Analytical Mechanics

This document provides an overview of the principles of Analytical Mechanics, including the Minimum Energy Principle and the Principle of Virtual Work. It also discusses Hamilton's equations of motion and Lagrangian equations in the context of conservative systems. The document offers a comprehensive study material for students studying Analytical Mechanics.

Analytical Mechanics

PART A
Question A.1
(i) The Minimum Energy Principle -
This principle refers to a restatement of II Law of Thermodynamics, which states that with
constant entropy and external parameters for a closed system, internal energy leads to reduce and
at equilibrium approach to minimum value. Such parameters include volume, constant magnetic
field and more. For example – Taking U(S, X1, X2, …) as total energy system, where S
represents entropy and extensive parameters by Xi, which varies with system approaches of
equilibrium then as per principle of maximum entropy –
ƏS = 0 … it states that entropy is at extremum
ƏX U
Ə2 S = 0 i.e., entropy is at maximum
ƏX2 U
As both equations follow the properties of exact differentiation, therefore, for a closed system,
entropy equation of state will be –
ƏU = – (ƏS/ƏX)U
ƏX S (ƏS/ƏU)X
= - T ƏS
ƏX U
Therefore, for keeping the energy at minimum,
Ə2 U = –T Ə2 S
ƏX2 S ƏX2 U
which is greater than or equal to zero.
(ii) Principle of virtual work
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The principle of virtual work as per law of Newton states that virtual work of any forces applied
in equilibrium to a system would be zero. In other words, the applied forces at equilibrium of
Static system are equal as well as opposite to constraint forces. The work done in this regard
during the virtual displacement, by a particular force F and acts on a particle δr refers to virtual
work, can be written as at equilibrium position –
ƏU = F. Ər
or,
ƏU = F. Əs. cosά
where, ά is angle between force and Ər, while Əs is magnitude of Ər.
In mechanism of static equilibrium, the most important advantage of applying principle of
virtual work is it includes only those forces which do work as the system moves
via virtual displacement, that helps in determining the mechanics of system.
Question A.3
As per d'Alembert's Principle,
R – T cosά = 0
or,
R – mg cosά = 0
here, R = mg cosά
then, work done against force = μR
= μ mg cosά
Question A.4
(a) Hamilton’s equation of motion for a conservative system is given by -
pj = – ƏH
Əqj
and,
qj = – ƏH
Əpj
L = KE – PE
where,
KE = ½ Iϴ2 = ½ ml2. Ӫ 2
PE = mgh = mg (l - lcosϴ)
Then,
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L = ½ ml2. Ӫ2 – mg (l - lcosϴ)
(i) Here, pj is the indexed to frame of reference of the system or conjugate momentum. In simple
pendulum, it can be represented as -
pj = ml2. Ӫ
in given diagram, l = 2a
pj = 4a2 m. Ӫ
(ii) Hamilton equation is given by –
H = ½ ml2. Ӫ2 – mg (l - lcosϴ)
= ½ x 4a2 Ӫ2 – 2amg (1 - cosϴ)
= 2a2 Ӫ2 – 2amg (1 - cosϴ)
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PART B
B.2
(a) The Lagrangian L in terms of generalised coordinates ƿ and Ф is given as –
L = ½ m(ῤ2 + ƿ2Ф’2) – U (ƿ)
(ii) By recognising the angular momentum of particle, as Ф is a cyclic coordinate where pФ is
conserved, then as per Newton’s equation of motion there will be no torque on the particle.
(iii) Taking partial derivative of Lagrangian equation
Lẟ = mṗ
ṗẟ
Lẟ = mƿ2Ф’
Фẟ ’
So, Lẟ = mƿФ’2 - U(ƿ,Ф)ẟ
ƿ ƿẟ ẟ
So, Lagrange equation will be
Lẟ – d Lẟ = 0
ƿ dt ṗẟ ẟ
and,
Lẟ – d Lẟ = 0
Ф dt Ф’ẟ ẟ
-mƿФ’’ – 2mṗФ’ = 0
or,
Ф’’ = 2ṗФ’
ƿ
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