10.5. Variational methods in physics


## 10.5. Variational methods in physics

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

### Classical dynamics

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 $$S= \int_{t_0}^{t_1} L(\mathbf{q}(t), \dot{\mathbf{q}}(t),t)\,dt \label{eq-10.5.1}$$ 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 $$\frac{\partial L}{\partial q_k}- \frac{d\ }{dt}\left(\frac{\partial L}{\partial \dot{q}_k}\right)=0 \qquad k=1,\ldots,N. \label{eq-10.5.2}$$

Next, $\mathbf{p}=\frac{\partial L}{\partial \dot{\mathbf{q}}}$ that means $$p_k=\frac{\partial L}{\partial \dot{q}_k}\qquad =1,\ldots, N \label{eq-10.5.3}$$ are generalized momenta and $$H:=\mathbf{p}\cdot\mathbf{q}-L= \sum_{k=1}^n \frac{\partial L}{\partial \dot{q}_k} \dot{q}_k -L \label{eq-10.5.4}$$ 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-10.5.5}\\ &\dot{p}_k = -\frac{\partial H}{\partial q_k}, &&k=1,\ldots,N. \label{eq-10.5.6} \end{align} This is a Hamiltonian formalism and $-\frac{\partial H}{\partial q_k}$ are called generalized forces. Another remarkable equality is $$\frac{d H}{dt}=\frac{\partial H}{\partial t} \label{eq-10.5.7}$$ 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-10.5.1}) along extremals we get $S=S(\mathbf{q},t)$. Then it satisfies Hamilton-Jacobi equation $$\frac{\partial S}{\partial t}+ H(\mathbf{q},\nabla S,t)=0 \label{eq-10.5.8}$$ which is a first order nonlinear PDE mentioned in Subsection 2.2.2.

Example 1. Very often $L=K-V$ where $$K=\frac{1}{2}\sum_{j,k} g_{jk}(q) \dot{q}_j\dot{q}_k \label{eq-10.5.9}$$ is a kinetic energy with positive defined symmetric matrix $(g_{jk})$ and $V=V(q)$ is a potential energy.

Then, Euler-Lagrange equations are $$\frac{d\ }{dt} \sum g_{jk}(q)\dot{q}_k= \sum_{lk} g_{lk, q_j}g_{jk}(q) \dot{q}_j\dot{q}_k- V_{q_j}(q). \label{eq-10.5.10}$$

Further, $$p_j =\sum g_{jk}(q)\dot{q}_k \label{eq-10.5.11}$$ and $$H(q,p) =\frac{1}{2}\sum g^{jk}(q)\dot{p}_jp_k +V(q), \label{eq-10.5.12}$$ $(g^{jk})=(g_{jk})^{-1}$ is also positive defined symmetric matrix.

In particular,

1. in Cartesian coordinates for a point $g_{jk} = m\delta_{jk}$ where $m$ is the mass, so $\boldsymbol{p}=m\dot{\boldsymbol{q}}$ is usual momentum.

2. On the other hand, for rotation of the solid about the center of the mass at $0$, $g{jk} =I{jk}$ is the inertia tensor, and $\boldsymbol{p}$ is the angular momentum.

Example 2. Let now $$L=K+\sum_j A_j(q)\dot{q}_j -V(q), \label{eq-10.5.13}$$ with the same $K,V$ as in Example 1.

Then, Euler-Lagrange equations are $$\frac{d\ }{dt} \sum g_{jk}(q)\dot{q}_k= \sum_{lk} g_{lk, q_j}g_{jk}(q) \dot{q}_j\dot{q}_k- V_{q_j}(q) + \sum_k F_{jk}\dot{q}_k \label{eq-10.5.14}$$ with antisymmetric matrix $$F_{jk}=A_{k,q_j}-A_{j,q_k}. \label{eq-10.5.15}$$ Here $\mathbf{A}=(A_1,\ldots,A_n)$ is a vector-potential, $F_{jk}$ is a tensor intensity of magnetic field. The last term in (\ref{eq-10.5.14}) is a Lorentz force.

For $n=3$ $$F = \begin{pmatrix} 0 &F_z &-F_y\\ -F_z &0 & F_x\\ F_y &-F_x &0 \end{pmatrix}, \label{eq-10.5.16}$$ $\mathbf{F}=\nabla \times \mathbf{A}$ and the last term becomes familiar $-F\times \dot{\mathbf{q}}$.

Further $$p_j =\sum g_{jk}(q)\dot{q}_k +A_j(q) \label{eq-10.5.17}$$ and and $$H(q,p) = \frac{1}{2}\sum g^{jk}(q)\bigl(p_j-A_j(q)\bigr) \bigl(p_k-A_k(q)\bigr) + V(q). \label{eq-10.5.18}$$

Note that $\mathbf{A}(q)\mapsto \mathbf{A}(q)+\nabla \varphi(q)$ does not affect Euler-Lagrange equations, $\mathbf{p}\mapsto \mathbf{p}-\nabla\varphi(q)$ and $S\mapsto S+ \varphi (q^1)-\varphi(q^0)$ where $q^0$, $q^1$ are initial and final points of trajectory.

### Continuum dynamics

Now the state of the system is described by $u(x;t)$ where $x=(x_1,\ldots,x_n)$ are spatial variables and the initial state $u(x;t_0)$ and the final state $u(x;t_1)$ are fixed and the action is defined by $$S= \int_{t_0}^{t_1} \mathcal{L} (u,u_t,t)\,dt := \int_{t_0}^{t_1} \iiint L(u ,u_x,\ldots, \, u_t, \ldots, t)\,d^nx\,dt \label{eq-10.5.19}$$ 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 $x=(x_0,x)=(x_0,x_1,\ldots,x_n)$, so we just borrow them from Section 10.3 and Section 10.4 (with the trivial modification).

On the other hand, defining generalized momenta and forces and defining Hamiltonian $t$ "sticks out": \begin{align} &\pi=\frac{\delta \mathcal{L}}{\delta u_t},\label{eq-10.5.20}\\ &\varphi=-\frac{\delta \mathcal{L}}{\delta u},\label{eq-10.5.21}\\ &\mathcal{H}= \iiint \pi u_t\,dx -\mathcal{L}. \label{eq-10.5.22} \end{align} Then we can develop Hamiltonian mechanics: \begin{align} &\frac{\partial u}{\partial t}= \frac{\delta \mathcal{H}}{\delta \pi},\label{eq-10.5.23}\\ &\frac{\partial \pi}{\partial t}= -\frac{\delta \mathcal{H}}{\delta u},\label{eq-10.5.24} \end{align} and $$\frac{d}{dt}\mathcal{H}=\frac{\partial}{\partial t}\mathcal{H}; \label{eq-10.5.25}$$ so $\mathcal{H}$ is constant provided it does not depent on $t$ explicitly.

Example 3. Let $$S=\frac{1}{2} \int_{t_0}^{t_1} \iiint \Bigl( \rho u_t^2 - K |\nabla u|^2 +2 fu \Bigr)\,d^nx dt; \label{eq-10.5.26}$$ Here $f=f(\mathbf{x},t)$ is a density of external force.

Then corresponding Lagrange equation is $$-(\rho u_t)_t + \nabla \cdot (K\nabla u) - f=0 \label{eq-10.5.27}$$ which for constant $\rho, K$ becomes a standard wave equation. Meanwhile as $$\mathcal{L}=\frac{1}{2} \iiint \Bigl( \rho u_t^2 - K |\nabla u|^2 + 2 fu \Bigr)\,d^nx \label{eq-10.5.28}$$ we have according to (\ref{eq-10.5.20})--(\ref{eq-10.5.22}) \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-10.5.29} \end{align} and $H$ is preserved as long as $\rho, K, f$ do not depend on $t$.

Example 4. Similarly, $$S=\frac{1}{2} \int_{t_0}^{t_1} \iiint \Bigl( \rho |\mathbf{u}_t|^2 - \lambda |\nabla \otimes \mathbf{u}|^2 - \mu |\nabla\cdot \mathbf{u}|^2 +2 \mathbf{f}\cdot\mathbf{u} \Bigr)\,d^nx dt \label{eq-10.5.30}$$ with constant $\rho,\lambda, \mu$ leads to elasticity equations $$-\rho \mathbf{u}_{tt} + \lambda \Delta \mathbf{u}+ \mu \nabla (\nabla\cdot \mathbf{u})+\mathbf{f}=0 \label{eq-10.5.31}$$ and $$H=\frac{1}{2} \iiint \Bigl( \rho |\mathbf{u}_t|^2 + \lambda |\nabla \otimes \mathbf{u}|^2 + \mu |\nabla\cdot \mathbf{u}|^2 -2 \mathbf{f}\cdot\mathbf{u} \Bigr)\,d^n. \label{eq-10.5.32}$$ We use notations $\nabla \otimes \mathbf{u}=(\partial_j u_k)$ is the matrix of derivatives and $|\nabla \otimes \mathbf{u}|^2=(\sum_{j,k}(\partial_j u_k)^2)^{1/2}$.

Example 5. Let $n=3$, then $|\nabla \otimes \mathbf{u}|^2= |\nabla \times \mathbf{u}|^2+|\nabla \cdot \mathbf{u}|^2$. Taking in Example 4 $\rho=1$, $\mu=-\lambda=-c^2$ and $\mathbf{f}=0$ we have \begin{gather} S=\frac{1}{2} \int_{t_0}^{t_1}\iiint \Bigl(|\mathbf{u}_t|^2 - c^2 |\nabla \times \mathbf{u}|^2 \Bigr)\,d^3 x dt, \label{eq-10.5.33}\\ -\mathbf{u}_{tt} - c^2 \nabla\times(\nabla\times \mathbf{u})=0 \label{eq-10.5.34} \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 6. Let $$S=\frac{1}{2} \int_{t_0}^{t_1}\iiint \Bigl(u_t^2 - \sum_{i,j} K u_{x_ix_j}^2 +2fu \Bigr)\,d^n x dt. \label{eq-10.5.35}$$ Then we arrive to $$-u_{tt} - K \Delta^2 u +f=0 \label{eq-10.5.36}$$ which is vibrating beam equation as $n=1$ and vibrating plate equation as $n=2$; further $$H=\frac{1}{2} \iiint \Bigl( \rho u_t^2 + \sum_{i,j} K u_{x_ix_j}^2 -2fu \Bigr)\,d^nx. \label{eq-10.5.37}$$

Example 7. Let $\psi$ be complex-valued function, $\bar{\psi}$ its complex-conjugate and $$S=\int_{t_0}^{t_1}\iiint \Bigl(i\hbar (\psi\bar{\psi}_t-\bar{\psi}\psi_t)- 2\bigl(\frac{\hbar^2}{2m}|\nabla \psi|^2+V(x)|\psi|^2\bigr) \Bigr)\,d^nx dt. \label{eq-10.5.38}$$

Then Lagrange equation is $-i\hbar\bar{\psi}_t+ (\frac{\hbar^2}{2m}\Delta -V )\bar{\psi}=0$ which is equivalent to the standard Schrödinger equation $$i\hbar \psi_t =\bigl(-\frac{\hbar^2}{2m} \Delta +V(x)\bigr)\psi. \label{eq-10.5.39}$$

Further $\pi (x)= -i\hbar\bar{\psi}$ and \begin{multline} \mathcal{H}=\iiint \Bigl(\frac{\hbar^2}{2m} |\nabla \psi|^2 +V(x)|\psi|^2 \Bigr)\,d^n x =\\ \iiint \Bigl(-\frac{\hbar^2}{2m} \Delta +V(x)\Bigr)\psi \cdot \bar{\psi}\,d^nx .\qquad \label{eq-10.5.40} \end{multline} Hint. Write $\psi=\psi_1+i \psi_2$ where $\psi_1=\Re \psi, u_2=\Im \psi$ and use $$\frac{\delta\ }{\delta \psi}=\frac{1}{2}\left(\frac{\delta\ }{\delta \psi_1}- i \frac{\delta\ }{\delta \psi_2}\right), \label{eq-10.5.41}$$ which corresponds conventions of Complex Variables. Integrate by parts with respect to $t$ to have only $\delta\psi$, but not $\delta\psi_t$.

Exercise 1. Write in the same way Lagrangian and Hamiltonian for the following equations:

$$-i\hbar \psi_t = \mathsf{H}\psi, \label{eq-10.5.42}$$ where $\mathsf{H}$ is

1. Schrödinger operator with magnetic field (14.3.3).
2. Schrödinger-Pauli operator with magnetic field (14.3.4).
3. Dirac operator with magnetic field (14.3.14).

### Equilibria

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 2. In Example 3, Example 4 and Example 6 write equations of equilibria.