Lagrangian Dynamics

when is F or H conserved

0 for an isolated system - an experiment done today will be equal to an experiment done tomorrow

A bead moves on a frictionless wire. How many degrees of freedom are there?

1

attwoods machine setup (inertia / DOF)

1 degree of freedom

orbit stability cont. omega value

A is positive assumed attractive central potential

lagrange, vary theta with r fixed

G = torque = generalised force dL/d theta dot = (angular momentum) rate of change of angular momentum is equal to the torque

Poisson Bracket - Transformation - symmetry

G defintion delta f chain rule sub in G

galilean relativity on H

Galileo's theory of relativity states that there is no absolute or preferred: 1. origin of time, 2. position in space, 3. orientation in space, 4. inertial frame of reference. 3. insensitive to the rotation

why you cant sub in J and get the lagrange to work

J is a generalised momenta (+combines r and phi) Lagrange should only contain generalised coordinates, generalised velocities and time

is any function of the generalised coords is time dependent

Poisson bracket + explicit dependence through partial derivative

double pendulum T

T quadratic n=2 DoF

lagrangian in polar coordinates L = T-V

Vr , theta fixed

arbitrary motion of a system

a n and phi n are determined by initial conditions

symmetries of lagrangian

adding a constant = change zero of PE multiplying by a const. = change of units can add a total derivative wrt time (use principle of least actions) parity

rewrite radius of orbit with constants

angular momentum = J A determines strength of inverse square law

Lagrangian for theta in central potential

angular momentum = J J is out of plane where motion is occurring

ball round cone L and H equations

angular momentum around axis of cone conserved

generating function

any function that is a function of coordinates momenta and time

vectors types of vector parity

axial vector (built as cross products) unchanged polar vector changes

We now consider a point particle of mass m sliding frictionlessly inside a conical vase of opening angle θ whose axis is vertical

azimuthal round the cone J is angular momentum

boost definition

brackets (X-Xdot t) is the x position of com @ t=0

[F,G]

change in F generated by the function G Poisson bracket * infinite quantity that controls the size of the transformation

virtual displacement

displacement of bead consistent with forces of constraint has to be perpendicular with the forces dot product = 0, no work is done

coupled systems

energy contribution of one part of the system depends on coords or velocities on another part of the system

bead on wire H example

energy not constant with time as driving force

generalised force Qj

force on particle dot product with derivative of position vector wrt to generalised coordinate

how is the bead held on the wire

forces of constraint - always act in direction that bead can be instantaneously moving

turning point (differentials)

function f(x) condition for max/min to first order f(x+dx)-f(x)=0 Taylor series near turning point is the position of turning point f'(x0) = 0 so it behaves quadratically near the turning point

Hamiltonian generalised

in Hamilton's we have 2N first order differential equations

find functional form of x(t) that minimises S

integrate by parts delta L

Lagrangian for r in central potential

mr𝜃˙^2 is a centrifugal term (not a real force) acts as a force in lagrange equation suppose no radial acceleration

virtual displacement equation given variation in generalised coordinate

no variation in time as forces in constraint would change

D'Alembert's Principle

only includes applied forces written as Fi or m double dot r For a system of mass of particles, the sum of the difference of the force acting on the system and the time derivatives of the momenta is zero when projected onto any virtual displacement.

lagrange in polar fixed theta

rate of change of momentum along r direction is equal to the radial force

spends more/less when lagrangian is large

spends least time when L is large

alternative from of lagrange (time independent)

total derivative dL/dt no explicit time dependence means terms in bracket is constant with time

particle of mass m in a central attractive potential

use