Classical Mechanics
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Think of a physical system as a map \(A \to B\), from a space \(A\) to a space \(B\).
As an example, a particle can be modelled with \(\phi : A \to B\), for \(A\) being time, and \(B\) being space, so that \(\phi(t)\) is the position of a particle at a given time.
A field theory has \(A\) be a position in space or spacetime, and \(B\) the same.
Least action¶
The fundamental idea of classical physics is that, from the space \(A \to B\) of all possible systems, and for a region \(T\) of space in \(A\), the actual system that describes nature is determined by minimizing a functional \(S : A \to (A \to B) \to \mathbb{R}\), i.e.1 \(\delta S(x) = 0\), where:
for the Lagrangian \(\mathcal{L} : (B, A \otimes B, A) \to \mathbb{R})\), which we can choose in different ways, depending on the system. One can also view \(S\) as the solution to an ODE.
Note
One can make this more precise by considering the manifold \(A \to B\) more carefully.
As a simple example with \(A = \R\) and \(B = \R^3\), i.e. \(A\) as time and \(B\) as space, consider \(\mathcal{L}(\phi,\dot \phi, t) = \frac{1}{2}m\dot \phi^2(t)\) and the time interval \(T\):
where in the last step, we integrate by parts, and crucially assume that the boundary term is \(0\). Since this holds for all \(f\), we have \(\ddot x = 0\).
The physical interpretation is that an object acted on by no forces (namely for the free Lagrangian, \(\frac{1}{2}m\dot x^2\)), an object maintains constant velocity. Or, if we think of the system as a line in spacetime, it is a straight line.
More generally, minimizing \(S\) yields the Euler-Lagrange condition that \(\forall t, (\frac{\partial L}{\partial q}(t) - \frac{d}{dt}\frac{\partial L}{\partial \dot q}(t)) = 0\). Note that in the physical setting, the first and second term have the same dimensions.
If \(S(q) = \int L(q)(t) dt\), and \(L(q)(t) = m\dot q(t)^2/2 - V(q)\) then we obtain Newton's equation of motion:
Invariants¶
Suppose that \(\phi(a : \R,x : A) : B\) is a family of configurations of a system, i.e for each choice of \(a\), \(\phi(a,x)\) describes the state of the system at every point in space and time. Additionally, suppose that \(x(a) : \R \to A\) is a family of transformations of the space \(A\).
Then we say that the system has a symmetry if \(\frac{d\mathcal{L}(\phi(a,x), \partial_\mu\phi(a,x), x(a))}{da}|_{a=0} = \partial_\mu J^\mu\), since in that case, to first order, a change in \(a\) will have no effect on the action, and therefore will map a solution of the equations of motion to a new solution.
Such symmetries imply an invariant. To see this, first note:
by virtue of the chain rule, commutation of partial derivatives and the product rule.
Suppose that the function \(x \mapsto \phi(0,x)\) is "on-shell", i.e. satisfies the equations of motion. In that case,
Then for \(j^\mu := K^\mu - J^\mu\), we evidently have \(\partial_\mu j^\mu = 0\), as long as \(x \mapsto \phi(0,x)\) follows the equations of motion. We say that \(j\) is the Noether current. Or in the language of forms, \(d\star j = 0\).
Advanced
This comment assumes familiarity with differential forms
One consequence is that \(Q(N) = \int_N \star J\), which is an integral over an \(M-1\) dimension surface (where \(M\) is the dimension of the full space, or spacetime) depends only on the homotopy class of \(N\), since for \(N\) and a deformation to \(N'\), \(Q(N)-Q(N') = \int_N \star J = \int d \star J = \int 0 = 0\).
An important special case is for constant time slices \(N\) of Minkowski space, in which case the homotopy statement amounts to conservation of \(Q\) over time.
Examples¶
For a particle theory, with \(\phi(a : \R^3, t : \R) : \R^3\), with \(\mathcal{L}(\phi, \dot \phi, t) = \frac{1}{2}m\dot \phi(a,t)^2\), let our transformation be \(\phi(a,t) = \phi(0,t)+a\). Then \(J^\mu = 0\), since \(\dot \phi(a,t) = \dot \phi(0,t)\) which does not depend on \(a\), and \(K^\mu = m\dot q\), so \(j^\mu = m\dot q\), and \(m\partial_\mu \dot q = 0 \Rightarrow \frac{d}{dt}\dot q = 0\) (noting that \(\partial_\mu = \pd{}{t}\)), so momentum is conserved.
Similarly, for the transformation \(\phi(a,t) = \phi(0,t+a)\) and \(t(a) = a + t(0)\), we find that \(\pd{\mathcal{L}(\phi(a,x), \partial_\mu\phi(a,x))}{a}|_{a=0} = \pd{\mathcal{L}}{t}\), so that \(J^\mu = \mathcal{L} = \frac{1}{2}m\dot q^2\). \(K^\mu = m\dot q^2\), so \(j^\mu = \frac{1}{2}m\dot q^2 = \mathcal{H}\), i.e. the Hamiltonian, or energy of the system.
Now consider a rotation, so \(\phi(a,t) = U(a)\phi(0,t)\), where \(U(r)\) is a rotation transform around some axis, say \(\hat n\). The Lie algebra of the 3D rotation group consists of transformations which in coordinates look like \(\phi(a,t) = \phi(0,t) + a \hat n \times \phi(0,t)\). This gives \(j^\mu = p \cdot (n \times \phi) - 0 = n \cdot (\phi \times p)\) which is the inner product of the angular momentum with the axis of rotation.
For fields, the same reasoning holds. Consider a spacetime translation \(x(a) = x + a\) and any Lagrangian \(\mathcal{L}\). Then \(j^\mu_\nu = \pd{\mathcal{L}}{(\partial_\nu \phi)}\pd{\phi}{x_\mu} - \delta^\mu_\nu\mathcal{L}\). In this context, \(T := j\) is known as the energy-momentum tensor. The corresponding conserved quantities are \(E = \int d^3 x T^0_0\) and \(P_i = \int d^3 x T^0_i\) .
Hamiltonian mechanics¶
The Lagrangian is a function of the tangent bundle. A Legendre transform gives us a function of the cotangent bundle, namely:
and
where \(\{f, g\} = \pd{f}{q}\pd{g}{p} - \pd{f}{p}\pd{g}{q}\) is the Poisson bracket.
This is the Hamiltonian formulation of classical mechanics: it describes a differential equation \(\frac{d}{dt} \begin{pmatrix} q \\ p \end{pmatrix} = \begin{pmatrix} \pd{H}{p} \\ -\pd{H}{q} \end{pmatrix}\).
Symplectic geometry¶
This is the geometrical structure associated with classical mechanics.
Geometrically, the configuration space, of which the Lagrangian is a function, is the tangent bundle, while phase space is the cotangent bundle. The evolution of a system can be described by a map from the cotangent bundle to itself.
Let \(\mathcal{M}\) be the \(n\) dimensional manifold on which states of your physical system live. Then \(TM\), the tangent bundle, is the configuration space, and \(T^*M\), the cotangent bundle, is the phase space. More concretely, points in \(T^*M\) are pairs \((p,q)\), for \(p : \mathcal{M}\) and \(q : T^*_pM\).
Recall that \(T^*M\) is itself a (\(2n\) dimensional) manifold, on which we may define differential forms.
In fact, there is a natural form \(\theta\):
where \(\pi_1(q,p)=q\). Examination shows that this is well-typed.
In coordinates, we find:
Under construction: derivation
Any particular Hamiltonian dynamics is a map from the cotangent bundle to itself, and is defined such that the differential form \(\omega := d\theta\) is preserved under the map.
Concretely, since we are in a category of smooth manifolds, the map will be a diffeomorphism \(f\), and it will act on \(\omega\) by the pullback \(f^*\omega\). In coordinates:
\(\omega\) is known as the symplectic form, and \(\Omega := \omega^n = \omega \wedge \omega..._n\wedge\omega\) is the volume form.
Now for the flow. Given a function \(f : T^*\mathcal{M}\to \R\), let \(L(f)\) be the map such that \(df(x) \mapsto \omega(L(f), x)\). Calling our Hamiltonian \(H\), we consider paths on the manifold that follow \(L(H)\).
Observe that in coordinates:
Also observe, if \(f\) follows the flow of some \(H\), that using the definition of \(L\), and the property of a one-form \(df(X) = X(f)\):
So we have:
so that \(\frac{\partial H}{\partial t} = L(H)(H) = \{H,H\} = 0\) (energy preservation).
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Here \(\delta\) refers to the variation (with respect to any function \(f : T \to C\)), so that \(\delta G(x) = \frac{d}{ds}G(x(t)+sf(t))|_{s=0} := \frac{d}{ds}G(x_s(t))|_{s=0}\). ↩