Poisson Brackets and Symmetries
Introduction
We look at yet another re-formulation of Physics - called Poisson Brackets. Using them, we see that a symmetry is an operation that preserves the Hamiltonian. We also derive a surprising relationship between Hamiltonians and conserved quantities.
0. Getting Started
If you are new to these series of posts, then visit this entry to find out the list of previous posts, sources, motivation etc.
If there are any errors or if you have any comments, do let me know.
OK, let us begin!
1. Origins
The physics of a system can be described by the positions (qs) and the momenta(ps). It follows that the time derivative of a function, A, which represents the physics of a system is given by:
Let us call this as equation (1)
We know from Hamilton’s equations that:
Call this as equation (2)
Call this as equation (3)
Substituting (2) and (3) in (1), we get:
Call this as equation (4)
The above equation is denoted as:
This is called as a Poisson Bracket. It is a mathematical operation that we have invented for our convenience. It applies not just to the Hamiltonian, but to any two arbitrary A and B that are functions of ps and qs.
Equation (4) is for the case of a single p and q. For the multi-dimensional case we can define the poisson bracket as:
Call this as equation (5)
2. General Properties
The following properties of poisson brackets can be trivially proved using the definition.
The following properties could be proved by using the definition, but we will prove them using other (simple) ways, just for fun.
2.1.1 Product Rule
Using the definition, it is easy to prove that:
Call this as equation (6)
But what about the case of {A,BC}? This is easy to do, if we use the anti-symmetry property.
2.1.2 Rule for taking derivatives
We have that:
Call this as equation (10)
and similarly,
Call this as equation (11)
Remember that A is any function of ps and qs. Let us now prove equation (10).
We need prove this holds only for monomials because:
Our method of proof will be induction.
Case: A = 1
We get (13) from (12) by using equation (9). Thus, this result holds good for the case of A = 1.
Case: A = p
Again, the result holds true, because it trivially follows from property 5.
Case: A = any function of ps and qs
Assume that result is true for pn, i.e. that {q,pn} = npn-1 is true. Then we have:
This is what we were looking for. Thus, the result holds good for any function of ps and qs.
Equation (11) can be proved in a similar manner - you have to simply use (6) instead of (9) to do it.
3. Canonical Transformations
Canonical transformations are changes in the co-ordinates of the system which do not screw up the Physics of the system.
All symmetries are canonical transformations, but the converse need not be true.
Our earlier trysts with changing the co-ordinates involved only changing the qs. We saw how to do this in the post on Noether’s Theorem. Now, we will look at changing even the ps.
3.1 What is happening here?
The motivation behind this section is that we want a mathematical condition for canonical transformations, as not all changes in co-ordinates, will preserve the physics of the system being studied.
How do we check that the physics is not altered? We will change the co-ordinates and if it still preserves the Poisson bracket structure, then we have a canonical transformation.
We also need a general way to perform co-ordinate transformations. To do this we will turn to our old friend, the infinitesimal transformation. (which we encountered in the post on Noether’s Theorem.) The advantage of this method is that when we are mucking around with the maths, we can drop higher powers. (we shall see this trick shortly.) This helps create a simple, elegant expression. It goes without saying that this stuff is not applicable to discrete transformations.
3.2 Condition for a Canonical Transformation
Let us define an infinitesimal transformation as follows:
Call this as equation (21)
and
Call this as equation (22)
Now, we have:
Equation (25) results from (24), because of we can drop higher powers i.e. we can drop products (recall the definition of poisson brackets given in (5)) of very small quantities.
In order to preserve the physics, the poisson bracket relations must hold, i.e., {Q,P} = {q,p} must be true. In order for this to happen, from (25), we get the following condition:
Call this as equation (26)
This is the condition for a transformation to be canonical. The question before us now is, how do we prove that an infinitesimal transformation is a canonical transformation?
3.2.1 Generators
Let us play with the above equation further. Now, any infinitesimal transformation can be represented in terms of some function called a Generator.
There exists some function G(∀q,∀p), called the generator, such that:
and
Some things to note here:
OK. Substituting (28) into (26), we get the LHS as:
Similarly, by substituting (31) into (26), we get the RHS as:
Thus, from the above we see that an infinitesimal transformation is also a canonical transformation.
4. Hamiltonians and Symmetries
Now, we come to the first surprising idea of this post - the Hamiltonian is a type of generator. To see this, we substitute into equation (1), the values of δq and δp, that we get from (28) and (31) respectively.
Upon comparing the equations we see that G does the same job as the Hamiltonian H. Thus, the Hamiltonian is a type of generator.
4.1 Symmetries
Recall in the post for Noether’s Theorem, we worked with the idea that a symmetry was any change in co-ordinates that did not change the Lagrangian. Now, we work with the idea that a symmetry is a change in co-ordinates that does not change the Hamiltonian.
From equation (38), we see that the time derivative of any function, A, is given by its poisson bracket with a generator. A symmetry is one where the time derivative of the Hamiltonian is zero.
But wait a minute! Equation (40) can be re-written as:
In other words, G is a conserved quantity.
If you thought that Noether’s theorem was weird, think again! There we saw that a conserved quantity was that which resulted from a symmetry. Here we see that a conserved quantity is a generator - for all we know, it could be the Hamiltonian of another system!
We look at yet another re-formulation of Physics - called Poisson Brackets. Using them, we see that a symmetry is an operation that preserves the Hamiltonian. We also derive a surprising relationship between Hamiltonians and conserved quantities.
0. Getting Started
If you are new to these series of posts, then visit this entry to find out the list of previous posts, sources, motivation etc.
If there are any errors or if you have any comments, do let me know.
OK, let us begin!
1. Origins
The physics of a system can be described by the positions (qs) and the momenta(ps). It follows that the time derivative of a function, A, which represents the physics of a system is given by:
Let us call this as equation (1)
We know from Hamilton’s equations that:
Call this as equation (2)
Call this as equation (3)
Substituting (2) and (3) in (1), we get:
Call this as equation (4)
The above equation is denoted as:
This is called as a Poisson Bracket. It is a mathematical operation that we have invented for our convenience. It applies not just to the Hamiltonian, but to any two arbitrary A and B that are functions of ps and qs.
Equation (4) is for the case of a single p and q. For the multi-dimensional case we can define the poisson bracket as:
Call this as equation (5)
2. General Properties
The following properties of poisson brackets can be trivially proved using the definition.
- Anti-symmetry
- Linearity
-
We also have:
Where F and G are only functions of qs.
-
We also have:
Where F and G are only functions of ps.
-
Here, δij is the Kronecker Delta Function.
The following properties could be proved by using the definition, but we will prove them using other (simple) ways, just for fun.
2.1.1 Product Rule
Using the definition, it is easy to prove that:
Call this as equation (6)
But what about the case of {A,BC}? This is easy to do, if we use the anti-symmetry property.
2.1.2 Rule for taking derivatives
We have that:
Call this as equation (10)
and similarly,
Call this as equation (11)
Remember that A is any function of ps and qs. Let us now prove equation (10).
We need prove this holds only for monomials because:
- this result holds true for all polynomials, on account of Linearity; and so
- this result will also hold true for all differentiable functions on account of Taylor’s Theorem.
Our method of proof will be induction.
Case: A = 1
We get (13) from (12) by using equation (9). Thus, this result holds good for the case of A = 1.
Case: A = p
Again, the result holds true, because it trivially follows from property 5.
Case: A = any function of ps and qs
Assume that result is true for pn, i.e. that {q,pn} = npn-1 is true. Then we have:
This is what we were looking for. Thus, the result holds good for any function of ps and qs.
Equation (11) can be proved in a similar manner - you have to simply use (6) instead of (9) to do it.
3. Canonical Transformations
Canonical transformations are changes in the co-ordinates of the system which do not screw up the Physics of the system.
All symmetries are canonical transformations, but the converse need not be true.
Our earlier trysts with changing the co-ordinates involved only changing the qs. We saw how to do this in the post on Noether’s Theorem. Now, we will look at changing even the ps.
3.1 What is happening here?
The motivation behind this section is that we want a mathematical condition for canonical transformations, as not all changes in co-ordinates, will preserve the physics of the system being studied.
How do we check that the physics is not altered? We will change the co-ordinates and if it still preserves the Poisson bracket structure, then we have a canonical transformation.
We also need a general way to perform co-ordinate transformations. To do this we will turn to our old friend, the infinitesimal transformation. (which we encountered in the post on Noether’s Theorem.) The advantage of this method is that when we are mucking around with the maths, we can drop higher powers. (we shall see this trick shortly.) This helps create a simple, elegant expression. It goes without saying that this stuff is not applicable to discrete transformations.
3.2 Condition for a Canonical Transformation
Let us define an infinitesimal transformation as follows:
Call this as equation (21)
and
Call this as equation (22)
Now, we have:
Equation (25) results from (24), because of we can drop higher powers i.e. we can drop products (recall the definition of poisson brackets given in (5)) of very small quantities.
In order to preserve the physics, the poisson bracket relations must hold, i.e., {Q,P} = {q,p} must be true. In order for this to happen, from (25), we get the following condition:
Call this as equation (26)
This is the condition for a transformation to be canonical. The question before us now is, how do we prove that an infinitesimal transformation is a canonical transformation?
3.2.1 Generators
Let us play with the above equation further. Now, any infinitesimal transformation can be represented in terms of some function called a Generator.
There exists some function G(∀q,∀p), called the generator, such that:
and
Some things to note here:
- ϵ is a tiny constant - it needs to be present to guarantee that the function always has a tiny value for all inputs.
- Do not get confused by the order of equation (31): we are used to seeing p as the second term in the poisson bracket, right? As we shall see shortly, this little wrinkle becomes crucial to proving that infinitesimal transformations are canonical. The point here is that we are always able to find some G that will work in the above fashion.
OK. Substituting (28) into (26), we get the LHS as:
Similarly, by substituting (31) into (26), we get the RHS as:
Thus, from the above we see that an infinitesimal transformation is also a canonical transformation.
4. Hamiltonians and Symmetries
Now, we come to the first surprising idea of this post - the Hamiltonian is a type of generator. To see this, we substitute into equation (1), the values of δq and δp, that we get from (28) and (31) respectively.
Upon comparing the equations we see that G does the same job as the Hamiltonian H. Thus, the Hamiltonian is a type of generator.
4.1 Symmetries
Recall in the post for Noether’s Theorem, we worked with the idea that a symmetry was any change in co-ordinates that did not change the Lagrangian. Now, we work with the idea that a symmetry is a change in co-ordinates that does not change the Hamiltonian.
From equation (38), we see that the time derivative of any function, A, is given by its poisson bracket with a generator. A symmetry is one where the time derivative of the Hamiltonian is zero.
But wait a minute! Equation (40) can be re-written as:
In other words, G is a conserved quantity.
If you thought that Noether’s theorem was weird, think again! There we saw that a conserved quantity was that which resulted from a symmetry. Here we see that a conserved quantity is a generator - for all we know, it could be the Hamiltonian of another system!