Generalized Displacement, Generalized Velocity, Generalized Acceleration

August 24, 2023

ADV-MATHEMATICS

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 $q_i (i=1,2,3,\dots,f)$ .

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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 $q_1,q_2,q_3,\dots,q_n$ . Then the time derivative of the generalized coordinates $q_j (j=1,2,3,\dots,n)$ 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

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

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

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

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

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 $q_1,q_2,q_3,\dots,q_n$ . 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),

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

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