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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 :