10.3. Variational methods in physics

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10.3. Variational methods in physics

  1. Classical dynamic
  2. Continuum dynamics
  3. Equilibria

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

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-10.3.1} \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-10.3.2} \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-10.3.3} \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-10.3.4} \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-10.3.5}\\ &\dot{p}_k = -\frac{\partial H}{\partial q_k}, &&k=1,\ldots,N. \label{eq-10.3.6} \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-10.3.7} \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-10.3.1}) 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-10.3.8} \end{equation} which is a first order nonlinear PDE mentioned in Subsection 2.2.2.

Continuum dynamics

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-10.3.9} \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-10.3.10}\\ &\varphi=-\frac{\delta \mathcal{L}}{\delta u},\label{eq-10.3.11}\\ &H= \iiint \pi u_t\,dx -\mathcal{L}. \label{eq-10.3.12} \end{align}

Example 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-10.3.13} \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-10.3.14} \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-10.3.15} \end{equation} we have according to (\ref{eq-10.3.10})--(\ref{eq-10.3.12}) \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-10.3.16} \end{align} and $H$ is preserved as long as $\rho, K, f$ do not depend on $t$.

Example 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-10.3.17} \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-10.3.18} \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-10.3.19} \end{equation}

Example 3. Let $n=3$, then $|\nabla \otimes \mathbf{u}|^2= |\nabla \times \mathbf{u}|^2+|\nabla \cdot \mathbf{u}|^2$. Taking in Example 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-10.3.20}\\ -\mathbf{u}_{tt} - c^2 \nabla\times(\nabla\times \mathbf{u})=0 \label{eq-10.3.21} \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.

Example 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-10.3.22} \end{equation} Then we arrive to \begin{equation} -u_{tt} - K \Delta^2 u +f=0 \label{eq-10.3.23} \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-10.3.24} \end{equation}

Example 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-10.3.25} \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-10.3.26} \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-10.3.27} \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-10.3.28} \end{equation} which corresponds conventions of CV.


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. In Example 1, Example 2 and Example 4 write equations of equilibria.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$