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<title>10.A. Variational methods in physics</title>

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##[10.A. Variational methods in physics](id:sect-10.A)

> 1.  [Classical dynamic](#sect-10.A.1)
> 2.  [Continuum dynamics](#sect-10.A.2)
> 3.  [Equilibria](#sect-10.A.3)

In Theoretical Physics equations of movement are frequently derived as Euler-Lagrange equations for a functional called *action* and traditionally denoted by $S$.

###[Classical dynamic](id:sect-10.A.1)

In the *classical dynamics* important role is played by *Lagrange formalism*. Positions of the system are described by *generalized coordinates* $\mathbf{q}=(q^1,q^2,\ldots, q^N)$ which are functions of *time* $t$: $\mathbf{q}=\mathbf{q}(t)$. Then their derivatives $\dot{\mathbf{q}}:=\mathbf{q}\_t$ are called *generalized velocities* (in physics upper dot traditionally is used for derivative with respect to $t$).

*Lagrangian* then is a function of $\mathbf{q}, \dot{\mathbf{q}},t$: $L=L(\mathbf{q}, \dot{\mathbf{q}},t)$ and usually $L=T-U$ where $T$ is a kinetic energy and $U$ is a potential energy of the system.

Finally, *action* $S$ is defined as
\begin{equation}
S= \int\_{t\_0}^{t\_1} L(\mathbf{q}(t), \dot{\mathbf{q}}(t),t)\,dt
\label{eq-10.A.1}
\end{equation}
and we are looking for extremals of $S$ as $\mathbf{q}(t\_0)$ (initial state) and $\mathbf{q}(t\_1)$ (final state) are fixed.

Then *Lagrange equations* are
\begin{equation}
\frac{\partial L}{\partial q\_k}-
\frac{d\ }{dt}\left(\frac{\partial L}{\partial \dot{q}\_k}\right)=0
\qquad k=1,\ldots,N.
\label{eq-10.A.2}
\end{equation}

Next $\mathbf{p}=\frac{\partial L}{\partial \dot{\mathbf{q}}}$ that means
\begin{equation}
p\_k=\frac{\partial L}{\partial \dot{q}\_k}\qquad =1,\ldots, N
\label{eq-10.A.3}
\end{equation}
 are *generalized momenta* and
\begin{equation}
H:=\mathbf{p}\cdot\mathbf{q}-L= \sum\_{k=1}^n
\frac{\partial L}{\partial \dot{q}\_k} \dot{q}\_k -L
\label{eq-10.A.4}
\end{equation}
is considered as an *energy* and if expressed through $\mathbf{q}, \mathbf{p}, t$ is called *Hamiltionian* $H=H(\mathbf{q}, \mathbf{p}, t)$.

Transition from generalized velocities to generalized momenta are called *Legendre transformation* and remarkable fact is that in
$(\mathbf{q}, \mathbf{p})$ movement equations are
\begin{align}
&\dot{q}\_k = \frac{\partial H}{\partial p\_k}, \label{eq-10.A.5}\\\\
&\dot{p}\_k = -\frac{\partial H}{\partial q\_k}, &&k=1,\ldots,N. \label{eq-10.A.6}
\end{align}
This is a *Hamiltonian formalism* and $-\frac{\partial H}{\partial q\_k}$ are called *generalized forces*. Another remarkable equality is
\begin{equation}
\frac{d H}{dt}=\frac{\partial H}{\partial t}
\label{eq-10.A.7}
\end{equation}
where in the left-hand expression $(\mathbf{q}, \mathbf{p})$ are considered as functions of $t$.

We will not pursue this road, just mention that if we fix $\mathbf{q}(t\_0)$ and calculate action $S$ defined by (\ref{eq-10.A.1}) along extremals we get $S=S(\mathbf{q},t)$. Then it satisfies *Hamilton-Jacobi equation*
\begin{equation}
\frac{\partial S}{\partial t}+ H(\mathbf{q},\nabla S,t)=0
\label{eq-10.A.8}
\end{equation}
which is a *first order nonlinear PDE* mentioned in [Subsection 2.2.2](../Chapter2/S2.2.html#sect-2.2.2).

###[Continuum dynamics](id:sect-10.A.2)

Now the state of the system is described by $u(\mathbf{x};t)$ where $\mathbf{x}=(x\_1,\ldots,x\_n)$ are *spatial variables* and the initial state $u(x;t\_0)$ and the final state $u(x;t\_1)$ are fixed and action is defined by
\begin{equation}
S= \int\_{t\_0}^{t\_1} \mathcal{L} (u,u\_t,t)\,dt :=
\int\_{t\_0}^{t\_1} \iiint L(u ,u\_x,\ldots,, u\_t, \ldots, t)\,d^nx\,dt
\label{eq-10.A.9}
\end{equation}
and *Lagrangian* $L$ in fact depends on $u$, $u\_x$ and may be higher derivatives of $u$ with respect to spatial variables and on $u\_t$ and may be its derivatives (including higher order) with respect to spatial variables.

Deriving Lagrange equations we treat $t$ as just one of the coordinates (so we have $x=(x\_0,\mathbf{x})=(x\_0,x]\_1,\ldots,x\_n)$ but defining generalized momenta and forces and defining  Hamiltonian $t$ "sticks out":
\begin{align}
&\pi=\frac{\delta \mathcal{L}}{\delta u\_t},\label{eq-10.A.10}\\\\
&\varphi=-\frac{\delta \mathcal{L}}{\delta u},\label{eq-10.A.11}\\\\
&H= \iiint \pi u\_t\,dx -\mathcal{L}.
\label{eq-10.A.12}
\end{align}

**[Example 1.](id:example-10.A.1)**
Let
\begin{equation}
S=\frac{1}{2} \int
\iiint \Bigl( \rho  u\_t^2 - K |\nabla u|^2 +2 fu \Bigr)\,d^nx dt;
\label{eq-10.A.13}
\end{equation}
Here $f=f(\mathbf{x},t)$ is a density of external force.

Then corresponding Lagrange equation is
\begin{equation}
-(\rho u\_t)\_t  + \nabla \cdot (K\nabla u) - f=0
\label{eq-10.A.14}
\end{equation}
which for constant $\rho, K$ becomes a standard *wave equation*. Meanwhile as
\begin{equation}
\mathcal{L}=\frac{1}{2} \iiint
\Bigl( \rho  u\_t^2 - K |\nabla u|^2 + 2 fu \Bigr)\,d^nx
\label{eq-10.A.15}
\end{equation}
we have according to (\ref{eq-10.A.10})--(\ref{eq-10.A.12})
\begin{align}
&\pi (x) =\rho u\_t,\notag\\\\
&\varphi (x)= \nabla \cdot (K\nabla u) + f,\notag\\\\
& H= \frac{1}{2} \iiint \Bigl( \rho  u\_t^2 + K |\nabla u|^2 -2 fu \Bigr)\,d^nx
\label{eq-10.A.16}
\end{align}
and $H$ is preserved as long as $\rho, K, f$ do not depend on $t$.
<!--\end{example}-->

**[Example 2.](id:example-10.A.2)**
Similarly,
\begin{equation}
S=\frac{1}{2} \int
\iiint \Bigl( \rho  |\mathbf{u}\_t|^2 - \lambda  |\nabla \otimes \mathbf{u}|^2 - \mu |\nabla\cdot \mathbf{u}|^2 +2 \mathbf{f}\cdot\mathbf{u} \Bigr)\,d^nx dt
\label{eq-10.A.17}
\end{equation}
with constant $\rho,\lambda, \mu$ leads to *elasticity equations*
\begin{equation}
-\rho \mathbf{u}\_{tt}  + \lambda \Delta \mathbf{u}+ \mu \nabla (\nabla\cdot \mathbf{u})+\mathbf{f}=0
\label{eq-10.A.18}
\end{equation}
and
\begin{equation}
H=\frac{1}{2}
\iiint \Bigl( \rho  |\mathbf{u}\_t|^2 + \lambda  |\nabla \otimes \mathbf{u}|^2 + \mu |\nabla\cdot \mathbf{u}|^2 -2 \mathbf{f}\cdot\mathbf{u} \Bigr)\,d^n.
\label{eq-10.A.19}
\end{equation}
<!--\end{example}-->


**[Example 3.](id:example-10.A.3)**
Let $n=3$, then $|\nabla \otimes \mathbf{u}|^2=
|\nabla \times \mathbf{u}|^2+|\nabla \cdot \mathbf{u}|^2$. Taking in
[Example 2](#example-10.A.2) $\rho=1$, $\mu=-\lambda=-c^2$ and $\mathbf{f}=0$ we have
\begin{gather}
S=\frac{1}{2} \int\iiint
\Bigl(|\mathbf{u}\_t|^2 - c^2 |\nabla \times \mathbf{u}|^2 \Bigr)\,d^3 x dt,
\label{eq-10.A.20}\\\\
-\mathbf{u}\_{tt}  - c^2 \nabla\times(\nabla\times \mathbf{u})=0
\label{eq-10.A.21}
\end{gather}
which is Maxwell equations without charges and currents for a *vector potential* $\mathbf{u}$, taking $\mathbf{E}=\mathbf{u}\_t$, $\mathbf{H}=\nabla \times \mathbf{u}$ we arrive to Maxwell equations in more standard form [Section 14.3](../Chapter14/14.3.html).
<!--\end{example}-->



**[Example 4.](id:example-10.A.4)**
Let
\begin{equation}
S=\frac{1}{2} \int\iiint
\Bigl(u\_t^2 - \sum\_{i,j}  K u\_{x\_ix\_j}^2 +2fu \Bigr)\,d^n x dt.
\label{eq-10.A.22}
\end{equation}
Then we arrive to
\begin{equation}
-u\_{tt}  - K \Delta^2 u +f=0
\label{eq-10.A.23}
\end{equation}
which is *vibrating beam equation* as $n=1$ and *vibrating plate equation* as $n=2$; further
\begin{equation}
H=\frac{1}{2}
\iiint \Bigl( \rho  u\_t^2 + \sum\_{i,j}  K u\_{x\_ix\_j}^2 -2fu \Bigr)\,d^n.
\label{eq-10.A.24}
\end{equation}
<!--\end{example}-->



**[Example 5.](id:example-10.A.5)**
Let $u$ be complex-valued function, $\bar{u}$ its complex-conjugate and
\begin{equation}
S=\int\iiint
\Bigl(-i\hbar u\_t\bar{u}-
\bigl(\frac{\hbar^2}{2m}|\nabla u|^2+V(x)|u|^2\bigr) \Bigr)\,d^nx dt.
\label{eq-10.A.25}
\end{equation}
It does not seem to be real-valued but it is in its essential part: integrating by parts by $t$ the first term, we get $i\hbar u\bar{u}\_t$ which is complex-conjugate to $-i\hbar u\_t\bar{u}$ plus terms without integration by $t$ (and with $t=t\_0,t\_1$).

Then Lagrange equation is
$-i\hbar\bar{u}\_t+ (\frac{\hbar^2}{2m}\Delta -V )\bar{u}=0$ which is equivalent to the standard Schrödinger equation
\begin{equation}
i\hbar u\_t =-\frac{\hbar^2}{2m} \Delta u  +V(x)u;
\label{eq-10.A.26}
\end{equation}
further  $\pi (x)= -i\hbar\bar{u}$ and
\begin{multline}
H=\iiint
\Bigl(\frac{\hbar^2}{2m} |\nabla u|^2 +V(x)|u|^2  \Bigr)\,d^n x =\\\\
\iiint
\Bigl(-\frac{\hbar^2}{2m} \Delta u +V(x)u \Bigr)\bar{u}\,d^n x .\qquad
\label{eq-10.A.27}
\end{multline}
*Hint.* Write $u=u\_1+i u\_2$ where $u\_1=\Re u, u\_2=\Im u$ and use
\begin{equation}
\frac{\delta\ }{\delta u}=\frac{1}{2}\left(\frac{\delta\ }{\delta u\_1}- i \frac{\delta\ }{\delta u\_2}\right)
\label{eq-10.A.28}
\end{equation}
which corresponds conventions of CV.
<!--\end{example}-->

###[Equilibria](id:sect-10.A.3)

Selecting $u=u(\mathbf{x})$ in the dynamical equations (and then $u\_t=0$) we get equations of equilibria. Equilibria which delivers to potential energy $U$ local minimum is stable; otherwise it is unstable.

**[Exercise 1.](id:exercise-10.A.1)**
In [Example 1](#example-10.A.1), [Example 2](#example-10.A.2) and [Example 4](#example-10.A.4) write equations of equilibria.
<!--\end{exercise}-->


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