Dr. Wang Xingbo

1
Mathematical Mechanical Method in Mechanical
Engineering
Dr. Wang Xingbo Fall,2005
2
Mathematical Mechanical Method in Mechanical
Engineering
Introduction to Calculus of Variations
Functional and Calculus of Variation
Find the shortest curve connecting P (a, y(a))
and Q (b, y(b)) in XY plane
3
Mathematical Mechanical Method in Mechanical
Engineering
Introduction to Calculus of Variations
The arclength is
The problem is to minimize the above integral
4
Mathematical Mechanical Method in Mechanical
Engineering
Introduction to Calculus of Variations
A function like J is actually called a functional
. y(x) is call a permissible function
A functional can have more general form
5
Mathematical Mechanical Method in Mechanical
Engineering
Introduction to Calculus of Variations
We will only focus on functional with integral
A increment of y(x) is called variation of y(x),
denoted as dy(x)
6
Mathematical Mechanical Method in Mechanical
Engineering
Introduction to Calculus of Variations
consider the increment of Jy(x) caused by dy(x)
?J y(x) J y(x)dy(x)- J y(x) ?J
y(x) Ly(x), dy(x)