Generalized Displacement, Generalized Velocity, Generalized Acceleration

The minimum number of independent coordinates or variables which is required to describe the motion of a dynamical system is known as generalized coordinates. For a system of N particles and for k constraints and d dimension, the number of independent coordinates f = dN-k . These ‘f’ number of minimum independent coordinates required to describe configuration and motion of a mechanical system are called generalized coordinates and are denoted by q_i (i=1,2,3,…,f) .

Generalized Displacement

Let us consider a N-particle system for which a small displacement δ(\overrightarrow{r_i}) is defined by change in position coordinates \overrightarrow{r_i}(i=1,2,,…,N) with time (t) kept as constant. The position vector \overrightarrow{r_i} of the i^{th} particle in the form of generalized coordinates can be written as

\overrightarrow{r_i}=\overrightarrow{r_i}((q_1, q_2, q_3, ... , q_f, t)

Using Euler’s theorem

δ\overrightarrow{r_i}=\frac{∂\overrightarrow{r_i}}{∂q_1}δq_1+\frac{∂\overrightarrow{r_i}}{∂q_2}δq_2+...+\frac{∂\overrightarrow{r_i}}{∂q_f}δq_f+\frac{∂\overrightarrow{r_i}}{∂t}δt\\

=>δ\overrightarrow{r_i}=\frac{∂\overrightarrow{r_i}}{∂q_1}δq_1+\frac{∂\overrightarrow{r_i}}{∂q_2}δq_2+...+\frac{∂\overrightarrow{r_i}}{∂q_f}δq_f\\

=>δ\overrightarrow{r_i}=\sum\limits_{j=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_j}δq_j

where δq_j represents generalized displacement.

Generalized Velocity

Let us consider a dynamical system at time t comprised of N particles.

Let each particle be specified by the n generalized coordinates q_1,q_2,q_3,…,q_n. Then the time derivative of the generalized coordinates q_j (j=1,2,3,…,n) is called the generalized velocity which is denoted by \dot{q_j }.

The position vector \overrightarrow{r_i} of the i^{th} particle in the form of generalized coordinates and time (t) can be written as

\overrightarrow{r_i}=\overrightarrow{r_i}((q_1, q_2, q_3, ... , q_f, t)

Using Euler’s theorem

d\overrightarrow{r_i}=\frac{∂\overrightarrow{r_i}}{∂q_1}dq_1+\frac{∂\overrightarrow{r_i}}{∂q_2}dq_2+...+\frac{∂\overrightarrow{r_i}}{∂q_f}dq_f+\frac{∂\overrightarrow{r_i}}{∂t}dt\\

=>\frac{d\overrightarrow{r_i}}{dt}=\frac{∂\overrightarrow{r_i}}{∂q_1}\frac{dq_1}{dt}+\frac{∂\overrightarrow{r_i}}{∂q_2}\frac{dq_2}{dt}+...+\frac{∂\overrightarrow{r_i}}{∂q_f}\frac{dq_f}{dt}+\frac{∂\overrightarrow{r_i}}{∂t}\frac{dt}{dt}\\

=>\frac{d\overrightarrow{r_i}}{dt}=\frac{∂\overrightarrow{r_i}}{∂q_1}\frac{dq_1}{dt}+\frac{∂\overrightarrow{r_i}}{∂q_2}\frac{dq_2}{dt}+...+\frac{∂\overrightarrow{r_i}}{∂q_f}\frac{dq_f}{dt}+\frac{∂\overrightarrow{r_i}}{∂t}\\

=>\overrightarrow{v}=\sum\limits_{j=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_j}\frac{dq_j}{dt}+\frac{∂\overrightarrow{r_i}}{∂t}

=>\overrightarrow{v}=\sum\limits_{j=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_j}\dot{q_j}+\frac{∂\overrightarrow{r_i}}{∂t}

where \dot{q_j} is called generalized velocity.

Generalized Acceleration

Let us consider a dynamical system at time t comprised of N particles.

Let each particle be specified by the n generalized coordinates q_1,q_2,q_3,…,q_n. Then the time derivative of the generalized velocity \dot{q_j} (j=1,2,3,…,n) is called the generalized acceleration which is denoted by \ddot{q_j}.

The velocity of the i^{th} particle in the form of generalized coordinates and time (t) can be written as

\overrightarrow{v}=\dot{\overrightarrow{r_i}}=\sum\limits_{k=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_k}\dot{q_k}+\frac{∂\overrightarrow{r_i}}{∂t}\qquad...(1)

Differentiating both sides w.r.t. ‘time’ we get

\dot{\overrightarrow{v}}=\ddot{\overrightarrow{r_i}}=\frac{d}{dt}\left[\sum\limits_{k=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_k}\dot{q_k}+\frac{∂\overrightarrow{r_i}}{∂t}\right]

=>\overrightarrow{a_i}=\frac{d}{dt}\left[\sum\limits_{k=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_k}\dot{q_k}\right]+\frac{d}{dt}\left[\frac{∂\overrightarrow{r_i}}{∂t}\right]

=>\overrightarrow{a_i}=\sum\limits_{k=1}^{f}\frac{\dot{∂\overrightarrow{r_i}}}{∂q_k}\dot{q_k}+\sum\limits_{k=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_k}\ddot{q_k}+\frac{∂\dot{\overrightarrow{r_i}}}{∂t}\qquad...(2)

Using equation (1) in equation (2) we get,

=>\overrightarrow{a_i}=\sum\limits_{k=1}^{f}\frac{∂}{∂q_k}\left[\sum\limits_{j=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_j}\dot{q_j}+\frac{∂\overrightarrow{r_i}}{∂t}\right]\dot{q_k}+\sum\limits_{k=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_k}\ddot{q_k}+\frac{∂}{∂t}\left[\sum\limits_{j=1}^{f}\frac{∂\overrightarrow{r_i}}{∂q_j}\dot{q_j}+\frac{∂\overrightarrow{r_i}}{∂t}\right]

which is the required generalized acceleration.

Generalized Force

Let us consider a N-particle system with no constraints imposed on the system.

Let a force \sum\limits_{i=1}^{N}F_i be acting on the system causing an arbitrary small displacement δ\overrightarrow{r_i} of the system by doing work δW which is given by

δW=\sum\limits_{i=1}^{N}F_i.δ\overrightarrow{r_i}\qquad...(1)

We have,

δ\overrightarrow{r_i}=\sum\limits_{j=1}^{3N}\frac{∂\overrightarrow{r_i}}{∂q_j}δq_j\qquad...(2)

Using (2) in (1) we get,

δW=\sum\limits_{i=1}^{N}F_i.\sum\limits_{j=1}^{3N}\frac{∂\overrightarrow{r_i}}{∂q_j}δq_j

=>δW=\sum\limits_{j=1}^{3N}Q_j.δq_j , where Q_j=\sum\limits_{i=1}^{N}F_i.\frac{∂\overrightarrow{r_i}}{∂q_j}

where Q_j is called generalized force.

Generalized Momentum

Let us consider a N-particle system with no constraints imposed on the system.

The K.E. T of the i^{th} particle having mass m_i and velocity v_i of a system is given by

T=\frac{1}{2}m_i {v_i}^2

=>T=\frac{1}{2}m_i (\frac{dx_i}{dt})^2

=>T=\frac{1}{2}m_i (\dot{x_i})^2\qquad...(1)

Differentiating equation (1) partially w.r.t. \dot{x_i} we get,

\frac{∂T}{∂\dot{x_i}}=m_i \dot{x_i}\qquad...(2)

Linear momentum p_i of the particle is given by

p_i=m_iv_i

=>p_i=m_i\frac{dx_i}{dt}

=>p_i=m_i\dot{x_i}\qquad...(3)

Comparing (2) and (3) we get,

p_i=\frac{∂T}{∂\dot{x_i}}\qquad...(4)

Similarly, linear momentum is associated with generalized coordinate q_k called generalized momentum p_k is given by

p_k=\frac{∂T}{∂\dot{q_k}}\qquad...(5)

First we derive the expression for K.E. (T) for a system of N-particles in terms of generalized velocities \dot{q_k}. K.E. of the system of N-particles free from constraints is

T=\sum\limits_{i=1}^{N}\frac{1}{2}m_i {v_i}^2=\sum\limits_{i=1}^{N}\frac{1}{2}m_i \dot{r_i}^2

T=\sum\limits_{i=1}^{N}\frac{1}{2}m_i (\dot{\overrightarrow{r_i}}.\dot{\overrightarrow{r_i}})\qquad...(6)

Now,

\dot{\overrightarrow{r_i}}=\sum\limits_{j=1}^{3N}\frac{∂\overrightarrow{r_i}}{∂q_j}\dot{q_j}+\frac{∂\overrightarrow{r_i}}{∂t}\qquad...(7)

Using (7) and (6),

T=\sum\limits_{i=1}^{N}\frac{1}{2}m_i \left[\sum\limits_{j=1}^{3N}\frac{∂\overrightarrow{r_i}}{∂q_j}\dot{q_j}+\frac{∂\overrightarrow{r_i}}{∂t}\right].\left[\sum\limits_{j=1}^{3N}\frac{∂\overrightarrow{r_i}}{∂q_j}\dot{q_j}+\frac{∂\overrightarrow{r_i}}{∂t}\right]