Tru Physics

d’Alembert’s Principle

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Introduction

D’Alembert’s principle, named after French mathematician and physicist Jean le Rond d’Alembert, is a fundamental concept in classical mechanics. This principle extends the Newtonian mechanics to systems with constraints and allows the derivation of the equations of motion for complex systems in a generalized and systematic way.

Basic Statement of d’Alembert’s Principle

D’Alembert’s principle states that the sum of the differences between the forces acting on a system of particles and their time derivatives is zero when calculated for the system’s virtual displacements. Mathematically, this is written as:

\displaystyle\sum_i \left(\vec{F}_i - \dot{\vec{p}}_i \right) \cdot \delta \vec{r}_i = 0

However, replacing the linear momentum the mass multipled by velocity for each particle, we get:

\displaystyle\sum_i \left(\vec{F}_i - m_i \vec{a}_i\right) \cdot \delta \vec{r}_i = 0

where \vec{F}_i is the force acting on the i-th particle, m_i is its mass, \vec{a}_i is its acceleration, and \delta \vec{r}_i is the virtual displacement of the i-th particle.

Generalized Forces

It is also crucial to understand the concept of generalized forces. In a system of particles, forces do not solely arise due to classical interactions, but can also come from constraints within the system itself. Such forces are termed as ‘generalized forces’. They do not necessarily have a physical counterpart but are convenient constructs that greatly simplify the equations of motion.

The concept of generalized forces becomes particularly relevant in systems where standard coordinates cannot be used, and generalized coordinates are introduced. These generalized forces then correspond to the non-conservative forces or potential energies in the system. As an extension, D’Alembert’s principle elegantly incorporates these forces, giving a more generalized view of dynamics. The components of some generalized force are denoted by Q_j where:

Q_j = \displaystyle\sum_i\vec{F}_i\cdot\dfrac{\partial\vec{r}_i}{\partial q_j}

These Q_j are part of the derivation of the virtual work done by the \vec{F}_i such that:

\displaystyle\sum_i\vec{F}_i\cdot\delta\vec{r}_i = \displaystyle\sum_{i,j}\vec{F}_i\cdot\dfrac{\partial\vec{r}_i}{\partial q_j}\delta q_j = \displaystyle\sum_j Q_j\delta q_j

It is important to note that the generalized coordinate q is not required to have units of length, nor is the generalized force Q required to have units of force. However, the product of Q_j\delta q_j must always result in units of work.

Also note that, when the forces can be derived via a scalar potential function (V), we can express the generalized work simply as:

Q_j=-\dfrac{\partial V}{\partial q_j}

D’Alembert’s Principle and Lagrange’s Equations

D’Alembert’s principle is an essential tool in the derivation of the Euler-Lagrange equations, which form the core of Lagrangian mechanics. Given a system’s Lagrangian L(q, \dot{q}, t), where q denotes the generalized coordinates and \dot{q} their time derivatives, the Euler-Lagrange equations are given by:

\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}_i}\right) - \dfrac{\partial L}{\partial q_i} = 0

With some effort, we can transform the basic statement of d’Alembert’s principle into the more complicated form below:

\displaystyle\sum_j\left\{\left[\dfrac{d}{dt}\left(\dfrac{\partial T}{\partial \dot{q}_j}\right)-\dfrac{\partial T}{\partial q_j}\right]-Q_j\right\}\delta q_j=0

where T is the kinetic energy of the system of particles. Consider the above expression valid for constraints which do no work for virtual displacements.

Application to Constrained Systems

One of the significant advantages of D’Alembert’s principle is its applicability to systems with constraints. Constraints can be incorporated by adding an appropriate term to the virtual work done by the forces. This enables the analysis of systems that would be difficult or impossible to handle using Newton’s laws alone.

D’Alembert’s Principle in Quantum Mechanics

D’Alembert’s principle also finds use in quantum mechanics, where it is invoked in the path-integral formulation. This approach, developed by Richard Feynman, involves integrating over all possible paths a system can take, with each path weighted by a phase factor derived from the action along the path.

Conclusion

D’Alembert’s principle is a cornerstone of classical mechanics, enabling systematic analysis of complex systems and forming the basis for Lagrangian and Hamiltonian mechanics. By extending the realm of Newtonian mechanics, it has proven to be an invaluable tool in both classical and quantum physics.

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