As we saw in the previous post, the equations of motion for a mechanical system can be cast into a 1st order form called Hamilton's equations, which are naturally interpreted as describing a path in the phase space \(T^\ast M\) associated to the configuration space \(M\). Let us investigate the geometry of \(T^\ast M\) see why Hamilton's equations are so nice.
Definition The canonical (or sometimes tautological) 1-form on the cotangent bundle \(T^\ast M\) is the 1-form \(\theta\) defined by
\[ \theta_{(q,p)}(X) = p(\pi_\ast X), \]
where \(\pi_\ast\) is the pushforward induced by the natural projection \(\pi: T^\ast M \to TM\). In other words, the form is defined by
\[ \theta_{(q,p)} = \pi^\ast p. \]
Definition The canonical symplectic form on the cotangent bundle \(T^\ast M\) is the 2-form \(\omega\) defined by
\[ \omega = -d\theta. \]
Let \(\omega_\flat: T M \to T^\ast M\) be the map given by \(X \mapsto \iota(X)\omega\).
Proposition The canonical symplectic form satisfies the following two conditions:
1. It is closed, i.e. \(d\omega = 0\).
2. It is nondegenerate, i.e. the map \(\omega_\flat\) is invertible with inverse \(\omega^\sharp: T^\ast M \to TM\).
Proof The first property follows from \(d^2 = 0\). To prove the second, suppose we have local coordinates \(q^i\) on \(M\) with cotangent coordinates \(p^i\). Then it is easily seen that
\[ \theta = p^i dq^i, \]
so that
\[ \omega = dq^i \wedge dp^i, \]
from which nondegeneracy is obvious.
Definition Any 2-form on a manifold \(N\) (not necessarily a cotangent bundle) which satisfies the above two properties will be called symplectic. A pair \((N, \omega)\) will be called symplectic if \(\omega\) is a symplectic 2-form on \(N\).
Definition Given a function \(H\) on a symplectic manifold \((N, \omega)\), the Hamiltonian vector field associated to \(H\) is the vector field \(X_H\) uniquely defined by
\[ dH = \omega_\flat X_H. \]
Proposition For \(N = T^\ast M\) a cotangent bundle with the canonical symplectic form, Hamilton's equations with respect to a Hamiltonian function \(H\) describe the flow of the vector field \(X_H\).
Proof Again pick local coordinates \(q\) and \(p\). Then the inverse map \(\omega^\sharp\) is given by
\[ dq \mapsto -\frac{\partial}{\partial p} \]
\[ dp \mapsto \frac{\partial}{\partial q} \]
Since
\[ dH = \frac{\partial H}{\partial q} dq + \frac{\partial H}{\partial p} dp, \]
we see that
\[ X_H = \frac{\partial H}{\partial p} \frac{\partial}{\partial q} - \frac{\partial H}{\partial q} \frac{\partial}{\partial p} \]
But then the equation describing the flow of \(X_H\) is (in components)
\[ \dot{q} = \frac{\partial H}{\partial p} \]
\[ \dot{p} = -\frac{\partial H}{\partial q} \]
which are exactly Hamilton's equations.
Thursday, August 25, 2011
Tuesday, August 23, 2011
Classical Mechanics 4: Hamilton's Equations
Recall from last time that a classical mechanical system consists of a manifold \(M\) (the configuration space) and a function \(L\) on the tangent bundle \(TM\). The equations of motion for a path \(x(t)\) in \(M\) are the 2nd order Euler-Lagrange equations:
\[ \frac{d}{dt}\left( \frac{\partial L}{\partial v}(x, \dot{x}) \right) = \frac{\partial L}{\partial x}(x, \dot{x}) \]
Hamilton discovered a way of recasting these equations as a first order system for a path in a related manifold, the cotangent bundle \(T^\ast M\). The benefits are twofold: in addition to reducing the equations to a first order system (at the cost of introducing new variables), it turns out that this framework makes it much easier to find conserved quantities and prove theorems about mechanical systems. So let's see what he did.
Theorem Under mild assumptions on \(L\), there is a function \(H\) on \(T^\ast M\) constructed canonically out of \(L\) such that solutions of the Euler-Lagrange equations are in 1-1 correspondence with solutions \(q(t), p(t)\) on \(T^\ast M\) of Hamilton's equations
\[ \dot{q} = \frac{\partial H}{\partial p}(q, p) \]
\[ \dot{p} = -\frac{\partial H}{\partial q}(q,p) \]
Furthermore, if the original Lagrangian function \(L\) is not explicitly time-dependent, then the function \(H\) is constant for any solution of the equations of motion.
To start with, introduce coordinates \(x, v\) on \(TM\) as before. We will define new coordinates \(q,p\) as follows:
\[ q(x,v) = x \]
\[ p(x,v) = \frac{\partial L}{\partial x}(x,v) \]
Our assumption on \(L\) will be the following: the above formulas can be inverted to obtain \(x\) and \(v\) as functions of \(q\) and \(p\). It is easily seen from the definition of \(p\) that under a coordinate transformation (of \(x\)), it behaves as a 1-form, so the coordinates \(q\) and \(p\) can be interpreted as local coordinates on the cotangent bundle \(T^\ast M\). We construct the Hamiltonian as
\[ H(q,p) = p\cdot v(q,p) - L(x(q,p), v(q,p)) \]
(this is the Legendre transform--more later). Of course, the above formula is not well-defined if we cannot solve for \(x\) and \(v\) in terms of \(p\) and \(q\)--hence the assumption. Now we check:
\[ \frac{\partial H}{\partial p} = v + p \frac{\partial v}{\partial p} - \frac{\partial L}{\partial x} \frac{\partial x}{\partial p} -\frac{\partial L}{\partial v}\frac{\partial v}{\partial p} = v + p\frac{\partial v}{\partial p} - p\frac{\partial v}{\partial p} = v\]
Since \(\dot{x} = v\), this is the first of Hamilton's equations.
For the second, we perform a similar computation:
\[ \frac{\partial H}{\partial q} = -\frac{\partial L}{\partial x} \frac{\partial x}{\partial q} - \frac{\partial L}{\partial v} \frac{\partial v}{\partial q} = -\frac{\partial L}{\partial x} \]
But the Euler-Lagrange equations say that
\[ \dot{p} = \frac{d}{dt}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x} = -\frac{\partial H}{\partial x}, \]
so we've obtained the second of Hamilton's equations.
For the last part, suppose that the Lagrangian does not depend explicitly on time, i.e.
\[ \frac{\partial L}{\partial t} = 0. \]
Then we compute:
\[ \frac{d}{dt}H = \frac{\partial H}{\partial q}\dot{q} + \frac{\partial H}{\partial p}\dot{p} = \frac{\partial H}{\partial q} \frac{\partial H}{\partial p} - \frac{\partial H}{\partial p} \frac{\partial H}{\partial q} = 0. \]
Hence \(H\) is automatically conserved. For this reason, \(H\) is often called the energy of the system.
Later, we will see that conserved quantities correspond to symmetries, and conservation of energy is a statement about the symmetry corresponding to time translation.
\[ \frac{d}{dt}\left( \frac{\partial L}{\partial v}(x, \dot{x}) \right) = \frac{\partial L}{\partial x}(x, \dot{x}) \]
Hamilton discovered a way of recasting these equations as a first order system for a path in a related manifold, the cotangent bundle \(T^\ast M\). The benefits are twofold: in addition to reducing the equations to a first order system (at the cost of introducing new variables), it turns out that this framework makes it much easier to find conserved quantities and prove theorems about mechanical systems. So let's see what he did.
Theorem Under mild assumptions on \(L\), there is a function \(H\) on \(T^\ast M\) constructed canonically out of \(L\) such that solutions of the Euler-Lagrange equations are in 1-1 correspondence with solutions \(q(t), p(t)\) on \(T^\ast M\) of Hamilton's equations
\[ \dot{q} = \frac{\partial H}{\partial p}(q, p) \]
\[ \dot{p} = -\frac{\partial H}{\partial q}(q,p) \]
Furthermore, if the original Lagrangian function \(L\) is not explicitly time-dependent, then the function \(H\) is constant for any solution of the equations of motion.
To start with, introduce coordinates \(x, v\) on \(TM\) as before. We will define new coordinates \(q,p\) as follows:
\[ q(x,v) = x \]
\[ p(x,v) = \frac{\partial L}{\partial x}(x,v) \]
Our assumption on \(L\) will be the following: the above formulas can be inverted to obtain \(x\) and \(v\) as functions of \(q\) and \(p\). It is easily seen from the definition of \(p\) that under a coordinate transformation (of \(x\)), it behaves as a 1-form, so the coordinates \(q\) and \(p\) can be interpreted as local coordinates on the cotangent bundle \(T^\ast M\). We construct the Hamiltonian as
\[ H(q,p) = p\cdot v(q,p) - L(x(q,p), v(q,p)) \]
(this is the Legendre transform--more later). Of course, the above formula is not well-defined if we cannot solve for \(x\) and \(v\) in terms of \(p\) and \(q\)--hence the assumption. Now we check:
\[ \frac{\partial H}{\partial p} = v + p \frac{\partial v}{\partial p} - \frac{\partial L}{\partial x} \frac{\partial x}{\partial p} -\frac{\partial L}{\partial v}\frac{\partial v}{\partial p} = v + p\frac{\partial v}{\partial p} - p\frac{\partial v}{\partial p} = v\]
Since \(\dot{x} = v\), this is the first of Hamilton's equations.
For the second, we perform a similar computation:
\[ \frac{\partial H}{\partial q} = -\frac{\partial L}{\partial x} \frac{\partial x}{\partial q} - \frac{\partial L}{\partial v} \frac{\partial v}{\partial q} = -\frac{\partial L}{\partial x} \]
But the Euler-Lagrange equations say that
\[ \dot{p} = \frac{d}{dt}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x} = -\frac{\partial H}{\partial x}, \]
so we've obtained the second of Hamilton's equations.
For the last part, suppose that the Lagrangian does not depend explicitly on time, i.e.
\[ \frac{\partial L}{\partial t} = 0. \]
Then we compute:
\[ \frac{d}{dt}H = \frac{\partial H}{\partial q}\dot{q} + \frac{\partial H}{\partial p}\dot{p} = \frac{\partial H}{\partial q} \frac{\partial H}{\partial p} - \frac{\partial H}{\partial p} \frac{\partial H}{\partial q} = 0. \]
Hence \(H\) is automatically conserved. For this reason, \(H\) is often called the energy of the system.
Later, we will see that conserved quantities correspond to symmetries, and conservation of energy is a statement about the symmetry corresponding to time translation.
Friday, August 19, 2011
Classical Mechanics 3: Hamilton's action principle.
We saw before that Newton's 2nd law can be written in a more general form as
\[ \frac{d}{dt} \frac{\partial L}{\partial v}(x, \dot{x}) = \frac{\partial L}{\partial x}(x, \dot{x}), \]
known as the Euler-Lagrange equations. Hamilton discovered a principle that explains the origin of these equations. Consider some path of the system given by a curve \(\gamma\), i.e.
\[ x(t) = \gamma(t) \]
\[ \dot{x}(t) = \frac{d}{dt}\gamma(t) \]
Then we may define a quantity associated with the path \(\gamma\):
\[ S = \int L(\gamma, \dot{\gamma})dt \]
called the action. Hamilton discovered the following.
Theorem The path taken by a mechanical system is one which extremizes the action.
To prove this, suppose we perturb the path a small amount, while leaving the endpoints fixed, i.e. \(\gamma \mapsto \gamma + \epsilon (\delta\gamma)\) with \(\epsilon > 0\) small and \(\delta\gamma\) a path that is \(0\) at its endpoints. Then
\[ L(\gamma + \epsilon\delta\gamma, \dot\gamma + \epsilon\delta\dot{\gamma}) = L(\gamma, \dot{\gamma}) + \epsilon \frac{\partial L}{\partial x}\delta\gamma + \epsilon \frac{\partial L}{\partial v} \delta\dot\gamma + o(\epsilon^2) \]
Thus
\[ S[\gamma + \epsilon\delta\gamma] = S[\gamma] + \epsilon \int \frac{\partial L}{\partial x} \delta \gamma dt + \epsilon \int \frac{\partial L}{\partial v} \delta \dot\gamma dt + o(\epsilon^2) \]
Integrating by parts, and using the fact that \(\delta\gamma\) is \(0\) on the endpoints, we have
\[ \int \frac{\partial L}{\partial v}\delta\dot\gamma dt = -\int \frac{d}{dt} \frac{\partial L}{\partial v} \delta\gamma dt \]
Combining the above, we have
\[ \frac{\delta S}{\delta \gamma}(\delta \gamma) = \int \left(\frac{\partial L}{\partial x} - \frac{d}{dt}\frac{\partial L}{\partial v} \right) \delta\gamma dt \]
Thus the variational derivative of \(S\) is
\[ \frac{\delta S}{\delta \gamma} = \frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L}{\partial v} \]
So a path \(\gamma\) is a critical point of \(S\) (i.e. it extremizes \(S\)) if and only if the Euler-Lagrange equations are satisfied.
\[ \frac{d}{dt} \frac{\partial L}{\partial v}(x, \dot{x}) = \frac{\partial L}{\partial x}(x, \dot{x}), \]
known as the Euler-Lagrange equations. Hamilton discovered a principle that explains the origin of these equations. Consider some path of the system given by a curve \(\gamma\), i.e.
\[ x(t) = \gamma(t) \]
\[ \dot{x}(t) = \frac{d}{dt}\gamma(t) \]
Then we may define a quantity associated with the path \(\gamma\):
\[ S = \int L(\gamma, \dot{\gamma})dt \]
called the action. Hamilton discovered the following.
Theorem The path taken by a mechanical system is one which extremizes the action.
To prove this, suppose we perturb the path a small amount, while leaving the endpoints fixed, i.e. \(\gamma \mapsto \gamma + \epsilon (\delta\gamma)\) with \(\epsilon > 0\) small and \(\delta\gamma\) a path that is \(0\) at its endpoints. Then
\[ L(\gamma + \epsilon\delta\gamma, \dot\gamma + \epsilon\delta\dot{\gamma}) = L(\gamma, \dot{\gamma}) + \epsilon \frac{\partial L}{\partial x}\delta\gamma + \epsilon \frac{\partial L}{\partial v} \delta\dot\gamma + o(\epsilon^2) \]
Thus
\[ S[\gamma + \epsilon\delta\gamma] = S[\gamma] + \epsilon \int \frac{\partial L}{\partial x} \delta \gamma dt + \epsilon \int \frac{\partial L}{\partial v} \delta \dot\gamma dt + o(\epsilon^2) \]
Integrating by parts, and using the fact that \(\delta\gamma\) is \(0\) on the endpoints, we have
\[ \int \frac{\partial L}{\partial v}\delta\dot\gamma dt = -\int \frac{d}{dt} \frac{\partial L}{\partial v} \delta\gamma dt \]
Combining the above, we have
\[ \frac{\delta S}{\delta \gamma}(\delta \gamma) = \int \left(\frac{\partial L}{\partial x} - \frac{d}{dt}\frac{\partial L}{\partial v} \right) \delta\gamma dt \]
Thus the variational derivative of \(S\) is
\[ \frac{\delta S}{\delta \gamma} = \frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L}{\partial v} \]
So a path \(\gamma\) is a critical point of \(S\) (i.e. it extremizes \(S\)) if and only if the Euler-Lagrange equations are satisfied.
Classical Mechanics 2: The Euler-Lagrange equations from Newton's 2nd law.
After the previous post, we are now familiar with Newton's 2nd law
\[ \mathbf{F} = m\mathbf{a}, \]
which (suitably interpreted) holds for any system of \(N\) particles. However, this equation requires the use of cartesian coordinates, which for many systems may not be the most convenient choice. Suppose we have some other coordinates \(q^i = q^i(x^j)\). What is the correct analogue of Newton's 2nd law for the \(q\)-coordinates?
To make life easier, we will assume for now that the force \(\mathbf{F}\) is conservative; i.e.
\[ \mathbf{F} = -\nabla V(x) \]
for some potential function \(V(x)\). Under this assumption, Newton's 2nd law is
\[ m\mathbf{a} + \nabla V(x) = 0. \]
Let us define the function \(T\) by
\[ T(x,v) = \frac{1}{2}m |v|^2, \]
and define the function \(L\) as
\[ L(x,v) = T(x,v) - V(x). \]
Then we have immediately that Newton's 2nd law is equivalent to
\[ \frac{d}{dt}\frac{\partial L}{\partial v}(x, \dot{x}) - \frac{\partial L}{\partial x}(x,\dot{x}) = 0. \]
Why go through the trouble of introducing these auxiliary functions and rewriting Newton's 2nd law in this way? The answer lies in the following theorem.
Theorem For any choice of coordinates \(y = y(x)\), Newton's 2nd law is equivalent to the equations
\[ \frac{d}{dt}\frac{\partial \tilde{L}}{\partial w}(y, \dot{y}) - \frac{\partial \tilde{L}}{\partial y}(y,\dot{y}) = 0, \]
where \(w = dY_x(v)\) and \(\tilde{L}(y,w) = L(x(y,w), v(y,w)\). These equations are called the Euler-Lagrange equations.
The proof of this theorem is a straightforward calculation using the chain rule. Let \(M\) denote the manifold \(\mathbb{R}^{3N}\) (or some open subset thereof). The coordinate change \(y = y(x) \) can be thought of as a diffeomorphism \(Y: M \to M\) given by \(x \mapsto y(x) \). The differential \( dY: TM \to TM\) is also a diffeomorphism. In coordinates, we have
\[ y = y(x) \]
\[ w = dY_x(v) = Jv \]
where \(y,w\) are coordinates on the target \(TM\).
Now we need to compute the derivatives of \(\tilde{L}\).
\[ \frac{\partial L}{\partial x} = \frac{\partial \tilde{L}}{\partial y}\frac{\partial y}{\partial x} + \frac{\partial \tilde{L}}{\partial w}\frac{\partial w}{\partial x} = \tilde{L}_y J + \tilde{L}_w H v \]
where \(J\) is the matrix of mixed partials and \(H\) is the Hessian matrix.
\[ \frac{\partial L}{\partial v} = \frac{\partial \tilde{L}}{\partial y} \frac{\partial y}{\partial v} + \frac{\partial \tilde{L}}{\partial w} \frac{\partial w}{\partial v} = \tilde{L}_w J \]
Then we have
\[ \frac{d}{dt}\left( \tilde{L}_w J \right) = \frac{d}{dt}\tilde{L}_w J + \tilde{L}_w H \dot{x} \]
Subtracting \(\frac{\partial L}{\partial x}\) from this and using the calculation above, we obtain
\[ \left( \frac{d}{dt} \tilde{L}_w - \tilde{L}_y \right) J \]
and since \(J\) is invertible, this is \(0\) if and only if \(\frac{d}{dt} \tilde{L}_w - \tilde{L}_y\) is.
But this is exactly what we wanted to prove!
Classical Mechanics 1: Newton's Laws.
This is a surpisingly hard topic, and someday I would like to make an honest attempt at it. But Feynman did it better than I ever could! See "What is a Force?" from the Feynman Lectures on Physics, available here: here (pdf).
I promise to update eventually.
I promise to update eventually.
Wednesday, August 17, 2011
Updated Outline
I was thinking about how to put this blog to use, and especially how to make it worthwhile for myself. In 2010, I ran a learning seminar on math/physics which in my opinion was extremely successful. I made detailed notes for nearly all of the talks, and I had several people ask me at the end of the seminar to give them copies of the notes I had made. We managed to do something very special in the seminar, and I think it's worth writing up notes from the best talks (and reorganizing/elaborating/etc. when necessary) and putting them online for all to see. It gives me a chance to practice my math writing, and if I ever run another seminar of this flavor (or teach a course, or write a book...) then a lot of the work will already be done for me.
Here is a rough outline of the topics we covered (or that need to be covered as background):
As you can see, it was an ambitious seminar! I will update the outline as I go through my old notes and start posting them.
Here is a rough outline of the topics we covered (or that need to be covered as background):
- Newton's Laws (review of high school physics).
- The Euler-Lagrange equations derived from Newton's 2nd law.
- Hamilton's action principle.
- Legendre transform, Hamilton's equations, basic symplectic geometry.
- Symmetry, Noether's theorem, and moment maps.
- Completely integrable systems.
- Special relativity.
- Classical field theory.
- Classical electrodynamics.
- General relativity.
- Canonical quantization
- Deformation quantization
- Geometric quantization.
- Path integrals in quantum mechanics.
- Quantization of free fields.
- Perturbation theory and Feynman diagrams.
- Feynman rules for QED.
- Berezin integration and quantization of fermionic fields.
- Gauge fields.
- Faddeev-Popov method for gauge fields.
- BRST.
- Path integral proof of the index theorem.
As you can see, it was an ambitious seminar! I will update the outline as I go through my old notes and start posting them.
Monday, August 8, 2011
Is it worth the trouble?
I've realized talking to fellow students and occasionally postdocs that I frequently have to explain the same few topics over and over. Since I have my own personal notes on these things, I might as well write them up for people to read, since then I can just point people towards the notes. And if I'm going through the trouble of writing things up, maybe I should post them on an indexed, google-searchable blog? Why not? Maybe I will.
MathJax
Even though I don't touch this blog, it occasionally gets some pageviews and I realized all my \(\LaTeX\) was broken! So now I've switched to MathJax and am trying to fix it.
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