
Page 19
Governing Equations
Mathematical Operations
d  L

dt   q i
 L

 q  0
i

L  T V
T 
1
m y
y
 y
m
V 
ky
k
2
y is measured from
the static position.
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L
y
1
ky
2
2
 m y
d  L

dt   y
2
2
m y 
2
2
L
2
1
L 
1

  m y

  ky
Equation of Motion
m y  ky  0
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Section 4 – Dynamic Simulation
Falling Mass Example
Module 6 – Lagrange’s Equation
Page 20
Governing Equations
d  L


dt   q i
Mathematical Operations
L 
 L

 q  0
i

L
 y
T 
1
m y
2
y
g
2
 m y
d  L

dt   y
L
2
V  mgy
m y  mgy
2
m
L  T V
1
y

  m y

  mg
x
Equation of Motion
m y  mg  0
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Section 4 – Dynamic Simulation
Module Summary
Module 6 – Lagrange’s Equation
Page 21

Lagrange’s equation has been derived from Hamilton’s Principle of
Least Action.

Finding the stationary value of a definite integral requires the solution
of a differential equation.

The differential equation is called “Lagrange’s equation” or the “EulerLagrange equation” or “Lagrange’s equation of motion.”

Lagrange’s equation will be used in the next module (Module 7) to
establish a systematic method for finding the equations that control
the motion of mechanical systems.