Master Variational Calculus & Advanced Mathematical Methods

derive how a particle crosses a potential step and relates theta one to theta two via mass and potential difference using a Lagrangian with a step function.

Derive angular momentum in cylindrical coordinates from the lagrangian. Get m_z = ∂L/∂θ̇ and use x = r cos θ, y = r sin θ.

Analyze how helical symmetry on a z-axis helix with pitch h yields the constant m_z + p_z h/(2π), from the Euler–Lagrange equations.

Explore conservation laws through Reynolds transport theorem, differentiating volume integrals over moving boundaries with velocity fields, the chain rule, and the divergence of the function with respect to time.

Explore canonical coordinates in a classical system by labeling positions q1 through qn and their momenta p1 through pn, emphasizing degrees of freedom.

We prove Liouville's theorem by showing the jacobian of a canonical transformation equals one, using a generating function that preserves Hamilton equations and the variational principle.

Explore how canonical transformations preserve dynamics and invariance of the variational principle. Learn how generating functions implement Legendre transformations to relate old and new variables and yield Hamilton's equations.

Derive Hamilton equations from a variational principle using the action, showing p_j = ∂L/∂q_dot_j and q_dot_j = ∂H/∂p_j, p_dot_j = -∂H/∂q_j.

Show Liouville's theorem via tracking a small square in phase space; prove area dq dp stays constant under hamiltonian flow, implying incompressible phase-space dynamics.

derive hamiltonian and lagrangian formalisms, introduce hamilton equations and poisson brackets, and reveal how angular momentum and coordinate–momentum brackets connect to quantum mechanics through commutators.

From a gravity-only setup, the video derives the time-minimizing path between fixed points using the Lagrangian and Euler–Lagrange equations, showing the curve is a cycloid with a parametric form.

Define the action S from the Lagrangian and rewrite H dt; show action invariance under canonical transformations via a generating function φ; derive Hamilton–Jacobi relations and constants of motion.

Explore tensors through geometry and general relativity to build intuition while mastering the mathematical tools used in tensor analysis.

Introduces tensors and the metric tensor, derives motion from the action principle with a Lagrangian, and links geodesics and ds^2 in inertial and non-inertial frames.

Shows another proof of the invariance of the spacetime interval ds^2 in special relativity using three frames, homogeneous isotropic spacetime, and the constant speed of light.

Einstein derives the Lorentz transformations from his 1905 paper on the electrodynamics of moving bodies, linking moving and stationary frames via x' = x - v t and tau.

Einstein's Lorentz transformations show a moving rigid sphere becomes an ellipsoid with the x-axis contracted by sqrt(1-v^2/c^2). Moving clocks tick slower, tau = t sqrt(1-v^2/c^2).

Explore invariant length element ds^2 = c^2 dt^2 − dx^2 and the moving-clock relation d tau = dt sqrt(1 − v^2/c^2). See how it underpins special relativity and general relativity.

Derive the relativistic Lagrangian from the invariant d tau in special relativity, yielding L = - m0 c squared sqrt(1 - v squared / c squared), linking to the Hamiltonian and energy.

Explore how non-inertial frames challenge classical coordinates and how covariance and differential geometry underpin general relativity, with Einstein, Grossman, and tensors introduced for gravitational fields.

Derive invariant ds^2 from metric eta_mu nu and x_mu using Einstein summation. Show g_alpha beta as a symmetric metric under changes and relate Lorentz transformations to gravity via equivalence principle.

Explain the transformation rules for differential coordinates dx_mu and tensors, covering contravariant and covariant indices, the chain rule, invariants, and tensor products with the metric g_mu nu.

Explore how tensors transform under index lowering and raising using the metric and inverse metric, including covariant and contravariant forms, index summation, and the Kronecker delta.

Show how Einstein proves the inverse metric tensor is a tensor by using g_mu_sigma g_nu_sigma = delta_mu_nu and rewriting ds^2 as g_sigma_tau dpsi_sigma dpsi_tau.

Einstein shows the invariance of the infinitesimal volume element by relating the determinant of the metric tensor to its transformed determinant via a Jacobian, yielding sqrt(-g') = J sqrt(-g).

Einstein derives the geodetic (geodesic) equation by varying the line element ds, parametrizes the geodesic with lambda, and obtains a tensor equation involving Christoffel symbols that describes geodesic motion.

Derive the geodesic equation from an inertial frame using proper time and a general metric, showing how Christoffel symbols yield the geodesic equation.

Einstein derives covariant derivatives differentiating scalars and tensors, defines a_mu = d phi/dx^mu, and shows chi_mu nu = d^2 phi/dx^mu dx^nu - Gamma_mu nu tau d phi/dx^tau is a tensor.

Explore the rule of differentiation of determinants by examining the differential of the metric determinant g. Show that dg/g equals g^{mu nu} dg_{mu nu} equals minus g_{mu nu} dg^{mu nu}.

Explore how Einstein derives the covariant divergence from the metric determinant and Christoffel symbols, using Einstein notation to express the covariant derivative of a four-vector.

Derive the covariant curl analogue in general relativity using antisymmetric tensors a_mu nu and a_mu nu sigma. Examine cyclic sums, cancellations, and divergence formulas involving Christoffel symbols and sqrt(-g).

Einstein derives the Riemann Christoffel tensor from the covariant derivative of a vector, defines g_mu sigma by contraction, decomposes it into r_mu sigma and s_mu sigma, and notes sqrt(-g)=1 simplifies.

Derives vacuum Einstein field equations by analyzing conditions in the absence of matter, using a constant metric and Christoffel symbols under sqrt(-g)=1 to obtain the gravitational equations.

Explore how Einstein derives the field equations from the action principle by varying the integral of h d tau over four-volume, linking them to momentum and energy conservation.

Derive the general relativistic field equations in the presence of matter by introducing the total energy tensor (matter plus gravity) and recover the vacuum form where Poisson's equation holds.

Einstein derives the general relativity field equations from a variational principle, using the metric tensor and matter fields, and demonstrates energy momentum conservation from the variational equations.

Explains geodesics on surfaces of revolution by deriving the geodesic equations from the metric, showing a constant c = r^2 dθ/ds, and noting meridians are geodesics while parallels are not.

Explore how curvature of a three-dimensional surface is described by the first and second fundamental forms, and how normal curvature arises from the Hessian and principal curvatures.

Explore how Gaussian curvature k equals det(B) over det(G) and remains intrinsic under frame changes, as Gauss's Theorema egregium links it to the Ricci tensor in two dimensions.

Explore how principal curvatures of a surface of rotation are the eigenvalues of the second fundamental form, linking gaussian and mean curvature through intuition from slicing and parametric surfaces.

Derive the strain tensor from displacement fields in elasticity, showing how dl prime squared relates to dl squared plus 2 u_jk dx_j dx_k, and its diagonalization.

Derive the thin plate theory from the action principle, using elastic energy per unit volume and stress–strain relations. Obtain the biharmonic equation nabla^4 w = 0 for the out-of-plane displacement.

Derive the strain tensor in spherical coordinates from cartesian coordinates using a practical coordinate transform, with a MATLAB script to compute ur, u_theta, and u_phi components.

Explore computing the gradient in spherical coordinates using tensor and matrix notation, including the jacobian transformation, unit vectors, and the relation between df/dx and df/dx′.

Derive the Laplacian in spherical coordinates by formulating the gradient with tensors and matrices, then translate to index notation and verify with a Matlab implementation.

Rewrite the action as the spacetime integral of the Lagrangian density for a field, vary the field with fixed endpoints, and obtain the Euler-Lagrange equation via vanishing boundary terms.

Derive the Euler Lagrange equations by taking functional derivative of the action with respect to the fields in a classical field theory, accounting for the Lagrangian density and its derivatives.

The lecture proves rigorously that the functional derivative of r_a(x) with respect to eta_b(x'') equals delta_ba times the derivative d_mu of the Dirac delta delta(x − x'').

Explains finding minimal surfaces with fixed boundary by optimizing the area functional for z(x,y). Presents two nontrivial solutions: the catenoid and the helicoid.

Explore the variation of the metric determinant and prove delta g = g g^{mu nu} delta g_{mu nu}. Apply to general relativity and field equations.

Show that for a diagonalizable matrix, the determinant equals the exponential of the trace of its logarithm, via diagonalization and the series definitions of exp and log.

Explore how the variation of a determinant relates to the trace of the inverse times the perturbation, leading to the Jacobi identity and its relevance to general relativity.

Explore the Neumann series for matrix inverses, deriving (I-b)^{-1} as an infinite series of powers of c = I - b and discussing convergence conditions.

Explore the variational derivation of thin-plate energy from the thickness integration, yielding bending and twisting moments m_x, m_y, m_xy, and the plate's equation of motion under external load p.

Derive the path integral from classical field theory using Schwinger's principle, linking the action to quantum amplitudes, then explain the measure over field configurations and the role of stationary action.

Promote the action to an operator to derive the Schrödinger equation from the Hamilton-Jacobi framework, linking momentum and energy operators to quantum dynamics.

Explore how functional derivatives extend Euler-Lagrange equations to quantum field theory through path integrals, yielding Schwinger-Dyson relations and Ehrenfest-like links between quantum and classical dynamics.

Explore complex functions f(z) with z = x + i y, express f as u(x, y) + i v(x, y), and map between C and R2 using f(z) = 1/z.

Explore complex functions, including conjugate, polynomials, and ratios, and derive representations of e^z, sine, and cosine in terms of x and y, with u(x,y) and v(x,y).

Explore the complex derivative and its limit definition, understand directional approach in the complex plane, and derive the Cauchy–Riemann conditions with u and v leading to the harmonic equation.

Examine how complex line integrals around closed curves vanish for continuous f(z), using u and v via Cauchy–Riemann relations and Stokes theorem.

Explore the extension of the Cauchy integral theorem: a continuous f inside a closed curve yields zero contour integral, with contour independence and the idea of holomorphic functions.

Derives Cauchy's integral formula for a pole at z0, showing the contour integral equals 2 pi i f(z0); highlights curve independence and that derivatives exist if f is continuous.

Explore Cauchy's integral formula and its generalization, deriving higher order derivatives as contour integrals of f(w)/(w - z)^{n+1} and noting winding number effects.

Explore Laurent series in complex calculus, a generalization of Taylor and Maclaurin series near z0, through contour integrals on gamma curves and Cauchy integral formula.

Explain the Laurent series in compact form, expressing f(z) as a sum c_n (z-z0)^n for n from minus to plus infinity, and compare with the Taylor series.

Derives the Fourier series from Laurent series by evaluating on a circle around z0, deriving coefficients via contour integration, and generalizes to any period T.

Generalize the Fourier series to any period T by substituting theta with 2π x / T and computing d_n via (1/T) ∫_0^T y(x) e^{-i n 2π x / T} dx.

Derive the Taylor series from the Laurent expansion for a function continuous at z0 by using a contour integral around gamma once, showing that the coefficients c_n equal f^(n)(z0)/n!.

Explore the residue concept, showing that the coefficient c_{-1} equals a contour integral around z0 and enables evaluating integrals via residues.

Explore the residue theorem, relate contour integrals to residues at poles including simple poles, and generalize to multiple discontinuities by summing residues around a closed curve.

Learn to compute residues and Laurent coefficients from derivatives, and use generalized Cauchy and Laurent series concepts to handle poles with multiplicity in complex functions.

Use contour integration around the real axis with a small indentation at zero to compute the principal value of e^{i ω z}/z, yielding i π and its cos/sin implications.

Demonstrate the Dirac delta’s representation as the limit of sin^2(t x)/(t x^2), giving pi times the delta distribution, a distribution with delta sub a approximations and unit normalization.

Prove the Dirac delta integral representation by applying the Fourier transform to 1/(t(t - x)) and evaluating the resulting integral via contour integration, using the signum of k.

Learn how contour integration using a four-segment path and the residue theorem evaluates ∫ e^{i a x^2} dx for a>0 and yields sqrt(π/a) e^{i π/4}.

Derive a simple, intuitive method to evaluate the third Gaussian-type integral from Feynman's path integral appendix, handling a 1/x^2 term and complex numbers.

Explore Fourier series and Fourier transform, using orthogonal sine and cosine and complex exponential bases to represent periodic and nonperiodic functions, with inner products, coefficients, and dual transforms.

Explain notational conventions for the Fourier transform and Fourier series, including space versus time domains and conjugate variables such as k and omega.

See how Dirac delta emerges as a limit of delta_a and defines inverse Fourier transform, linking f_hat(k) = ∫ f(x) e^{-ikx} dx to f(x) = ∫ f_hat(k) e^{ikx} dk/(2π).

Derive the uncertainty principle via Fourier analysis, defining time and frequency variances from f(t) and its Fourier transform, and show delta omega delta t ≥ 1/2.

Derive the Dirac delta as a Fourier integral and show how a train of impulses becomes a Fourier series, leading to the inverse Fourier transform from the Fourier transform.

Derive and apply the Fourier transform to the derivative of a function, showing that F{f'}(ω) = i ω F{f}(ω) and enabling conversion between PDEs and ODEs in engineering and physics.

Explore the Gibbs phenomenon in Fourier series, showing how truncating harmonics causes overshoots at discontinuities and how limits reduce oscillations, with a MATLAB demonstration.

Solve a Dirac delta driven integral via triple convolution and Fourier transforms, yielding a gaussian density with variance linked to sigma, under the condition sigma ≥ 1.

Derive the energy-bandwidth bound for a band-limited signal via Fourier transforms and Parseval’s theorem, showing |h(t)| ≤ sqrt(e ω_naught / π) and the link to energy and bandwidth.

Relate the Laplace transform to the Fourier transform by damping x(t) with e^{-sigma t}, define s = sigma + i omega, and note unilateral and bilateral forms.

Apply Laplace in time and Fourier in space to solve diffusion equation with a delta initial condition, obtaining Green's function f(x,t) = (4 pi d t)^{-3/2} exp(-|x|^2/(4 d t)).

Solve a second-order non-homogeneous linear differential equation with damping using the Fourier transform, deriving the forced response from f(t) and the initial-condition homogeneous solution with y(0)=y0 and y'(0)=y0prime.

Derive the inverse Laplace transform of (6 s^2 + 8) / (s^2 (s^2 + 4)^2) using contour integration and residues, obtaining the result t sin^2(t).

Derive a time-dependent sine-series expansion for h(x,t) on 0 to L using h_n(t)=2/L ∫ h(x,t) sin(nπx/L) dx. Explore an intuitive approximation ∑ sin(nπx/L)/n ≈ π/4 and boundary effects.

Represent q(x,y) and w(x,y) with a double Fourier sine series for a thin plate with zero boundary loads, solving nabla^4 w = q/d and noting high modes decay.

Apply path integrals with imaginary time, use Fourier transforms to turn Gaussian time-slice convolutions into the Schrodinger equation for a free particle.

Visualize perturbation theory with Feynman diagrams derived from the path integral propagator, showing how the Green's function evolves from initial to final state through potential interactions.

Explore divergent series and their asymptotic use in physics, showing how a t-expansion and gamma function relate to approximating a non divergent path integral from quantum field theory.