Proving Symmetry and Finding Conserved Quantities for Lagrangian Functions

In summary: Does this clarify?Yes, thank you!In summary, when given L(q, dq/dt, t) and a translation q ---> q + e (e is infinitesimal constant), we want to show that if ∂L/∂q = 0, then L is symmetry under the above translation. To do this, we can consider the expansion of L in the infinitesimal parameter e and use the Euler-Lagrange equation to find a conserved quantity.
  • #1
sunmaggot
62
4

Homework Statement


Given L (q, dq/dt, t).
translation: q ---> q + e (e is infinitesimal constant)
show that if ∂L/∂q = 0, then L is symmetry under the above translation.
then find conserved quantity.

Homework Equations


S = ∫ L dt

The Attempt at a Solution


My attempt is nothing... because I don't know the proper procedure to prove a symmetry. Do I simply prove L' - L = 0? But then the next part askes me to find conserved quantity. I have no idea how to find it...
So, can anyone spare me some hints...?[/B]
 
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  • #2
Let ##L_1=L_1(q, \dot{q}, t)## and ##L_2## that with the replacement ##q\rightarrow q+ \epsilon##. Now consider the expansion of ##L_2## in the infinitesimal parameter ##\epsilon##.
 
  • #3
CAF123 said:
Let ##L_1=L_1(q, \dot{q}, t)## and ##L_2## that with the replacement ##q\rightarrow q+ \epsilon##. Now consider the expansion of ##L_2## in the infinitesimal parameter ##\epsilon##.
do you mean taylor expansion??
 
  • #4
sunmaggot said:
But then the next part askes me to find conserved quantity. I have no idea how to find it...

A conserved quantity means null time derivative of that quantity. Look at the Euler-Lagrange equation.
 
  • #5
sunmaggot said:
do you mean taylor expansion??
Yup :)
 
  • #6
CAF123 said:
Yup :)
cool, I will try it first!
 
  • #7
CAF123 said:
Yup :)
the taylor series has a term (x-a), what should be the a?
 
  • #8
sunmaggot said:
the taylor series has a term (x-a), what should be the a?

What should be the x?
 
  • #9
ddd123 said:
What should be the x?
x is q?
 
  • #10
That's not what CAF123 suggested you to do. L is a function of phase space: you are interested in a small perturbation of the position variable, which changes the whole function L. So this means taking a McLaurin expansion in the infinitesimal translation parameter.
 

1. What is the Lagrangian symmetry problem?

The Lagrangian symmetry problem is a question in theoretical physics that asks whether a physical system with a certain type of symmetry can be described by a Lagrangian, which is a mathematical function that describes the dynamics of a system.

2. Why is the Lagrangian symmetry problem important?

The Lagrangian symmetry problem is important because it allows scientists to understand the fundamental principles that govern the behavior of physical systems. By studying symmetries and their relationship to Lagrangians, we can make predictions about how a system will behave and make sense of complex physical phenomena.

3. What is an example of a physical system that exhibits Lagrangian symmetry?

An example of a physical system that exhibits Lagrangian symmetry is a pendulum. The pendulum has rotational symmetry, meaning that it looks the same when rotated about its axis of rotation. This symmetry can be described by a Lagrangian function that governs the motion of the pendulum.

4. What are some challenges associated with the Lagrangian symmetry problem?

One challenge associated with the Lagrangian symmetry problem is that it can be difficult to determine whether a given physical system has a Lagrangian description. This requires a deep understanding of symmetries and mathematical techniques for constructing Lagrangians. Additionally, even if a Lagrangian exists, it may be very complex and difficult to work with.

5. How does the Lagrangian symmetry problem relate to other areas of physics?

The Lagrangian symmetry problem is closely related to other areas of physics, such as classical mechanics, quantum mechanics, and field theory. It provides a framework for understanding the behavior of physical systems and has applications in a wide range of fields, including particle physics, cosmology, and condensed matter physics.

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