Analytical Mechanics — formula sheet
Every boxed result from the 18 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
D'Alembert's Principle and Generalized Forces
The applied forces and the reversed inertial forces are in balance against every allowed virtual displacement; projected onto each independent coordinate this becomes an equation of motion in which the generalized inertia (the T-terms) equals the generalized force. Constraint reactions never appear. When the applied forces are derivable from a potential, Qj=−∂V/∂qj and the pair collapses to the Euler–Lagrange equations with L=T−V.
The Euler-Lagrange Equation
One second-order equation per degree of freedom. The first term is the rate of change of the generalised momentum conjugate to qi; the second is the generalised force. So the equation says exactly what Newton's second law says — but in coordinates you chose for convenience, with constraint forces already eliminated.
Recovery of Newton's Second Law from L = T - V
Choosing L = T − V makes the velocity-slot derivative ∂L/∂ẋ the linear momentum and the position-slot derivative ∂L/∂x the force. The Euler-Lagrange equation then says "rate of change of momentum equals force" — Newton's second law, term for term. The minus sign in T − V is essential: it is what puts −∂V/∂x (an attractive force toward lower potential) on the correct side.
Constraint Forces via Lagrange Multipliers
The generalized force of constraint on coordinate qi is a superposition of the constraint-surface normals ∂fα/∂qi, each weighted by its own multiplier λα. The multiplier is therefore the strength of that reaction: a single scalar per constraint that measures how hard the surface must push to keep fα=0. Because the force points along ∇fα it is normal to the constraint manifold, which is exactly why an ideal constraint does no work on motions tangent to that manifold.
Noether's Theorem for Time Translation
The conserved quantity is not defined as the energy; it emerges from the symmetry and then turns out to equal T + V under the conditions below. The object H is the Hamiltonian, arriving here as a by-product before it is given its own role in D-004.
Noether's Theorem for Continuous Symmetries
To each continuous symmetry attach its generators (τ for time, ζi for the coordinates, F for the boundary term of a quasi-symmetry). Contract the coordinate generator with the conjugate momentum, subtract the energy weighted by the time generator, subtract the boundary term, and the result is a number that does not change as the system evolves. Setting τ = 1, ζ = 0, F = 0 gives Q = −h (energy conservation); a uniform spatial shift ζi = constant gives total momentum; a rotation generator ζ = n × r gives the angular-momentum component along n.
Cyclic Coordinates and Conserved Conjugate Momenta
A coordinate that the Lagrangian ignores cannot exert any generalised force on its own momentum, so that momentum is a constant of the motion. Each cyclic coordinate supplies one first integral pk(q, q̇) = ck, lowering the effective order of the problem by one.
The Central-Force Orbit Equation
A driven harmonic equation with constant forcing, solved by a constant plus a cosine. The eccentricity is fixed by the energy alone: E < 0 gives e < 1, an ellipse; E = 0 gives a parabola; E > 0 gives a hyperbola. Kepler's first law is a corollary, not a separate discovery.
Effective Potential and Radial Reduction
The radius obeys Newton's second law for a fictitious particle of mass m in a one-dimensional potential Veff. The extra term ℓ2/(2m r2) is the centrifugal barrier: it is the rotational kinetic energy re-expressed through the conserved ℓ, and being positive and steeply rising as r → 0, it repels the particle from the origin whenever ℓ ≠ 0. Turning points are where Veff(r) = E; a minimum of Veff is a stable circular orbit.
Kepler Orbits and the Laplace-Runge-Lenz Vector
The orbit is a conic with the force centre at one focus. Energy sets the shape: E < 0 gives an ellipse (e < 1, semi-major axis a = −k/2E), E = 0 a parabola (e = 1), E > 0 a hyperbola (e > 1). The vector A lies in the orbital plane, points from focus to perihelion, and has fixed length mke; its constancy is exactly the statement that the perihelion never moves, so a bound orbit closes on itself after one revolution.
Normal Modes as a Generalized Eigenproblem
The general small-amplitude motion is a superposition of n independent harmonic oscillations. In the normal-coordinate basis both M and K are diagonal simultaneously, and the system is literally n uncoupled oscillators. The coupling in the original coordinates was an artefact of the basis, not of the physics.
Decoupling into Normal Coordinates
One linear change of variables turns n coupled oscillators into n uncoupled ones. Each normal coordinate ξr obeys the equation of a single free harmonic oscillator at its own frequency ωr, with amplitude Cr and phase φr fixed by initial conditions. The physical motion is the superposition q(t) = Σr a(r) ξr(t): the columns of A are the mode shapes, the ξr are how strongly each mode is excited. Modes with ωr = 0 are free translations/rotations obeying ξ̈r = 0.
The Legendre Transform to the Hamiltonian
Motion becomes a flow in a 2n-dimensional phase space. The state is a point; the Hamiltonian is a function on that space; the dynamics is the vector field it generates. The antisymmetry between the two equations is the seed of the symplectic structure and of D-005.
Hamilton's Canonical Equations
The rate of change of a coordinate is the gradient of the energy with respect to its conjugate momentum, and the rate of change of that momentum is minus the gradient of the energy with respect to the coordinate. The two equations are mirror images up to a single sign — that antisymmetric sign is the whole content of the symplectic structure. A coordinate missing from H gives ṗ = 0 (conservation); a Hamiltonian with no explicit t gives dH/dt = 0 (energy conservation).
Poisson Brackets and the Algebra of Observables
The Hamiltonian generates time translations. Any quantity with no explicit time dependence is conserved if and only if its bracket with H vanishes. Conservation laws are now a computation, not an insight — which is what makes the formalism worth learning.
Liouville's Theorem on Phase-Space Volume
Hamiltonian flow is incompressible. A droplet of representative points in phase space may be sheared and folded into ever more contorted filaments, but its total 2N-volume never changes; equivalently, an observer riding along any single trajectory sees the local density of neighbouring systems stay fixed. Phase space behaves like an ideal, frictionless fluid whose velocity field has zero divergence everywhere.
Canonical Transformations and Generating Functions
F3(p,Q): q=−∂pF3, P=−∂QF3 • F4(p,P): q=−∂pF4, Q=∂PF4
and in all four cases K = H + ∂F/∂t
Pick any one old variable and any one new variable as the two independent arguments of a single scalar F. Its partial derivatives then define the remaining two variables and thereby the whole map; the transformation is guaranteed canonical by construction. When F has no explicit time dependence the Hamiltonian is simply carried over, K = H expressed in the new variables.
The Hamilton-Jacobi Equation
A single first-order, generally non-linear PDE for Hamilton's principal function S replaces the whole set of equations of motion. Its complete integral generates the trajectories by pure differentiation: pi = ∂S/∂qi gives the momenta, and βi = ∂S/∂αi gives the orbits. S is a field over configuration space, not a number; its level surfaces are the wavefronts of the motion.