Hamiltonian Mechanics
Introduction
Using elementary maths, we derive Hamilton’s equations and show, via Liouville's Theorem, how all physical processes in nature are of the type that preserve information. In more sensationalist prose, this means that you can run the Universe forwards/backwards to an arbitrary save-point, because all processes in nature are reversible.
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. Deriving Hamilton’s equations
Hamilton’s equations are derived using a standard mathematical operation called as the Legendre transform. The Legendre transform finds heavy use in Thermodynamics.
In this section, we will derive Hamilton’s equations. For now, take the occurrence of the Legendre transform as it is. Do not worry; we will explain the conceptual reasoning behind it in the next section.
1.1 Using the Legendre transform
We know from previous posts, that the Lagrangian is a function of all the positions and velocities, i.e.:
Let us call this as equation (1)
Let us now define the following operation:
Let us call this as equation (2)
Now, let us define the Hamiltonian, H using the Legendre transform as follows:
Let us call this as equation (3)
Let us now consider a small change in the Hamiltonian. This small change can be spread across all of its components in the following manner (Explanation follows after this):
1.1.1 Explanation
We are now able to get the following result from the final equation (8):
This is true because of the following ideas:
Similarly, using the same ideas on equation (8) we get this result:
Equations (10) and (12) are the Hamilton’s equations that we have been searching for.
In the above derivation, we simply used the Legendre transform as a black-box. In the next section we will examine the intuition behind it.
2. Legendre transform
How were we able to come up with equation (2)?
The answer is apparent when we look at the single-variable case.
2.1 Single-variable case
Consider two variables p and v which are single-valued functions of each other.
Now, because of their property of being single-valued, we can define the following functions L(v) and H(p), such that the following two equations hold:
Now, we integrate the above equations on both sides, to get:
Here, we have assumed that the graph of the function between p and v passes through the origin. Even if it does not, it can be moved there, without any loss in generality.
Adding equations (15) and (16), we get:
The above result follows from the definition of a definite integral as the limit of Riemann sums; something that you learn in high-school calculus.
From equation (18), we get:
Let us call this as equation (19)
Does the above equation seem familiar to you? Resembles the definition of the Hamiltonian given in equation (3) an awful lot, does it not?
This is because it is the Legendre transform extended to the multi-variable case.
2.2 Multi-variable case
In the case of multi-variable functions, the following changes need to be made:
All of the above was general mathematics - the Legendre transform can be applied to any pair of variables that are single valued functions of each other.
In the case of physics, velocity (
) and momentum (p) are single-valued functions of each other. Thus, we can apply the Legendre transform to define the Hamiltonian.
There is one little fine point that we have to consider.
The declaration of the Lagrangian, given in equation (1), tells us that it is a function of both the positions (qs) as well as the velocities (
s).
Until now, we have not talked about the positions much. But this does not matter. That is because the position is independent from the velocity and momenta. Thus, the equations still hold - the qs simply go along for the ride.
3. Information Preservation
Now that we have Hamilton’s Equations, we will now prove that all processes in nature are of the kind that preserve information. Though the mathematics is simple, the concepts behind the proof require an explanation. This is what we shall do below.
3.1 High-Level View
The central idea behind the proof is to show that the flow constituted by the Hamiltonian is an incompressible flow. This proves that information is preserved. Now, since the Hamiltonian is derived from the Lagrangian, which in turn is constructed using the Principle of Least Action, this result is true for all physical processes that occur in nature. (every process in nature has a least action formulation.)
3.2 Why Flows?
In the above section, where did the idea of “flows” come? The reasoning is as follows:
OK, now that we have understood why the Hamiltonian of a system, forms a flow in phase space, we need to understand why we want to prove that it is an incompressible flow. The answer is given below.
3.3 Why Incompressibility?
An incompressible flow is one which does not change the density of the fluid, at any point in time.
Why does an incompressible flow in phase space mean that information is preserved? The reasoning is as follows:
With all the conceptual baggage out of the way, we are now ready to do some maths. The theorem that we shall be proving below is called Liouville's Theorem.
4. Condition for Incompressibility
We want to show that Hamiltonian flow constitutes an incompressible flow. In order to do that, we need to arrive at a condition for incompressibility, that we can use to check the Hamiltonian. This is what we are going to do here. Let us start off.
4.1 Preliminaries
Recall that an incompressible flow is one which does not change the density of the fluid. Now, we can check the density of the fluid in 2 equivalent ways:
We will be using the second method for our calculations.
Now, the density is given by the formula:
Here, V is the volume of the region being tested. Because of our method of testing, this is a constant.
Now M is the total mass of all the fluid particles in the region. It is given by:
Here, C are the total number of fluid particles, and m is the mass of each individual fluid particle. Now, m is a constant because of our assumption that all fluid particles are infinitesimal. (and hence identical.)
Thus, we get the condition that
Let us call this as equation (20)
Now, we already have the condition for incompressibility as ΔD = 0. Using equation (20), the condition now becomes:
Let us call this as equation (21)
This is true because any constant gets eaten up by the zero.
We will now use this equation in order to arrive at a general expression for incompressibility. Our method of choice - mathematical induction over the dimensions of the physical system.
4.2 One Dimensional Case
Here we look at fluid flow only along a single direction (say the x direction,) in a single dimension.
We have that:
Here, Cx(i) is the number of particles passing through an arbitrary point i, in the x direction. And Vx(i) is the x component of the velocity of the fluid at point i.
Equation (23) follows from equation (22), when we apply the general condition for incompressibility (21); any constants get swallowed up by the zero.
Assuming an infinitesimal interval Δx, we have:
Let us call this as equation (24)
Equation (24) follows from the definition of the derivative:
If we assume that Δf = {f(x + Δx) - f(x)}, then we have:
OK?
Finally, by applying (23) to (24), we get the condition for incompressibility in the one dimensional case as:
Let us call this as equation (25)
4.3 Two Dimensional Case
Now let us look at the case of two dimensions. The results that we derive here, we shall be extending to the general case using mathematical induction.
Let us consider the flow of particles in the x direction. The two dimensional case is similar to the one dimensional case, except that we now have to also consider the flow of particles in the y dimension as well.
In other words, not only do we have (from before):
But also:
Thus, we get:
Let us call this as equation (26)
Similarly, we also have for flows in the y direction:
Let us call this as equation (27)
The total change in the number of points is given by:
Given that we are looking at an instantaneous snapshot in time, ΔCx is independent of the flow in the y direction. Similarly, ΔCy is independent of the flow in the x direction.
Thus, if ΔCx and ΔCy are independent of each other, then the only way that equation (28) can be satisfied is if both the terms are equal to zero.
So we have:
And similarly:
Thus, equation (28) can be re-written as:
This is the condition for an incompressible flow. A couple of minor points to note:
The above result can be extended by induction to any number of dimensions, via induction; The only change is that an extra term gets incorporated for each new dimension.
For example in the case of three dimensions, equation (31) becomes:
The rest of the derivation remains the same.
5. Hamiltonian Flow is Incompressible
For each dimension i, the component of velocity in the qth direction is given by:
Similarly, the component of the velocity in the pth direction is given by:
Thus, the divergence of the velocity in the qth direction in all the dimensions is given by:
Similarly, the divergence of the velocity in the pth direction in all the dimensions is given by:
Note, that equation (47) results from (46), because it does not matter what order you take the partial derivatives. (this is an old “trick” that we have seen many times before in previous posts.)
The total divergence is given by the sum of the terms in (44) and (47).
This is the result we were looking for. Thus, any Hamiltonian flow is incompressible.
Using elementary maths, we derive Hamilton’s equations and show, via Liouville's Theorem, how all physical processes in nature are of the type that preserve information. In more sensationalist prose, this means that you can run the Universe forwards/backwards to an arbitrary save-point, because all processes in nature are reversible.
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. Deriving Hamilton’s equations
Hamilton’s equations are derived using a standard mathematical operation called as the Legendre transform. The Legendre transform finds heavy use in Thermodynamics.
In this section, we will derive Hamilton’s equations. For now, take the occurrence of the Legendre transform as it is. Do not worry; we will explain the conceptual reasoning behind it in the next section.
1.1 Using the Legendre transform
We know from previous posts, that the Lagrangian is a function of all the positions and velocities, i.e.:
Let us call this as equation (1)
Let us now define the following operation:
Let us call this as equation (2)
Now, let us define the Hamiltonian, H using the Legendre transform as follows:
Let us call this as equation (3)
Let us now consider a small change in the Hamiltonian. This small change can be spread across all of its components in the following manner (Explanation follows after this):
1.1.1 Explanation
- The first two terms in equation (4) are the change in the product term,
ipi. The answer is given by the sum of two terms. (just like in the case of the product rule for derivatives.)
The two terms in the Lagrangian are because of equation (1); we need to record the effect of all the variables that make up the Lagrangian.
- We get equation (8) from equation (7), by substitution from the Euler-Lagrange equation:
We are now able to get the following result from the final equation (8):
This is true because of the following ideas:
- δx is analogous to ∂x.
- q and p are independent variables.
- pi is independent from pj, when i ⁄= j. Similarly in the case of q.
Similarly, using the same ideas on equation (8) we get this result:
Equations (10) and (12) are the Hamilton’s equations that we have been searching for.
In the above derivation, we simply used the Legendre transform as a black-box. In the next section we will examine the intuition behind it.
2. Legendre transform
How were we able to come up with equation (2)?
The answer is apparent when we look at the single-variable case.
2.1 Single-variable case
Consider two variables p and v which are single-valued functions of each other.
Now, because of their property of being single-valued, we can define the following functions L(v) and H(p), such that the following two equations hold:
Now, we integrate the above equations on both sides, to get:
Here, we have assumed that the graph of the function between p and v passes through the origin. Even if it does not, it can be moved there, without any loss in generality.
Adding equations (15) and (16), we get:
The above result follows from the definition of a definite integral as the limit of Riemann sums; something that you learn in high-school calculus.
From equation (18), we get:
Let us call this as equation (19)
Does the above equation seem familiar to you? Resembles the definition of the Hamiltonian given in equation (3) an awful lot, does it not?
This is because it is the Legendre transform extended to the multi-variable case.
2.2 Multi-variable case
In the case of multi-variable functions, the following changes need to be made:
- Equation (19) has to include a sum term (∑ i) for all the variables.
- Equations (13) and (14) now use partial derivatives, instead of ordinary derivatives.
All of the above was general mathematics - the Legendre transform can be applied to any pair of variables that are single valued functions of each other.
In the case of physics, velocity (
) and momentum (p) are single-valued functions of each other. Thus, we can apply the Legendre transform to define the Hamiltonian.
There is one little fine point that we have to consider.
The declaration of the Lagrangian, given in equation (1), tells us that it is a function of both the positions (qs) as well as the velocities (
s).
Until now, we have not talked about the positions much. But this does not matter. That is because the position is independent from the velocity and momenta. Thus, the equations still hold - the qs simply go along for the ride.
3. Information Preservation
Now that we have Hamilton’s Equations, we will now prove that all processes in nature are of the kind that preserve information. Though the mathematics is simple, the concepts behind the proof require an explanation. This is what we shall do below.
3.1 High-Level View
The central idea behind the proof is to show that the flow constituted by the Hamiltonian is an incompressible flow. This proves that information is preserved. Now, since the Hamiltonian is derived from the Lagrangian, which in turn is constructed using the Principle of Least Action, this result is true for all physical processes that occur in nature. (every process in nature has a least action formulation.)
3.2 Why Flows?
In the above section, where did the idea of “flows” come? The reasoning is as follows:
- As mentioned in a previous post, the “phase space” of all systems in nature, consists of positions and velocities.
- By constructing Hamilton’s equations, we have changed the phase space to positions and momenta.
- A point in phase space (q1,p1), is represented by a physical system at position q1 and momentum p1.
- When you apply a force on the system, its position and momentum may change to a different point, say (q2, p2).
- Thus, this force (and consequently all physical process) may be thought of as a flow from point (q1,p1) to (q2,p2).
- In this way, the Hamiltonian, which represents the characteristics of a physical system, also represents a flow in phase space.
OK, now that we have understood why the Hamiltonian of a system, forms a flow in phase space, we need to understand why we want to prove that it is an incompressible flow. The answer is given below.
3.3 Why Incompressibility?
An incompressible flow is one which does not change the density of the fluid, at any point in time.
Why does an incompressible flow in phase space mean that information is preserved? The reasoning is as follows:
- We know that phase space can be thought of as a fluid, and the Hamiltonian can be thought of as a flow in it.
- If the density of a fluid remains the same, then it means no compression or decompression occurs.
- For a given volume of fluid, if more number of particles (or in this case, phase space points) are added to it, then the fluid becomes compressed.
Similarly, if the number of particles are reduced, then the fluid undergoes decompression.
- On the other hand, if the number of points in the phase space fluid remain constant, (as it is the case with an incompressible flow) then you do not lose information; all the states of the physical system (i.e. the phase space points) remain preserved for eternity.
- This means that information is preserved and theoretically, you can revert the physical system, back to any point in time i.e., you have a reversible process.
With all the conceptual baggage out of the way, we are now ready to do some maths. The theorem that we shall be proving below is called Liouville's Theorem.
4. Condition for Incompressibility
We want to show that Hamiltonian flow constitutes an incompressible flow. In order to do that, we need to arrive at a condition for incompressibility, that we can use to check the Hamiltonian. This is what we are going to do here. Let us start off.
4.1 Preliminaries
Recall that an incompressible flow is one which does not change the density of the fluid. Now, we can check the density of the fluid in 2 equivalent ways:
- By going throughout all regions of the fluid, testing a fixed volume of it and seeing if its density has changed, since the flow started;
- By parking our asses at some arbitrary point and periodically testing a fixed volume of the fluid. The idea here is that because there is a fluid flow, new fluid particles keep flowing into our testing region. We keep testing the density of this region repeatedly, if there is no change in the value, then the fluid flow is incompressible.
We will be using the second method for our calculations.
Now, the density is given by the formula:
Here, V is the volume of the region being tested. Because of our method of testing, this is a constant.
Now M is the total mass of all the fluid particles in the region. It is given by:
Here, C are the total number of fluid particles, and m is the mass of each individual fluid particle. Now, m is a constant because of our assumption that all fluid particles are infinitesimal. (and hence identical.)
Thus, we get the condition that
Let us call this as equation (20)
Now, we already have the condition for incompressibility as ΔD = 0. Using equation (20), the condition now becomes:
Let us call this as equation (21)
This is true because any constant gets eaten up by the zero.
We will now use this equation in order to arrive at a general expression for incompressibility. Our method of choice - mathematical induction over the dimensions of the physical system.
4.2 One Dimensional Case
Here we look at fluid flow only along a single direction (say the x direction,) in a single dimension.
We have that:
Here, Cx(i) is the number of particles passing through an arbitrary point i, in the x direction. And Vx(i) is the x component of the velocity of the fluid at point i.
Equation (23) follows from equation (22), when we apply the general condition for incompressibility (21); any constants get swallowed up by the zero.
Assuming an infinitesimal interval Δx, we have:
Let us call this as equation (24)
Equation (24) follows from the definition of the derivative:
If we assume that Δf = {f(x + Δx) - f(x)}, then we have:
OK?
Finally, by applying (23) to (24), we get the condition for incompressibility in the one dimensional case as:
Let us call this as equation (25)
4.3 Two Dimensional Case
Now let us look at the case of two dimensions. The results that we derive here, we shall be extending to the general case using mathematical induction.
Let us consider the flow of particles in the x direction. The two dimensional case is similar to the one dimensional case, except that we now have to also consider the flow of particles in the y dimension as well.
In other words, not only do we have (from before):
But also:
Thus, we get:
Let us call this as equation (26)
Similarly, we also have for flows in the y direction:
Let us call this as equation (27)
The total change in the number of points is given by:
Given that we are looking at an instantaneous snapshot in time, ΔCx is independent of the flow in the y direction. Similarly, ΔCy is independent of the flow in the x direction.
Thus, if ΔCx and ΔCy are independent of each other, then the only way that equation (28) can be satisfied is if both the terms are equal to zero.
So we have:
And similarly:
Thus, equation (28) can be re-written as:
This is the condition for an incompressible flow. A couple of minor points to note:
- The ∇ in equation (38) is the divergence operator.
- The Δx and Δy can be canceled from equation (36) to give (37), as they are just numbers.
The above result can be extended by induction to any number of dimensions, via induction; The only change is that an extra term gets incorporated for each new dimension.
For example in the case of three dimensions, equation (31) becomes:
The rest of the derivation remains the same.
5. Hamiltonian Flow is Incompressible
For each dimension i, the component of velocity in the qth direction is given by:
Similarly, the component of the velocity in the pth direction is given by:
Thus, the divergence of the velocity in the qth direction in all the dimensions is given by:
Similarly, the divergence of the velocity in the pth direction in all the dimensions is given by:
Note, that equation (47) results from (46), because it does not matter what order you take the partial derivatives. (this is an old “trick” that we have seen many times before in previous posts.)
The total divergence is given by the sum of the terms in (44) and (47).
This is the result we were looking for. Thus, any Hamiltonian flow is incompressible.