Lagrangian mechanics

1. Lagrangian Mechanics
Lagrangian Mechanics is the reformulation of Classical Mechanics introduced by Italian
French Mathematician and Astronomer “Joseph-Louis Lagrange” in 1788.
Lagrangian is a function of generallized coordinate, their time derivative and time and
contains the information about the dynamics of the system.
Generallized Coordinates
Minimum no. of coordinates to specify the system.
Any set of variables which are used to specify the configuration of a system (of particles) are
called Generallized Coordinates.
Degree of Freedom:
Degree of freedom of a mechanical system is
“ The number of independent parameters that defines its configuration.”
For Example
i) Particle in a plane of two coordinates can be specified by its location, and has 2
degree of freedom.
ii) A single particle in space has degree of freedom of order 3.
iii) Two particles in space have combined degree of freedom of order 6.
iv) Two particles in space constrained to maintain a constant distance between them
have degree of freedom of order 5.
General Lagrangian Equation
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=
Standard Form of Lagrangian Equation
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= 0
Where = −

2. Mass Spring System
Since the particle is constrained to move along x-axis. So degree of freedom of this
system is 1. Proper set of generallized coordinate is “x” only, which is independent variable.
Equation of Motion by Classical Mechanics
From Hook’s Law
From Newton’s 2nd
Law
Comparing above equations we have
The solution of this differential Equation is
Equation of Motion by Lagrangian Mechanics
Lagrangian is defined as = −
= =
So above equation becomes
=
1
2
2 −
1
2
2
As degree of freedom of this system is 1, so there is only 1 Lagrangian Equation, which is
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−
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= 0

3. Simple Pendulum
A simple pendulum consists of a point mass “m” suspended
by a massless, inextensible string of length “l” is constrained to
oscillate in a vertical plane.
Degree of freedom of this system is 1, and the proper set of
generallized coordinate is only Ө(angular position of bob).
Lagrangian is defined as = −
= = .
= ℎ = ( − )