Title: Part IB / Variational Principles
Tags: Hamiltons_Principle Euler-Lagrane Legendre_Transform Second_Variation Constrained_Variation Convexity Symmetries Variational_Principles Variational_Principles_PDEs

Q: Claim. Let \(\lambda_i\) be the eigenvalues of the Hessian matrix H. What are the different cases for different types of stationary points.
A: All \(\lambda_i\) positive --> local minimum
All \(\lambda_i\) negative --> local maximum
Some positive, some negative --> saddle point
Otherwise --> degenerate and need to investigate higher order terms

Q: Claim. For a function f that is everywhere twice differentiable, the convexity condition is equivalent to the positivity of the Hessian matrix.
A: Necessity follows from the first-order condition. Choose \(\bf y = x+h\). \[\mathbf{h} \cdot [\nabla f(\mathbf{x+h}) - \nabla f(\mathbf{x})] \geq 0\]Now use the Taylor approximsation on \(\nabla f(\mathbf{x+h})\) to get \[h_i h_j H_{ij} + \mathcal{O}(h^3) \geq 0\]

Q: Definition. The Legendre transform of a function \(f: \mathbb{R}^n \to \mathbb{R}\) is a new function f* defined by \[[...]\]
A: f^*(\mathbf{p}) = \sup_{\mathbf{x} } [\mathbf{p \cdot x} - f(\mathbf{x})]
Back Extra: The domain is the subset of \(\mathbb{R}^n\) for which a supremum exists.

Q: Theorem. If f is a convex differentiable function with Legendre transform f* then f** = f.
A: Given \(f^* (\mathbf{p}) = \mathbf{p \cdot x} - f(\mathbf{x})\) where \(\mathbf{x}\) is the solution of \(\nabla f(\mathbf{x}) = \mathbf{p}\), it follows from the chain rule that \(\frac{\partial f^*(\mathbf{p})}{\partial p_i} = x_i\).
We have shown that the function \(\mathbf{p(x)}\) inverse to \(\mathbf{x(p)}\), satisfies \[\nabla f^* (\mathbf{p}) = \mathbf{x}\]where the derivative is taken with respect to p.
Now we take the Legendre transform of f*:
 \[f^{**} (\mathbf{x}) = \mathbf{x \cdot p(x)} - f^* (\mathbf{p(x)}) = \mathbf{x \cdot p(x)} - [\mathbf{p(x) \cdot x} - f(\mathbf{x(p(x))})] = f(\mathbf{x})\]

Q: Definition. Lagrange Multiplier. How do we turn a constrained variational problem into an unconstrained variational problem?
A: We solve for the stationary points of \[\phi(x,y, \lambda) = f(x,y) - \lambda p(x,y)\]This ensures we satisfy the constraints while solving the variational problem.

Q: Definition. Euler-Lagrange Equation. For a functional \(F(\mathbf{y}) = \int_{\alpha}^{\beta} f(\mathbf{y,y'}, x) \mathrm{d}x\), how do we find stationary points.
A: The equations are \(\frac{\partial f}{\partial y_i} - \frac{\mathrm{d}}{\mathrm{d}x}\left ( \frac{\partial f}{\partial y_i'}\right )\) for \(i = 1, \dots, n,\;\;\; \alpha \leq x \leq \beta\).
This comes from taking a small variational in the functional and choosing boundary conditions on the function y so that the boundary term vanishes.

Q: Claim. What can we rewrite the Euler-Lagrange equations as if \(\frac{\partial f}{\partial x} = 0\)?
A: By the chain rule, \(\frac{\mathrm{d}f}{\mathrm{d}x} = \frac{\partial f}{\partial x} + y' \frac{\partial f}{\partial y} + y'' \frac{\partial f}{\partial y'}\) which can be rewritten as \[ \frac{\mathrm{d}f}{\mathrm{d}x} = \frac{\partial f}{\partial x} + y' \left [\frac{\partial f}{\partial y} - \frac{\mathrm{d}}{\mathrm{d}x} \left ( \frac{\partial f}{\partial y'}\right )\right]+ \frac{\mathrm{d}}{\mathrm{d}x} \left ( y'\frac{\partial f}{\partial y'} \right )\]which we can rearrange to get \[\frac{\mathrm{d}}{\mathrm{d}x}\left (f - y' \frac{\partial f}{\partial y'} \right ) = \frac{\partial f}{\partial x}\]from which our result follows.

Q: Definition. What is the action for a path in configuration space between from A and point B?
A: \[I[q] = \int_{t_a}^{t_b} L(t) \mathrm{d}t\] where L is the Lagrangian \[L = T - V\]

Q: Definition. What is the Hamiltonian?
A: \[H(\mathbf{x, p};t) = [\mathbf{p \cdot v} - L(\mathbf{x,v})]_{\mathbf{v = v(p)}}\]where \(\mathbf{v(p)}\) is the solution to \(\partial L / \partial \mathbf{v} = \mathbf{p}\)

Q: Definition. What are Hamilton's equations?
A: These come from the EL equations for the 'phase-space action' functional \(I[\mathbf{x,p}] = \int {\mathbf{\dot{x} \cdot p} - H(\mathbf{x,p}; t) \mathrm{d}t}\)
\[\mathbf{\dot{x}} = \frac{\partial H}{\partial \mathbf{p}}, \mathbf{\dot{p}} = -\frac{\partial H}{\partial \mathbf{x}} \]

Q: Theorem. What is Noether's Thoerem(for Lagrangian mechanics)?
A: If \(\mathbf{q \to Q_s}\) is a one-parameter family of symmetries for a dynamical system with Lagrangian L, then \[\mathbf{\xi}(t) \cdot \frac{\partial L}{\partial \mathbf{\dot{q}} (t)} - k(t)\]is a constant of the motion. Where \[\xi (t) = [d \mathbf{Q}_s / ds]_{s=0}\]

Q: Claim. How do find stationary points of functionals for functions of more than one independent variable?
A: We calculate the \(\delta F\) for the functional manually and end up with PDEs.

Q: Definition. What is the second variation of a functional?
A: We take the expansion of a function to second order in \(\delta y\). We write \(\delta y(x) = \epsilon \xi (x)\) and expand the integrand to second order in \(\epsilon\). \[f(y + \epsilon \xi , y' + \epsilon \xi ' ; x) = f(y, y'; x) + \epsilon \left \{ \xi \left [ \frac{\partial f}{\partial y} - \frac{d}{dx} \left ( \frac{\partial f}{\partial y'} \right ) \right ] + \frac{d}{dx} \left[\xi \frac{\partial f}{\partial y'} \right ]\right \} + \frac{1}{2} \epsilon^2 \left \{ \xi^2 \frac{\partial^2 f}{\partial y^2} + 2\xi \xi' \frac{\partial^2 f}{\partial y \partial x} + \xi'^2 \frac{\partial^2 f}{\partial y'^2}\right \} + \mathcal{O}(\epsilon^3)\]Then do the regular method and calclaute \(\delta^2 F\) by integrating by parts.

Q: Definition. What is the Legendre condition for second variations?
A: If \(y_0(x)\) is a local minimum of \(F[y]\), then it must satisfy this condition.
\[\left . \frac{\partial^2 f}{\partial y' \partial y'} \right |_{y_0} \geq 0\]