# 4. Some examples.

Exempel 1: (Harmonisk oscillator.) Consider a mass m suspended in a spring with spring constant k>0. We release the mass from a starting point at time 0 and let it swing oscillate around an equilibrium point. If y=y(t) denotes the distance from the equilibrium point, then the kinetic energy is and the potential energy is The Lagrangian is thus and the action integral We have when Lagrange’s equation is fulfilled, that is, when This is exactly Newton’s second law for the harmonic oscillator. Hamilton’s principle states that the motion will occur according to the above differential equation, with solution Finally we note that the Hamiltonian is the sum of the potential and kinetic energies, that is, the total energy.

Example 2: (Simple pendulum.) Consider a mass m suspended in a pendulum of length l. We denote the angle from the vertical line with and the arc length from the equilibrium point with s The arc length s is connected to the angle by Then we have the kinetic energy and the potential energy Hamilton’s principle states that the motion will be such that the action integral has a stationary value. Lagrange’s equation is in this case This implies that the equation of motion for the pendulum is The Hamiltonian is in this case that is, the total energy of the system.

Example 3: (Motion in a central force field.) Let us examine the motion in the plane of a mass m that is affected by a tensile force towards that is reciprocal to the square of the distance to origo, that is As generalized coordinates we choose the polar coordinates r=r(t) and . They are connected to the cartesian coordinates x and y by This means that the kinetic energy for the system is and the potential energy is (up to an additive constant) (since the force is the negative gradient of the potential energy.) The Lagrangian is thus Hamilton’s principle now states that we should find where In this case, Lagrange’s equations become that is This coupled system can be solved exactly (leading to Kepler’s laws), but it is out of the scope of this course.