In summary: G## as a "general function of the coordinates, momenta, and time, G(q,p,t)" in Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws. They explain that ##G## is used to generate transformations, which are maps from one configuration of coordinates and momenta to another. This can be seen through the equations ##\delta q_\alpha = \partial G / \partial p_\alpha \ \delta \lambda## and ##\delta p_\alpha = -\partial G / \partial q_\alpha \ \delta \lambda##, where ##\delta \lambda## represents a change in the coordinates and momenta
  • #1
sophiatev
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In Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws (starting on page 291 here), the authors introduce the concept of a generator function G, where the transformation generated by G is given by (equation 12.29 on page 292 in the text)

##\delta q_\alpha = \partial G / \partial p_\alpha \ \delta \lambda##
##\delta p_\alpha = -\partial G / \partial q_\alpha \ \delta \lambda##

They seem to introduce G as a "general function of the coordinates, momenta, and time, G(q,p,t)", where q and p range over all n generalized coordinates ##q_\alpha## and ##p_\alpha##. But if the above equations are true, they imply that

## \partial G / \partial p_\alpha \ \delta p_\alpha = -\partial G / \partial q_\alpha \ \delta q_\alpha ##

This property does not seem generally true at all, and so I don't see why it would apply to a "general function" G. Am I missing something?
 
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  • #2
Why do you think the authors imply the last equation?
 
  • #3
vanhees71 said:
Why do you think the authors imply the last equation?
I don't have that book and I am no good at Lagrangian mechanics, but isn't the last equation an algebraic consequence of the first two equations? Solve both equations for ##\delta \lambda## and equate what you get.

P.S I have no idea what ##\delta \lambda ## is so I don't know if I can treat it algebraically.
 
  • #4
Hi. Yes, ##G## is a perfectly general function.

The book is discussing transformations of the state. A transformation is just a map from one configuration of coordinates and momenta to another configuration. For example, in the usual cartesian coordinates ##x,y,z##, if you shift everything by ##\delta \lambda## in the x-direction, then the transformation is given by:

##x \rightarrow x + \delta \lambda##
##y \rightarrow y##
##z \rightarrow z##
##p^x \rightarrow p^x##
##p^y \rightarrow p^y##
##p^y \rightarrow p^y##

This transformation changes the x coordinate but nothing else.

Absolutely any mapping counts as a transformation, although the ones we are interested in for physics have some physical significance.

In general a transformation function would require one function for each coordinate and momenta:

##\delta q^j = Q^j(q,p,t) \delta \lambda##
##\delta p_j = P_j(q,p,t)\delta \lambda##

But an interesting type of transformation is one given by a generating function ##G(q,p, t)##, which defines the transformations via

##\delta q^j = \dfrac{\partial G}{\partial p_j} \delta \lambda##

##\delta p_j = - \dfrac{\partial G}{\partial q^j} \delta \lambda##

##G## is any function at all. It is just a way of generating a transformation.

For example, in one dimension, the transformation

##x \rightarrow x + \delta \lambda##
##p \rightarrow p##

is given by the generating function

##G = p##

Then ##\dfrac{\partial G}{\partial x} = 0## so ##\delta p = 0##. ##\dfrac{\partial G}{\partial p} = 1##, so ##\delta x = \delta \lambda##

The function ##G## is the x-component of momentum, and the effect is to shift ##x##. This is what is meant when they say that ##p## is the generator of translations.

A more interesting case is rotations. In two dimensions, a rotation is the transformation

##x \rightarrow x - y \delta \lambda##
##y \rightarrow y + x \delta \lambda##
##p_x \rightarrow p_x -p_y \delta \lambda##
##p_y \rightarrow p_y + p_x \delta \lambda##

(Note: this is an infinitesimal rotation, where we are allowed to approximate ##sin(\delta \lambda)## by ##\delta \lambda## and ##cos(\delta \lambda)## by 1. A real rotation is made up by summing many infinitesimal rotations.)

The generator for this transformation is:
##G = x p_y - y p_x##, which is just the angular momentum.

[Edit: was ##G = x p_y - z p_x##]

These sorts of transformations are interesting because if the transformation leaves the system unchanged, then the corresponding generator is a constant.
 
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  • #5
stevendaryl said:
Hi. Yes, ##G## is a perfectly general function.

The book is discussing transformations of the state. A transformation is just a map from one configuration of coordinates and momenta to another configuration. For example, in the usual cartesian coordinates ##x,y,z##, if you shift everything by ##\delta \lambda## in the x-direction, then the transformation is given by:

##x \rightarrow x + \delta \lambda##
##y \rightarrow y##
##z \rightarrow z##
##p^x \rightarrow p^x##
##p^y \rightarrow p^y##
##p^y \rightarrow p^y##

This transformation changes the x coordinate but nothing else.

Absolutely any mapping counts as a transformation, although the ones we are interested in for physics have some physical significance.

In general a transformation function would require one function for each coordinate and momenta:

##\delta q^j = Q^j(q,p,t) \delta \lambda##
##\delta p_j = P_j(q,p,t)\delta \lambda##

But an interesting type of transformation is one given by a generating function ##G(q,p, t)##, which defines the transformations via

##\delta q^j = \dfrac{\partial G}{\partial p_j} \delta \lambda##

##\delta p_j = - \dfrac{\partial G}{\partial q^j} \delta \lambda##

##G## is any function at all. It is just a way of generating a transformation.

For example, in one dimension, the transformation

##x \rightarrow x + \delta \lambda##
##p \rightarrow p##

is given by the generating function

##G = p##

Then ##\dfrac{\partial G}{\partial x} = 0## so ##\delta p = 0##. ##\dfrac{\partial G}{\partial p} = 1##, so ##\delta x = \delta \lambda##

The function ##G## is the x-component of momentum, and the effect is to shift ##x##. This is what is meant when they say that ##p## is the generator of translations.

A more interesting case is rotations. In two dimensions, a rotation is the transformation

##x \rightarrow x - y \delta \lambda##
##y \rightarrow y + x \delta \lambda##
##p_x \rightarrow p_x -p_y \delta \lambda##
##p_y \rightarrow p_y + p_x \delta \lambda##

(Note: this is an infinitesimal rotation, where we are allowed to approximate ##sin(\delta \lambda)## by ##\delta \lambda## and ##cos(\delta \lambda)## by 1. A real rotation is made up by summing many infinitesimal rotations.)

The generator for this transformation is:
##G = x p_y - z p_x##, which is just the angular momentum.

These sorts of transformations are interesting because if the transformation leaves the system unchanged, then the corresponding generator is a constant.
Was that last equation meant to be ##G = x p_y - y p_x##? But otherwise, I think that makes sense, thank you. Sounds like I was interpreting the ##\delta q^j## and ##\delta p_j## as the differential changes in the respective variables when taking a derivative (i.e. ##\delta G = \sum \partial G / \partial q^j \ \delta q^j + \partial G / \partial p_j \ \delta p_j##) rather than the differential changes produced by the transformation.
 
  • #6
sophiatev said:
Was that last equation meant to be ##G = x p_y - y p_x##?

Yes, you're right.
 

1. What is a symmetry in Lagrangian mechanics?

A symmetry in Lagrangian mechanics is a transformation that leaves the equations of motion invariant. In other words, the system behaves the same way before and after the transformation is applied.

2. How do symmetries affect the Lagrangian equations of motion?

Symmetries can simplify the Lagrangian equations of motion by reducing the number of independent variables needed to describe the system. This can make it easier to solve the equations and understand the behavior of the system.

3. What is Noether's theorem and how does it relate to symmetries in Lagrangian mechanics?

Noether's theorem states that for every continuous symmetry in a system, there exists a corresponding conserved quantity. In Lagrangian mechanics, this means that symmetries can lead to the conservation of energy, momentum, and angular momentum.

4. Can symmetries be broken in Lagrangian mechanics?

Yes, symmetries can be broken in Lagrangian mechanics if the system is subject to external forces or if there are constraints that limit the motion of the system. In these cases, the equations of motion may no longer be invariant under certain transformations.

5. How are symmetries used in Lagrangian mechanics to solve problems?

Symmetries can be used to simplify the equations of motion and reduce the number of variables needed to solve a problem. This can make it easier to find analytical solutions or to use numerical methods to solve the equations. Symmetries can also provide insights into the behavior of a system and help identify conserved quantities.

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