# Symmetries in Lagrangian Mechanics

• sophiatev
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

#### sophiatev

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?

Delta2
Why do you think the authors imply the last equation?

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.

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.

Last edited:
Delta2 and vanhees71
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.

sophiatev said:
Was that last equation meant to be ##G = x p_y - y p_x##?

Yes, you're right.

## 1. What are symmetries in Lagrangian mechanics?

Symmetries in Lagrangian mechanics refer to the invariance of the equations of motion under certain transformations. These transformations can be spatial translations, rotations, or time translations, among others. In other words, if a system's Lagrangian remains unchanged under a particular transformation, the system is said to possess a symmetry.

## 2. Why are symmetries important in Lagrangian mechanics?

Symmetries play a crucial role in Lagrangian mechanics because they lead to conserved quantities, known as Noether's Theorem. This means that if a system has a symmetry, there exists a corresponding conserved quantity, such as energy or momentum, which can simplify the equations of motion and provide insights into the system's behavior.

## 3. How do symmetries affect the Lagrangian and equations of motion?

If a system has a symmetry, the Lagrangian will remain unchanged under the corresponding transformation. This leads to a simplification of the equations of motion, as the conserved quantity can be used to eliminate some variables. In other words, symmetries can reduce the complexity of the equations of motion and make them easier to solve.

## 4. Can a system have multiple symmetries?

Yes, a system can have multiple symmetries. In fact, most physical systems possess multiple symmetries, which can lead to multiple conserved quantities. This can provide a deeper understanding of the system's dynamics and behavior.

## 5. Are symmetries unique to Lagrangian mechanics?

No, symmetries are not unique to Lagrangian mechanics. They are also present in other branches of physics, such as quantum mechanics and general relativity. However, the application of symmetries in Lagrangian mechanics is particularly useful in simplifying the equations of motion and understanding the behavior of physical systems.