Conservation of Energy Using Lagrange Equation

                           Conservation of Energy Using Lagrange Equation

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In the blog post, we discussed the conservation of energy using Newton's Mechanics. Now, we’d discuss the conservation of energy using the Lagrange equation.

It involves the following steps:

1. Write lagrangian in Generalized coordinates.

L = L (q,q dot, t)

2. Find the differential of L, dL.

3. Divide dL by dt.

4. Write Euler Lagrangian Equation. And put the value of dL/dt obtained in step 3 in the equation.

5. The expression for Lagrange Equation, has two terms on the right side. We sum these terms up in a derivative using the linearity of the derivative. 

6. From the while expression we take, d/dt as common. The expression in parenthesis is supposed as a constant H which is called Hamiltonian is the system.

7. We know L=T-U

We differentiate the above equation w.r.t qj dot.

8. Potential is not velocity dependent. So the term with a derivative of U w.r.t qj dot equals zero. So, we are left with two terms in the expression. 

 

9. We have the value of Hamiltonian supposed in step 6. 

10. From the expression of Generalized kinetic energy, we find the term with the differential equation to 2T.

11. Rearranging we get H=T+U. 

Important Note

Hamiltonian H of a system is equal to the total energy E of the system only if two conditions are satisfied.

1. The transformation equations that connect rectangular and Generalized coordinates must be time-independent. The constraints must be schleronomic and this ensures K.E. is a homogeneous function of Generalized coordinates.

2. The potential energy U should be velocity independent. And it depends only on position.

U=U(qj)

Hence, the derivative of potential w.r.t qj dot must equal zero.