Generating Function for Lagrangian Invariant System

In summary, the conversation discusses finding the generating function G for a system with a Lagrangian and Hamiltonian that are invariant under a given transformation. The attempt at a solution involves using the canonical momentum to determine the change in momentum, with the assumption that the Lagrangian is of a specific form. However, this assumption may not be valid and further exploration is needed to find the correct approach.
  • #1
Physgeek64
247
11

Homework Statement


Given a system with a Lagrangian ##L(q,\dot{q})## and Hamiltonian ##H=H(q,p)## and that the Lagrangian is invariant under the transformation ##q \rightarrow q+ K(q) ## find the generating function, G.

Homework Equations

The Attempt at a Solution


##\delta q = \{ q,G \} = \frac{\partial G}{\partial p} ##
##\delta p = \{ p, G \} = -\frac{\partial G}{\partial q} ##

in this case ##\delta q =K(q)##
Hence
##G= pK + c_1(q) ##
Assuming the Lagrangian is of the form ##L=\frac{1}{2} m q^2 -V(q)##
then ##p=m\dot{q}##
and ##\delta p = m\dot{K} = mK' \dot{q}= pK'##

##\delta p = pK'=-\frac{\partial G}{\partial q} ##
## G= -pK +c_2(p)##

I know this is not right because i have two different expressions that can't be matched, but i can't think of another way to do this.

Many thanks
 
Last edited:
Physics news on Phys.org
  • #2
Physgeek64 said:
Assuming the Lagrangian is of the form
What makes you think you can assume this?

Edit: Also note that { and } are LaTeX delimiters used for grouping. To actually get the brackets of the Poisson bracket you need to use \{ and \}.
 
  • #3
Orodruin said:
What makes you think you can assume this?

Edit: Also note that { and } are LaTeX delimiters used for grouping. To actually get the brackets of the Poisson bracket you need to use \{ and \}.
Ahh okay then, so I guess i can't make that assumption. In which case i don't know how to proceed, any tips?

Thank you, i will change them :)
 
  • #4
Consider how the canonical momentum ##P = \partial L/\partial \dot Q## connected to ##Q = q + K(q)## must change relative to ##p##. This should give you ##\delta p##.
 
  • #5
Orodruin said:
Consider how the canonical momentum ##P = \partial L/\partial \dot Q## connected to ##Q = q + K(q)## must change relative to ##p##. This should give you ##\delta p##.

## \frac{\partial L}{\partial \dot{Q}} = \frac{\partial L}{\partial \dot{q}} \frac{\partial \dot{q}}{\partial \dot{Q}} =\frac{\partial L}{\partial \dot{q}} \frac{1}{1+\dot{K}'} = p \frac{1}{1+\dot{K}'} ##

?? is this along the right lines?

Many thanks
 

What is a generating function for Lagrangian invariant system?

A generating function for Lagrangian invariant system is a mathematical tool used to find and analyze symmetries in physical systems described by Lagrangian mechanics. It is a function that encodes the equations of motion and the symmetries of the system in a compact and elegant way.

Why is a generating function useful in Lagrangian mechanics?

A generating function allows us to identify and analyze symmetries in a physical system, which can provide insights into the underlying dynamics and behavior of the system. It also simplifies the process of finding conserved quantities, such as energy and momentum, in a Lagrangian invariant system.

How is a generating function related to Noether's theorem?

Noether's theorem states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. A generating function provides a systematic way to find these conserved quantities by exploiting the symmetries of the system. In essence, a generating function is the mathematical realization of Noether's theorem for Lagrangian invariant systems.

What are the different types of generating functions for Lagrangian invariant systems?

There are three main types of generating functions: type 1, type 2, and type 3. Type 1 generating functions are used for systems with one degree of freedom, type 2 for systems with multiple degrees of freedom, and type 3 for systems with both holonomic and non-holonomic constraints. Each type has its own specific form and is used for different types of physical systems.

Can a generating function be used to solve for the equations of motion in a Lagrangian invariant system?

No, a generating function does not directly provide the equations of motion. However, it can be used to simplify the process of finding the equations of motion by exploiting the symmetries of the system. The equations of motion can be obtained by taking appropriate derivatives of the generating function.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
1K
Replies
19
Views
1K
Replies
5
Views
1K
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
636
  • Classical Physics
Replies
1
Views
497
  • Classical Physics
Replies
21
Views
1K
Replies
27
Views
2K
  • Introductory Physics Homework Help
Replies
11
Views
774
Replies
3
Views
829
Back
Top