March 18th 2002
Modeling of Mechanical Systems II (RW)
MM2
A) Calculus of Variations - Corner Conditions
B) Lagrangian Mechanics
Place and time: Tuesday March 19th 2002 at 12.30 in C3-204
During the last lecture we discussed general issues related to the calculus of variations. An example the Fremat
principle was addressed. Finding a trajectory of a mechanical system is reduced to a solution of minimum/maximum
problem. The tools of variation calculus were analyzed and the Euler equation was derived. On Tuesday we shall look
upon the remaining issue: the corner conditions, which tell whether the solution curve is C 1 or D1. The second topic of
is modeling of mechanical systems using the calculus of variations. The key notion is the Lagrange equation.
Topics
1)
2)
3)
4)
5)
6)
Continuation of the introduction to calculus of variations – Corner Conditions
Hamiltonian Principle
Lagrange Equations
Connection between classical Newtons mechanics and variational mechanics
Generalized Forces
Conservation theorems and symmetry
Literature
1) Goldstein Herbert, Classical Mechanics, Addison-Wesley 1980 pp. 21-24, 35-37, 43-45, 54-63
2) John C. Clegg, Calculus of Variations, Oliver and Boyd Ltd 1968, pp. 29-40
Exercises:
Continue exercises from the last lecture:
1) Find minimum surface of revolution. Suppose we form a surface of revolution by taking some curve passing
between two fixed end points (x1, y1) and (x2, y2) and rotating it about the y axis (see figure below). The
problem then is to find that curve for which the surface are is a minimum.
Y
(x2,y2)
X
2) The brachistochrone problem. This well known problem posed first by John Bernoulli is to find the curve
joining two points, along which a particle falling from the rest under the influence of gravity travels from the
higher to the lower point in the least time.
X
1
V
Y
2
New exercise:
3) The term “generalized mechanics” has come to designate a variety of classical mechanics in which the
Lagrangian contains time derivatives of qi higher than the first. By applying the methods of the calculus of
, t ) and the Hamilton’s principle holds with
variations, show that if there is a Lagrangian of the form L(q, q , q
zero variation of both
 L  d  L  L

  
 
 0,
 qi  dt  q i  qi
i  1,2,..., n
Apply this result to the Lagrangian L  
m
k
qq  q 2 .
2
2
d2
dt 2
Best Regards,
Rafael
q i and q i at the end points, then the corresponding Euler-Lagrange equations are