A system of particles is subject to holonomic constraints

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

Action and Symmetry, Ch Constraints, generalised coordinates and Lagrange's equations twice over

Problem

A system of

N

particles is subject to holonomic constraints, leaving

n

independent generalised coordinates

q_1,\ldots,q_n

. For any applied forces

\mathbf F_i^{\rm app}

(whether or not they come from a potential), d'Alembert's principle gives

\frac{\mathrm d}{\mathrm dt}\frac{\partial T}{\partial \dot q_k} - \frac{\partial T}{\partial q_k} = \mathcal Q_k, \qquad \mathcal Q_k = \sum_i \mathbf F_i^{\rm app}\cdot\frac{\partial \mathbf r_i}{\partial q_k}, \qquad k=1,\ldots,n,

where

T

is the kinetic energy written in terms of

q,\dot q,t

. ((a)) The coordinates are relabelled by an invertible transformation

Q_j = Q_j(q_1,\ldots,q_n,t)

j=1,\ldots,n

, allowed to depend on

t

, with inverse

q_k = q_k(Q_1,\ldots,Q_n,t)

. Starting from the chain rule for

\dot q_k

in terms of

Q,\dot Q,t

, show that

\dfrac{\partial \dot q_k}{\partial \dot Q_j} = \dfrac{\partial q_k}{\partial Q_j}

. ((b)) Hence show that the boxed equation above keeps exactly the same form when written for the

Q_j

, namely

\dfrac{\mathrm d}{\mathrm dt}\dfrac{\partial T}{\partial \dot Q_j} - \dfrac{\partial T}{\partial Q_j} = \tilde{\mathcal Q}_j

, and give

\tilde{\mathcal Q}_j

in terms of the original

\mathcal Q_k