davidbrown3 : Lagrangian Dynamics

Problem Definition

In a cartesian co-ordinate frame of reference, a system of nn particles are subject to hh 'holonomic' constraints and rr 'non-holonomic' constraints

m=h+r m = h + r

fi(x,t)=0,i=1,2,...,h f_i(x, t) = 0, \quad i = 1,2,...,h

Given hh holonomic constraints, it is possible to express hh components of vector xx in terms of the remaining (3n−h)(3n-h) components, regarding these components as independent and able to fully define the system.

∑j=1j=3ndij(x,t)x˙j+gi(x,t)=0,i=1,2,...,m \sum_{j=1}^{j=3n}d_{ij}(x, t)\dot x_{j} + g_{i}(x, t) = 0, \quad i=1, 2,...,m

dij(x,t)=∂fi(x,t)∂xj d_{ij}(x,t) = \frac{\partial f_i(x, t)}{\partial x_j}

gi(x,t)=∂fi(x,t)∂t g_i(x, t) = \frac{\partial f_i(x, t) }{\partial t}

Expressed in matrix form:

D(x,t)x˙j=g D(x, t)\dot x_j = g

Equation of Motion

Ma=F Ma = F

Mx¨=F+Fc M \ddot x = F + F^c

Virtual Displacements

For a given system position of x(t)x(t), if a velocity x˙(t)\dot x(t) satisfies all the conditions of the constraints, it is is called a "possible velocity". There are infinitely many possible velocities.

Integrating this with time we obtain a "possible infinitesimal displacement".

δx(t)=x˙dt \delta x(t) = \dot xdt

Dδx=0 D \delta x = 0

Ideal Constraints

(Also known as "D'Alembarts principle", "Basic Equation of Analytical Mechanics")

The physical work done by a constraint force in the vector direction of a possible motion is zero.

δxTFc=δxT(Mx¨−F)=0 \delta x^T F^c = \delta x^T \left( M \ddot x - F \right) =0

Generalised Coordinate Transforms

The (3n−h)(3n-h) co-ordinates used to describe a system needn't be Cartesian.

xi=f(q1,q2,...,q3n−h,t),i=1,2,...,3n x_i = f(q_1, q_2, ..., q_{3n-h}, t), \quad i=1,2,...,3n

δxi=∑j=1j=s∂xi∂qj⋅δqj \delta x_{i} = \sum_{j=1}^{j=s}\frac{\partial x_{i}}{\partial q_{j}}\cdot\delta q_{j}

x˙i=∑j=1s∂xi∂qj⋅q˙j+∂xi∂t \dot x_{i} = \sum_{j=1}^{s} \frac{\partial x_{i}}{\partial q_{j}}\cdot \dot q_{j} + \frac{\partial x_{i}}{\partial t}

∂x˙i∂q˙j=∂xi∂qj \frac{\partial \dot x_{i}}{\partial \dot q_{j}} = \frac{\partial x_{i}}{\partial q_{j}}

∂x˙i∂qj=∑m=1s∂2xi∂qm∂qjq˙m+∂2xi∂t∂qj=ddt(∂xi∂qj) \frac{\partial \dot x_{i}}{\partial q_{j}} = \sum_{m=1}^{s} \frac{\partial^2 x_{i}}{\partial q_{m} \partial q_{j}}\dot q_{m} + \frac{\partial^2 x_{i}}{\partial t \partial q_{j}} = \frac{d}{dt}\left( \frac{\partial x_{i}}{\partial q_{j}}\right)

ddt(x˙i∂xi∂qj)=x¨i∂xi∂qj+x˙i(ddt∂xi∂qj) \frac{d}{dt}\left( \dot x_{i} \frac{\partial x_{i}}{\partial q_{j}}\right) = \ddot x_{i} \frac{\partial x_{i}}{\partial q_{j}} + \dot x_{i} \left(\frac{d}{dt} \frac{\partial x_{i}}{\partial q_{j}}\right)

x¨i⋅∂xi∂qj=ddt(x˙i∂xi∂qj)−x˙i∂x˙i∂qj=ddt(x˙i∂x˙i∂q˙j)−x˙i∂x˙i∂qj \ddot x_{i}\cdot\frac{\partial x_{i}}{\partial q_{j}} = \frac{d}{dt}\left( \dot x_{i} \frac{\partial x_{i}}{\partial q_{j}}\right) - \dot x_{i} \frac{\partial \dot x_{i}}{\partial q_{j}} = \frac{d}{dt}\left( \dot x_{i} \frac{\partial \dot x_{i}}{\partial \dot q_{j}}\right) - \dot x_{i} \frac{\partial \dot x_{i}}{\partial q_{j}}

Forces in generalized co-ordinates:

Qj=∑i=13nFi∂xi∂yj Q_{j} = \sum_{i=1}^{3n} F_{i} \frac{\partial x_{i}}{\partial y_{j}}

Expressions for Kinetic Energy

Note: these are not several stages of a calculation. They are derived independently.

T=12∑i=13nmixi˙2 T = \frac{1}{2}\sum_{i=1}^{3n}m_{i}\dot{x_{i}}^2

∂T∂q˙j=∑i=13nmi(xi˙∂x˙i∂q˙j) \frac{\partial T}{\partial \dot q_{j}} = \sum_{i=1}^{3n}m_{i}\left(\dot{x_{i}} \frac{\partial \dot x_{i}}{\partial \dot q_{j}}\right)

ddt(∂T∂q˙j)=ddt∑i=13nmi(xi˙∂x˙i∂q˙j) \frac{d}{dt}\left(\frac{\partial T}{\partial \dot q_{j}}\right) =\frac{d}{dt} \sum_{i=1}^{3n}m_{i}\left(\dot{x_{i}} \frac{\partial \dot x_{i}}{\partial \dot q_{j}}\right)

∂T∂qj=∑i=13nmi(xi˙∂x˙i∂qj) \frac{\partial T}{\partial q_{j}} = \sum_{i=1}^{3n}m_{i}\left(\dot{x_{i}} \frac{\partial \dot x_{i}}{\partial q_{j}}\right)

∑i=13nmix¨i∂xi∂qj=ddt∑i=13nmi(x˙i∂x˙i∂q˙j)−∑i=13nmix˙i∂x˙i∂qj \sum_{i=1}^{3n}m_{i}\ddot x_{i}\frac{\partial x_{i}}{\partial q_{j}} = \frac{d}{dt} \sum_{i=1}^{3n}m_{i}\left( \dot x_{i} \frac{\partial \dot x_{i}}{\partial \dot q_{j}}\right) - \sum_{i=1}^{3n}m_{i}\dot x_{i} \frac{\partial \dot x_{i}}{\partial q_{j}}

∑i=13nmix¨i∂xi∂qj=ddt∂T∂q˙j−∂T∂qj \sum_{i=1}^{3n}m_{i}\ddot x_{i}\frac{\partial x_{i}}{\partial q_{j}} = \frac{d}{dt}\frac{\partial T}{\partial \dot q_{j}} - \frac{\partial T}{\partial q_{j}}

Lagrange Equations of Motion for Unconstrained Systems

δxT(Mx¨−F)=∑i=13n(mix¨i−Fi)δx=0 \delta x^T \left( M \ddot x - F \right) = \sum_{i=1}^{3n}\left( m_{i} \ddot x_{i} - F_{i}\right) \delta x = 0

∑i=13n(mix¨i−Fi)∑j=1j=s∂xi∂qjδqj=0 \sum_{i=1}^{3n}\left( m_{i} \ddot x_{i} - F_{i}\right) \sum_{j=1}^{j=s}\frac{\partial x_{i}}{\partial q_{j}}\delta q_{j} = 0

∑j=1s[∑i=13n(mixi¨−Fi)∂xi∂yj]δqj=0 \sum_{j=1}^{s} \left[ \sum_{i=1}^{3n} \left(m_{i} \ddot{x_{i}} - F_{i} \right) \frac{\partial x_{i}}{\partial y_{j}}\right]\delta q_{j} = 0

∑j=1s[ddt∂T∂q˙j−∂T∂qj−Qj]δqj=0 \sum_{j=1}^{s} \left[\frac{d}{dt}\frac{\partial T}{\partial \dot q_{j}} - \frac{\partial T}{\partial q_{j}} - Q_{j} \right]\delta q_{j} = 0

If the constraints are only of the holonomic type and the minimum number of generalized coordinates are used (3n−h)(3n-h)

ddt∂T∂q˙j−∂T∂qj−Qj=0 \frac{d}{dt}\frac{\partial T}{\partial \dot q_{j}} - \frac{\partial T}{\partial q_{j}} - Q_{j} = 0

Generalized potential energy function

V=f(q1,q2,...,qs,q˙1,q˙2,...,q˙s) V = f(q_1, q_2, ..., q_s, \dot q_1, \dot q_2, ..., \dot q_s)

Qjc=−∂V∂qj+ddt(∂V∂q˙j) Q_{j}^{c} = - \frac{\partial V}{\partial q_j} + \frac{d}{dt}\left(\frac{\partial V}{\partial \dot q_j}\right)

Qj=Qjc+Qjnc Q_j = Q_j^c+ Q_j^{nc}

L=(T−V) L = (T-V)

Plugging in expression for the Lagrangian & generalized force:

ddt(∂L∂q˙j+∂V∂q˙j)−(∂L∂qj+∂V∂qj)−[−∂V∂qj+ddt(∂V∂q˙j)]−Qjnc=0 \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_{j}} + \frac{\partial V}{\partial \dot q_{j}}\right) - \left(\frac{\partial L}{\partial q_{j}} + \frac{\partial V}{\partial q_{j}}\right) - \left[-\frac{\partial V}{\partial q_j} + \frac{d}{dt}\left(\frac{\partial V}{\partial \dot q_j}\right)\right] - Q_{j}^{nc} = 0

ddt(∂L∂q˙j)−(∂L∂qj)−Qjnc=0 \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_{j}} \right) - \left(\frac{\partial L}{\partial q_{j}}\right) - Q_{j}^{nc} = 0