Lagrangian Equation using Hamilton’s Principle

Lagrangian Equation using Hamilton’s Principle 

Photo by Anoushka Puri on Unsplash

In the blog post, I discussed the Generalized coordinates and the need for Hamiltonian and Lagrangian. 

In this blog post, I discussed Hamilton’s Principle in detail. Go through it, to follow through Lagrangian Equation using Hamilton’s Principle.

1. Step 1 involves writing Hamilton’s Principle.

2. Langrangian depends on q, q dot and t so

L= L (q,q dot, t)

But if we use scheleronomic constraints, the time term is ignored.

3. Write the expression of dL using the expression in step 2.

 Put the expression obtained in step 3 in Hamilton’s Principle expression.

4. Open up the bracket such that dt comes with both terms.

5. The second term (the term containing dq dot) should be solved using integration by parts. 

6. If there’s no variation in q dot, we put it equal to zero.

7. The remaining expressions are rearranged by taking fast common.

8. dqdt is not equal to zero, so the remaining expression must be equal to zero.

9. On rearranging, we get the Lagrangian equation.  

It can be written for any variable other than q. For example, theta, psi, etc.