Chapter 7
Lagrangian and Hamiltonian Mechanics
Abstract Chapter 7 is devoted to problems solved by Lagrangian and Hamiltonian
mechanics.
7.1 Basic Concepts and Formulae
Newtonian mechanics deals with force which is a vector quantity and therefore difficult to handle. On the other hand, Lagrangian mechanics deals with kinetic and
potential energies which are scalar quantities while Hamilton’s equations involve
generalized momenta, both are easy to handle. While Lagrangian mechanics contains n differential equations corresponding to n generalized coordinates, Hamiltonian mechanics contains 2n equation, that is, double the number. However, the
equations for Hamiltonian mechanics are linear.
The symbol q is a generalized coordinate used to represent an arbitrary coordinate x, θ , ϕ, etc.
If T is the kinetic energy, V the potential energy then the Lagrangian L is
given by
L = T − V
(7.1)
Lagrangian Equation:
d
dt
dL
d ˙
q K
−
∂ L
∂q K
= 0 (K = 1, 2 . . .)
(7.2)
where it is assumed that V is not a function of the velocities, i.e.
∂v
∂ ˙
q K
= 0. Eqn (2)
is applicable to all the conservative systems.
When n independent coordinates are required to specify the positions of the
masses of a system, the system is of n degrees of freedom.
Hamilton
H =
r
s=1 p s ˙
q s − L
(7.3)
where p s is the generalized momentum and ˙
q K is the generalized velocity.
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Lagrangian and Hamiltonian Mechanics
Abstract Chapter 7 is devoted to problems solved by Lagrangian and Hamiltonian
mechanics.
7.1 Basic Concepts and Formulae
Newtonian mechanics deals with force which is a vector quantity and therefore difficult to handle. On the other hand, Lagrangian mechanics deals with kinetic and
potential energies which are scalar quantities while Hamilton’s equations involve
generalized momenta, both are easy to handle. While Lagrangian mechanics contains n differential equations corresponding to n generalized coordinates, Hamiltonian mechanics contains 2n equation, that is, double the number. However, the
equations for Hamiltonian mechanics are linear.
The symbol q is a generalized coordinate used to represent an arbitrary coordinate x, θ , ϕ, etc.
If T is the kinetic energy, V the potential energy then the Lagrangian L is
given by
L = T − V
(7.1)
Lagrangian Equation:
d
dt
dL
d ˙
q K
−
∂ L
∂q K
= 0 (K = 1, 2 . . .)
(7.2)
where it is assumed that V is not a function of the velocities, i.e.
∂v
∂ ˙
q K
= 0. Eqn (2)
is applicable to all the conservative systems.
When n independent coordinates are required to specify the positions of the
masses of a system, the system is of n degrees of freedom.
Hamilton
H =
r
s=1 p s ˙
q s − L
(7.3)
where p s is the generalized momentum and ˙
q K is the generalized velocity.
287