Lagrangian Mechanics

Potential and Kinetic energies in terms of generalized coordinates.

Hamilton’s Principle

• The equation of motion will be derived.• Called Lagrange’s equations.• More fundamental than Newton’s equations • Because Hamilton’s principle can be applied wider range

of physical phenomena.• Hamilton’s principle state that the motion of a system

from time t₁ to t₂ is such that the line integral.J =

Where L = T – U T= kinetic energyU= potential energy

Hamilton’s Principle

J =

equation of motion will be

derived – Lagrange’s equations.

More fundamental

than Newton’s equations

Where L = T – V V = potential energyT = kinetic energy

state that the motion of a system from time t₁ to t₂ is such that the line

integral.

Example :

ly

x

𝜃

Lagrange’s equationL = T - V

Solution :

In Cartesian coordinates the kinetic and potential energy, and the Lagrangian are :

T = m - m

V = mgy

L = T – V

= m - m - mgy

Relations:

x= l

y= - l

• Taking time derivatives, we find

• Only one generalized coordinates for this problem, the angle . To find the equation of motion :

• and finally :