On page 26, Section 1.3, of his book on linear dynamical systems1, Professor Roger Brockett asks:
If $$\dot{\mathbf{x}}(t) = A(t) x(t) , \qquad \mathbf{x}(0) = \mathbf{x}_0$$ and $$\dot{\mathbf{p}}(t) = -A^T(t) \mathbf{p}(t), \qquad \mathbf{p}(0) = \mathbf{p}_0 \quad (\mbox{Adjoint System}),
$$ show that $$ \langle \mathbf{x}(t), \mathbf{p}(t) \rangle = \langle \mathbf{x}_0, \mathbf{p}_0 \rangle \quad\mbox{for all} \ t \in
\mathbf{R}
$$
My take
An easy way to show that a certain function $\psi(t)$ is a constant function is to show that $\dot{\psi}(t) \equiv 0$.
Thus, we find that
$$
{d \over dt} \langle \mathbf{x}(t), \mathbf{p}(t) \rangle \, = \, \langle \dot{\mathbf{x}}(t), \mathbf{p}(t) \rangle
+ \langle \mathbf{x}(t), \dot{\mathbf{p}}(t) \rangle
$$
That is,
$$
{d \over dt} \langle \mathbf{x}(t), \mathbf{p}(t) \rangle \, = \, \langle A(t) \mathbf{x}(t), \mathbf{p}(t) \rangle
+ \langle \mathbf{x}(t), – A^T(t) \mathbf{p}(t) \rangle
$$
Simplifying, we get
$$
{d \over dt} \langle \mathbf{x}(t), \mathbf{p}(t) \rangle \, = – \mathbf{x}^T(t) A^T(t) \mathbf{p}(t) + \mathbf{x}^T(t) A^T(t) \mathbf{p}(t) \equiv 0
$$
This shows that
$$
\langle \mathbf{x}(t), \mathbf{p}(t) \rangle = \langle \mathbf{x}_0, \mathbf{p}_0 \rangle \ \ \mbox{for all} \ t \in
\mathbf{R}
$$
I hope that the calculations are correct.
I would like to learn more on the adjoint of a linear dynamical system. Is there any geometric interpretation for the adjoint of a linear dynamical system and the identity established in the Brockett's exercise problem?
References
- Roger W. Brockett, Finite Dimensional Linear Systems, Wiley, 1970.
Best Answer
I think the clearest geometric interpretation of the adjoint system is as the cotangent lift of the original differential equation.
Consider a (time-independent) differential equation $\dot{x} = f(x)$ on a manifold $M$, specified by a vector field $f$ which is a section of the tangent bundle to $M$. This generates a flow $\Phi_{t}$ on $M$; the adjoint system is a differential equation on the cotangent bundle $T^*M$ whose flow is given by the cotangent lift of the flow $\Phi_{t}$; i.e., the flow of the adjoint system is given by $(\Phi_{-t})^*$ where $^*$ denotes the pullback. The vector field on $T^*M$ describing the adjoint system ODE on the cotangent bundle is given by the cotangent lift of the vector field $f$.
In fact, the adjoint system corresponds to a symplectic flow on the cotangent bundle equipped with its canonical symplectic form. To see this, define the Hamiltonian $H(x,p) = \langle p, f(x)\rangle$ where $\langle \cdot,\cdot\rangle$ denotes the duality pairing between $T^*M$ and $TM$. Hamilton's equations for this Hamiltonian is precisely the adjoint system,
$$ \dot{x} = \partial H/\partial p = f(x),\ \dot{p} = -\partial H/\partial x = - (Df(x))^T p$$
For an intrinsic definition of $H$, let $\hat{f}$ be any lift of $f$ to a vector field on the cotangent bundle; then, define $H = i_{\hat{f}}\theta$ where $\theta$ is the tautological one-form on the cotangent bundle, which you can see gives the same coordinate expression as above. This is independent of the choice of lift because $\theta$ is a horizontal one-form with respect to the bundle $T^*M \rightarrow M$. Then, the adjoint system is defined intrinsically as the differential equation corresponding to the Hamiltonian vector field $X_H$ satisfying $i_{X_H} \Omega = dH$, where $\Omega$ is the canonical symplectic form on the cotangent bundle.
In terms of the conservation property you discussed, if $(\Phi_t)_*$ denotes the pushforward of the flow of the original differential equation onto the tangent bundle, then for $v(t) = (\Phi_t)_* v_0$, $v_0 \in T_xM$, and $p(t) = (\Phi_{-t})^*p_0$, $p_0 \in T_x^*M$, we have
$$ \langle p(t), v(t) \rangle = \langle (\Phi_{-t})^*p_0, (\Phi_{t})_*v_0\rangle = \langle p_0, (\Phi_{-t} \circ \Phi_t)_* v_0\rangle = \langle p_0, v_0\rangle. $$
This property can actually be interpreted as symplecticity of the flow on $T^*M$ (see the linked paper below for details).
In your case, the differential equation you started with is linear, so you can identify $v$ with $x$ and this gives the same relation that you derived (modulo the fact that your equation was time-dependent; you can use the formalism of time-dependent Hamiltonian mechanics to derive an analogous theory, but I stuck to the time-independent case for simplicity).
For more details, you can see the paper Geometric Methods for Adjoint Systems.
Regarding your comment on the terminology "adjoint system" vs "adjoint equation", the adjoint equation refers to the differential equation for $p$ whereas adjoint system refers to the combined system for $(x, p)$. Note that the adjoint equation is then coordinate-dependent, so either requires a trivial cotangent bundle (e.g. $M$ is a vector space) or a connection to make sense of in a coordinate-independent way. The adjoint system, on the other hand, is a coordinate-independent concept, as explained above (it corresponds to a Hamiltonian system on the cotangent bundle).