This document discusses variational principles and Lagrange's equations. It introduces Hamilton's principle, which states that a holonomic and conservative dynamical system will follow the path in configuration space that makes the action integral stationary. The action integral depends on how the generalized coordinates change over time and the shape of the path in configuration space. Configuration space is defined as the space described by the generalized coordinates of a system. Hamilton's principle can be derived from D'Alembert's principle using the kinetic energy and applied forces. Lagrange's equations can then be derived from Hamilton's principle by considering how the action integral changes for nearby virtual paths compared to the actual path.

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Chapter 2
Variational Principles & Lagrange’s Equations
In Last chapter Lagrangian equation is derived by considering instantaneous state of the system
and small virtual displacement about the instantaneous state, i-e from Differential Principle
such as “D’Alembert’s Principle”.
In this chapter Lagrangian equation is obtained from Principle that consider entire motion of
the system between time ., and small virtual variations of this motion from actual
motion. A principle of this nature is known as an “Integral Principle”.
“Motion of the system between ” is the instantaneous configuration of a system by the
value of the n generallized coordinates … … … . , and corresponds to a particular point in a
Cartesian hyperspace where the q’s from the n coordinate axis. This n-dimensional space is
therefore known as Configuration space. Astime goes on, the state of the system is changes and
the system point’s moves in configuration space tracing out a curve described as “the path of
motion of the system”. The motion of the system refers to the motion of the system point
along this path in configuration space. Time can be considered as a parameter of the curve; to
each point on the path there is associated one or more values of the time. Note that
configuration space has no necessary connection with the physical three-dimensional space,
just as the generallized coordinates are not necessarily position coordinates. The path of
motion in configuration has no resemblance to the path in space of any actual particle; each
point on the path represents the entire system configuration at some given instant of time.
The integral Hamilton’s principle describes the motion of those mechanical systems
for which all forces (except the forces of constraint) are derivable from a generallized scalar
potential that may be a function of the coordinates, velocities, and time. Such systems will be
denoted as monogenic. Where the potential is an explicit function of position coordinated
only, then a monogenic system is also conservative.
Hamilton’s Principle
For the monogenic system, the motion of the system from the time , such that
the line integral (called the Action Integral)
=
has the stationary value for the actual path of motion.
OR
The actual path which a particle follows between two points 1 and 2 in a given time interval
, is such that the action integral

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=
Is stationary when taken along the actual path.
OR
“Hamilton Principle states that a Holonomic and conservative dynamical system moves over
the path in configuration space corresponding to which the value of action is stationary.”
(Variation Zero)
Value of integration Depends on
(i) How changes with time.
(ii) Shape of the path of dynamical system in configuration space.
(iii) Time.
Configuration Space
The space defined by the generallized coordinates of a system is called configuration of
the system.
The dimensions of this space are equal to the number of generallized coordinates.
Corresponding to any point in the configuration space, we have a unique value of generallized
coordinates, which represents a unique configuration of space. Therefore a point in the
configuration space represents the configuration of the system.
A point in the configuration space is specified by specifying all the generallized coordinates. A
path in the configuration space has shown the evolution of the system. The shape of the path
depends on time

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Derivation of Hamilton’s Principle from D’ Alembert’s Principle
Consider the system of n-particles whose configuration relative to the initial form is specified by
the vector , , , … … … , .
According to D’ Alembert’s Principle
Where
( )
represents the applied force acting on the ith particle and ___ is the virtual displacement.
So
The Kinetic Energy of the system is

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Derivation of Lagrange Equation from Hamilton’s
Principle
Consider a function ( , , )defined on a path =
( )between two values and , where = . For a
particular path y(x) such that the line integral J of the function
between and
Has a stationary value relative to paths differing infinitesimally
from the correct function ( ). The variable x plays the role of the parameter t.
Here ( ) = ( ) = . The figure does not represents the configuration space.
Since J have a stationary value for the correct path relative to any neighboring paths labeled by an
infinitesimal parameter , such that ( , ) with ( , 0) represents the correct path.
Consider a function ȵ(x) that vanishes at = and = , then the possible set of varied path
is given by
( , ) = ( , 0) + ȵ(x)
The action integral then becomes

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