Deriving Lagrange's & Hamilton's Equations in 1-Dimension

  • Thread starter Ed Quanta
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In summary, there are several textbooks on classical mechanics, including Landau and Lifschitz, that could provide a derivation of Lagrange's equations in one dimension. Another recommended book is "Mathematical Methods for Physics" by Vladimir Arnold. The principle of stationary action can be used to derive both Lagrange's and Hamilton's equations, which can then be connected through a Legendre transformation. The Euler-Lagrange equation can be obtained for a single particle with a single degree of freedom, and can be extended to systems with more degrees of freedom using indices.
  • #1
Ed Quanta
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Can someone direct me towards or provide me with a derivation of Lagrange's equations in one dimension? Are Hamilton's equations derived in a similar manner?
 
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  • #2
I think you could probably find both these things in any reasonable textbook on classical mechanics. Goldstein, Marion & Thornton, Corben & Stehle, Landau, any of those.
 
  • #3
Landau and Lifschitz

StatMechGuy said:
I think you could probably find both these things in any reasonable textbook on classical mechanics. Goldstein, Marion & Thornton, Corben & Stehle, Landau, any of those.

I was just going to suggest Landau and Lifschitz, Mechanics, Pergamon, 1976. This classic textbook is short and sweet, highly readable, has many excellent problems, and is often cited and should be widely available e.g. via amazon.com
 
  • #4
Yeah, an eighth of the book is homage to Landau.
 
  • #5
Yes, Landau is brilliant ! Another book, for the more mathematically inclined, is Arnold.
 
  • #6
i suggest you also "Mathematical methods for physics" Vladimir Arnold (i think this is the english translation.).
I think is the best because of its wide mathematical explanations. very good appendix.
To derive Lagrange's or hamilton's equation you just need the least action principle, or better the principle of stationary action.
applying this principle you get the right equations of motion even for fileds theory.
then if the hessian determinant of the Lagrangian or Hamiltonian is different form zero you can connect them via Legendre trasformation...
For a particle:

remeber that [tex]L=L(q,\dot{q})[/tex]
THE PRINCIPLE STATES THAT:

[tex]\delta\int L dt=\int\delta L dt=0[/tex]

[tex]\int\frac{\partial L}{\partial q}\delta q+ \frac{\partial L}{\partial \dot{q}}\delta\dot{q} dt=0[/tex]

now integrating by parts the second term of this integral and making the assumption that the little variations

[tex]\deltaq[/tex]

vanishes at the boundary. you get:

[tex]\int(\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}})\deltaq dt=0[/tex]

so if it is zero for arbitrary [tex]\delta q[/tex] it must be:

[tex]\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}=0[/tex]

wich is the Eulero Lagrange equation for a single particle with a single degree of freedom. The equations for three degree of freedom are soon obtained using indices, the same for a system of particles.

bye MArco
 
Last edited:
  • #7
I hope u can read i don't know what's going with latex,
i don't know if i understood right how to put symbols inside thge forum.

bye bye :-)
 
  • #8
You forgot to add a space after the code. Click on this symbol for the code.
[tex]\delta q[/tex]
 

1. What are Lagrange's and Hamilton's equations in 1-Dimension?

Lagrange's and Hamilton's equations are mathematical expressions that describe the motion of a particle or system in one dimension. They are based on the principle of least action, which states that a system will follow a path that minimizes the action (a measure of the energy) required for its motion.

2. What is the difference between Lagrange's and Hamilton's equations?

Lagrange's equations describe the motion of a system in terms of generalized coordinates, while Hamilton's equations describe the motion in terms of canonical coordinates. Lagrange's equations are based on the principle of least action, while Hamilton's equations are derived from Hamilton's principle, which states that the action is stationary along the path of motion.

3. How are Lagrange's and Hamilton's equations derived?

Lagrange's and Hamilton's equations can be derived using the calculus of variations, which involves finding the path that minimizes the action for a given system. This involves setting up the Lagrangian function (a combination of the kinetic and potential energies of the system) and then using the Euler-Lagrange equations to find the equations of motion. Hamilton's equations can also be derived from the Hamiltonian function, which is the sum of the kinetic and potential energies in terms of the canonical coordinates.

4. What are the applications of Lagrange's and Hamilton's equations?

Lagrange's and Hamilton's equations are used extensively in classical mechanics to describe the motion of particles and systems. They are also applied in fields such as quantum mechanics, electromagnetism, and even economics. They are powerful tools for analyzing the dynamics of complex systems and have numerous practical applications in engineering and physics.

5. Can Lagrange's and Hamilton's equations be extended to multiple dimensions?

Yes, Lagrange's and Hamilton's equations can be extended to multiple dimensions, allowing for the analysis of more complex systems. In multiple dimensions, there are more generalized and canonical coordinates, and the equations become more complex, but the basic principles and methods of derivation remain the same.

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