Lagrange's equations to Newton's equations

  • Thread starter Reshma
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In summary, Lagrange's equations can be reduced to Newton's equation of motion by taking Cartesian coordinates as the generalized coordinates. By substituting the expressions for kinetic energy and potential energy into the Lagrangian, we can obtain the Lagrange's equations for each variable. This equation shows the relationship between mass, acceleration, and the potential force. The terms involving the potential and the non-conservative forces represent different types of forces, with the potential force being conservative and the non-conservative forces including friction. Therefore, by setting the non-conservative forces to zero, we can derive Newton's second law of motion from the Lagrange's equations.
  • #1
Reshma
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I want to show the Lagrange's equations reduces to Newton's equation of motion if we take the Cartesian coordinates as the generalised coordinates.

So let T be the K.E. of the system and V be the P.E. of the system. So the Lagrangian is L=T-V.

So [tex]T = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)[/tex]

& [tex]V = V(x,y,z)[/tex]
Help me proceed with the proof :frown:.
 
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  • #2
Actually,

[tex]
T = \frac{1}{2}m \left( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 \right).
[/tex]

What do you get when you substitute [itex]T[/itex] and [itex]V[/tex] into Lagrange's equations?

Regards,
George
 
  • #3
Sorry for the typo and thank you replying. I made the substitution in Lagrange's equation.
[tex]\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q_i}}\right) - \frac{\partial L}{\partial \dot{q_i}}= Q_i'[/tex]

Setting q1=x,q2=y,q3=z

I get the Lagrange's equation for variable x as:
[tex]m\ddot{x} + \frac{\partial V}{\partial x} = Q_1'[/tex]

How does this equation show Newton's second law of motion?
 
  • #4
What is

[tex]
-\frac{\partial V}{\partial x}?
[/tex]

Regards,
George
 
  • #5
Umm...is it part of the net force?
 
  • #6
Reshma said:
Umm...is it part of the net force?

Yes. What is the force that results from a potential [itex]V=V\left(x,y,z\right)[/itex], and what type of force is it?

Regards,
George
 
  • #7
YOUR LAGRANGE EQUATION IS NOT RIGHT. THERE SOULD BE ZERO ON THE WRITE HAND SIDE NOT Q. AGAIN IF YOU PUT Q=0 IN YOUR SECOND EQUATION YOU GET;
MASS X ACCELERATION = - THE DIREVATIVE OF THE POTENTIAL WITH RESPECT TO x.
= FORCE.
ISN'T THIS THE SECOND LAW OF NEWTON?
 
  • #8
samalkhaiat said:
YOUR LAGRANGE EQUATION IS NOT RIGHT. THERE SOULD BE ZERO ON THE WRITE HAND SIDE NOT Q. AGAIN IF YOU PUT Q=0 IN YOUR SECOND EQUATION YOU GET;
MASS X ACCELERATION = - THE DIREVATIVE OF THE POTENTIAL WITH RESPECT TO x.
= FORCE.
ISN'T THIS THE SECOND LAW OF NEWTON?

The terms involving the potential and the terms involving the Q' s represent different types of forces. This is what I was trying to point the way towards in my last post.

Regards,
George
 
  • #9
Thank you very much for all your replies.
George Jones said:
Yes. What is the force that results from a potential [itex]V=V\left(x,y,z\right)[/itex], and what type of force is it?

Regards,
George
There is part of the force derivable from PE and the other part independent of the PE. So one has to be the conservative part and the other the non-conservative part. So Q's represent the non-conservative parts and V(x,y,z) represent the conservative part, right? Can you elaborate more on these forces?
 
  • #10
samalkhaiat said:
YOUR LAGRANGE EQUATION IS NOT RIGHT. THERE SOULD BE ZERO ON THE WRITE HAND SIDE NOT Q. AGAIN IF YOU PUT Q=0 IN YOUR SECOND EQUATION YOU GET;
MASS X ACCELERATION = - THE DIREVATIVE OF THE POTENTIAL WITH RESPECT TO x.
= FORCE.
ISN'T THIS THE SECOND LAW OF NEWTON?
Be easy on the Caps :wink:!
 
  • #11
so you wrote down Lagrange equation with a source on the right hand side of it. Then you ask about deriving Newton second law from it. But you actually did derive it.
The forces as you said conservative (derivable from potential) and non-conservative (which are not) like friction forces etc.
 

1. What are Lagrange's equations?

Lagrange's equations, also known as the Euler-Lagrange equations, are a set of equations used to describe the motion of a system in classical mechanics. They are derived from the principle of least action, and provide an alternative formulation to Newton's laws of motion.

2. How do Lagrange's equations relate to Newton's equations?

Lagrange's equations are equivalent to Newton's equations, meaning that they describe the same physical motion of a system. However, they are often preferred in more complex systems as they provide a more concise and elegant mathematical representation.

3. What are the advantages of using Lagrange's equations over Newton's equations?

One advantage of Lagrange's equations is that they can be easily generalized to systems with more complicated constraints or coordinate systems. They also often result in simpler equations of motion, making them easier to solve.

4. Can Lagrange's equations be used in all situations where Newton's equations are applicable?

Yes, Lagrange's equations can be used in any situation where Newton's equations are applicable. However, they are most commonly used in systems with many degrees of freedom or non-Cartesian coordinate systems.

5. Are Lagrange's equations applicable to all branches of science?

No, Lagrange's equations are specifically used in classical mechanics to describe the motion of physical systems. They are not applicable to other branches of science, such as quantum mechanics or relativity.

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