Variational Principles

The equations of motion in classical mechanics arise as solutions of variational problems: A general mechanical system possesses both kinetic and potential energy. The quantity that is minimized is the mean value of kinetic minus potential energy.

Let $M$ be an n-dimensional manifold. Its tangent bundle $TM$ is a 2n-dimensional manifold. Let $F:TM\to \mathbb{R}$ be a smooth function. If $\gamma :[a,b]\to M$ is a smooth curve on $M$, define the lift of $\gamma$ to $TM$ to be the smooth curve on $TM$ given by $\begin{array}{rl}\tilde{\gamma }:[a,b]& ⟶TM\\ t& ⟼\left(\gamma (t),\frac{d\gamma }{dt}(t)\right).\end{array}$ The action of $\gamma$ is ${\mathcal{A}}_{\gamma }:={\int }_a^b({\tilde{\gamma }}^{\ast }F)(t)dt={\int }_a^{b}F\left(\gamma (t),\frac{d\gamma }{dt}(t)\right)dt.$For fixed $p,q\in M$, let $\mathcal{P}(a,b,p,q):=\{\gamma :[a,b]⟶M\mid \gamma (a)=p,\gamma (b)=q\}.$

Lemma

Suppose that ${\gamma }_0:[a,b]\to M$ is minimizing. Let $[a_1,b_1]$ be a sub-interval of $[a,b]$ and let $p_1={\gamma }_0(a_1)$, $q_1={\gamma }_0(b_1)$. Then ${\gamma }_0{|}_{[a_1,b_1]}$ is minimizing among the curves in $\mathcal{P}(a_1,b_1,p_1,q_1)$.

Let $c_1,\dots ,c_n\in C^{\infty }([a,b])$ be such that $c_i(a)=c_i(b)=0$. Let ${\gamma }_{\epsilon }:[a,b]⟶\mathcal{U}$ be the curve

$${\gamma }_{\epsilon }(t)=({\gamma }_1(t)+\epsilon c_1(t),\dots ,{\gamma }_n(t)+\epsilon c_n(t)).$$

For $\epsilon$ small, ${\gamma }_{\epsilon }$ is well-defined and in $\mathcal{P}(a,b,p,q)$.

Euler-Lagrange equations.

Let ${\mathcal{A}}_{\epsilon }={\mathcal{A}}_{{\gamma }_{\epsilon }}={\int }_a^{b}F\left({\gamma }_{\epsilon }(t),\frac{d{\gamma }_{\epsilon }}{dt}(t)\right)dt$. If ${\gamma }_0$ minimizes $\mathcal{A}$, then $\frac{d{\mathcal{A}}_{\epsilon }}{d\epsilon }(0)=0.$ $\begin{array}{rl}\frac{d{\mathcal{A}}_{\epsilon }}{d\epsilon }(0)& ={\int }_a^b\sum _i\left[\frac{\partial F}{\partial x_i}\left({\gamma }_0(t),\frac{d{\gamma }_0}{dt}(t)\right)c_i(t)+\frac{\partial F}{\partial v_i}\left({\gamma }_0(t),\frac{d{\gamma }_0}{dt}(t)\right)\frac{d{c}_i}{dt}(t)\right]dt\\ & ={\int }_a^b\sum _i\left[\frac{\partial F}{\partial x_i}(\dots )-\frac{d}{dt}\frac{\partial F}{\partial v_i}(\dots )\right]c_i(t)dt=0\end{array}$ where the first equality follows from the Leibniz rule and the second equality follows from integration by parts. Since this is true for all $c_i$‘s satisfying the boundary conditions $c_i(a)=c_i(b)=0$, we conclude that$\frac{\partial F}{\partial x_i}\left({\gamma }_0(t),\frac{d{\gamma }_0}{dt}(t)\right)=\frac{d}{dt}\frac{\partial F}{\partial v_i}\left({\gamma }_0(t),\frac{d{\gamma }_0}{dt}(t)\right).$ These are the Euler-Lagrange equations.

Legendre condition

Legendre condition $\det \left(\frac{{\partial }^{2}F}{\partial v_i\partial v_j}\right)\ne 0$
Letting $G_{ij}(x,v)={\left(\frac{{\partial }^{2}F}{\partial v_i\partial v_j}(x,v)\right)}^{-1}$, the Euler-Lagrange equations become $\frac{d^2{\gamma }_j}{d{t}^2}={\sum }_i{G}_{ji}\frac{\partial F}{\partial x_i}\left(\gamma ,\frac{d\gamma }{dt}\right)-{\sum }_{i,k}{G}_{ji}\frac{{\partial }^{2}F}{\partial v_i\partial x_k}\left(\gamma ,\frac{d\gamma }{dt}\right)\frac{d{\gamma }_k}{dt}\text{ .}$ This second order ordinary differential equation has a unique solution given initial conditions $\gamma (a)=p\text{and}\frac{d\gamma }{dt}(a)=v\text{ .}$

The reason we define Legendre condition like this comes from following analysis:

Case 1: Suppose that $F(x,v)$ does not depend on $v$. The Euler-Lagrange equations become $\frac{\partial F}{\partial x_i}\left({\gamma }_0(t),\frac{d{\gamma }_0}{dt}(t)\right)=0⟺\text{the curve }{\gamma }_0\text{ sits on the critical set of }F\text{ .}$ For generic $F$, the critical points are isolated, hence ${\gamma }_0(t)$ must be a constant curve. \Case 2: Suppose that $F(x,v)$ depends affinely on $v$: $F(x,v)=F_0(x)+{\sum }_{j=1}^n{F}_j(x)v_j\text{ .}$ $\begin{array}{rl}\text{LHS of E-L}:& \frac{\partial F_0}{\partial x_i}(\gamma (t))+\sum _{j=1}^n\frac{\partial F_j}{\partial x_i}(\gamma (t))\frac{d{\gamma }_j}{dt}(t)\\ \text{RHS of E-L}:& \frac{d}{dt}{F}_i(\gamma (t))=\sum _{j=1}^n\frac{\partial F_i}{\partial x_j}(\gamma (t))\frac{d{\gamma }_j}{dt}(t)\end{array}$ The Euler-Lagrange equations become $\frac{\partial F_0}{\partial x_i}(\gamma (t))={\sum }_{j=1}^n\underset{n\times n\text{ matrix}}{\underbrace{\left(\frac{\partial F_i}{\partial x_j}-\frac{\partial F_j}{\partial x_i}\right)}}(\gamma (t))\frac{d{\gamma }_j}{dt}(t)\text{ .}$ If the $n\times n$ matrix $\left(\frac{\partial F_i}{\partial x_j}-\frac{\partial F_j}{\partial x_i}\right)$ has an inverse $G_{ij}(x)$, then $\frac{d{\gamma }_j}{dt}(t)={\sum }_{i=1}^n{G}_{ji}(\gamma (t))\frac{\partial F_0}{\partial x_i}(\gamma (t))$ is a system of first order ordinary differential equations. Locally it has a unique solution through each point $p$. If $q$ is not on this curve, there is no solution at all to the Euler-Lagrange equations belonging to $\mathcal{P}(a,b,p,q)$. Therefore, we need non-linear dependence of F on the $v$ variables in order to have appropriate solutions.

Theorem

For every sufficiently small subinterval $[a_1,b_1]$ of $[a,b]$, ${\gamma }_0$ satisfying Euler-Lagrange equation,${\gamma }_0{|}_{[a_1,b_1]}$ is locally minimizing in $\mathcal{P}(a_1,b_1,p_1,q_1)$ where $p_1={\gamma }_0(a_1),q_1={\gamma }_0(b_1)$.