# Why Schrödinger’s equation?

For any pair of points $$A,\ B\in\mathbf{R}^n$$ consider the space of
paths $$\gamma\colon [0,T]\to\mathbf{R}^n$$ between $$A$$ and $$B$$. Let $$L$$
be a Lagrangian and
$I(\gamma)=\int_0^TL(t,\gamma(t),\dot{\gamma}(t))dt$ be the action
of that path. A classical path of time $$T$$ joining $$A$$ and $$B$$ is a
solution to the corresponding Euler-Lagrange equation. Let's suppose
we're in an ideal situation: for any $$T$$ and any pair of points $$A$$,
$$B$$ there is a unique classical path $$\gamma_{A,B,T}$$ of time $$T$$
joining $$A$$ and $$B$$.

Fix $$A$$, but allow $$B$$ and $$T$$ to vary. Define the function
$W(B,T)=I(\gamma_{A,B,T}).$

The Hamilton-Jacobi equation is a PDE satisfied by this
function. Let's first compute the derivatives of $$W$$ with respect to
$$B$$. Replace $$B$$ by $$B+b$$ and suppose that
$$\gamma_{A,B+b,T}(t)=\gamma_{A,B,T}(t)+\eta(t)$$. Then, for small $$b$$,
writing $$\gamma_{A,B,T}(t)=(x_1(t),\ldots,x_n(t))$$, we have
$I(\gamma_{A,B+b,T})=I(\gamma_{A,B,T})+\int_0^T\left(\frac{\partial L}{\partial x_i}-\frac{d}{dt}\frac{\partial L}{\partial\dot{x}_i}\right)dt+\left[\frac{\partial L}{\partial \dot{x}_i}\eta_i(t)\right]_0^T+\cdots$ by the usual Euler-Lagrange
argument for computing the first variation of $$I$$. Since
$$\gamma_{A,B,T}$$ is the classical path, the first term vanishes. Since
$$\eta_i(0)=0$$ (the point $$A$$ is fixed) the only remaining term is
$$\frac{\partial L}{\partial \dot{x}_i}(T)\eta_i(T)$$. Since
$$b_i=\eta_i(T)$$, this means that the first variation of $$W$$ is
$\frac{\partial W}{\partial B_i}=\frac{\partial L}{\partial\dot{x}_i}(T)$ By Hamilton's equations, $$\frac{\partial L}{\partial\dot{x}_i}=p_i$$ so this says that $$\partial W/\partial B_i$$
is the ith component of momentum at the endpoint of the path.

Now, by the fundamental theorem of calculus:
$\frac{dW}{dt}=L(T,B_i,\dot{B}_i)$ but by the chain rule
$\frac{dW}{dT}=\frac{\partial W}{\partial T}+\sum_i\frac{\partial W}{\partial B_i}\dot{B}_i$ This gives $\frac{\partial W}{\partial T}=L-\sum_ip_i\dot{B}_i$ where $$p_i$$ is the momentum at the
endpoint. Since the Hamiltonian $$H$$ and Lagrangian $$L$$ are related by
a Legendre transform, we have
$L(T,B_i,\dot{B}_i)-\sum_ip_i\dot{B}_i=-H(T,B_i,p_i)=-H\left(T,B_i,\frac{\partial W}{\partial B_i}\right)$ so we see that $$W$$ satisfies the
Hamilton-Jacobi equation $\frac{\partial W}{\partial T}=-H(T,B,\nabla W).$