Classical Mechanics

Authors:Gilbert Gede, Luke Peterson, Angadh Nanjangud

Abstract

In this documentation many components of the physics/mechanics module will be discussed. mechanics has been written to allow for creation of symbolic equations of motion for complicated multibody systems.

Mechanics

In physics, mechanics describes conditions of rest (statics) or motion (dynamics). There are a few common steps to all mechanics problems. First, an idealized representation of a system is described. Next, we use physical laws to generate equations that define the system’s behavior. Then, we solve these equations, sometimes analytically but usually numerically. Finally, we extract information from these equations and solutions. The current scope of the module is multi-body dynamics: the motion of systems of multiple particles and/or rigid bodies. For example, this module could be used to understand the motion of a double pendulum, planets, robotic manipulators, bicycles, and any other system of rigid bodies that may fascinate us.

Often, the objective in multi-body dynamics is to obtain the trajectory of a system of rigid bodies through time. The challenge for this task is to first formulate the equations of motion of the system. Once they are formulated, they must be solved, that is, integrated forward in time. When digital computers came around, solving became the easy part of the problem. Now, we can tackle more complicated problems, which leaves the challenge of formulating the equations.

The term “equations of motion” is used to describe the application of Newton’s second law to multi-body systems. The form of the equations of motion depends on the method used to generate them. This package implements two of these methods: Kane’s method and Lagrange’s method. This module facilitates the formulation of equations of motion, which can then be solved (integrated) using generic ordinary differential equation (ODE) solvers.

The approach to a particular class of dynamics problems, that of forward dynamics, has the following steps:

  1. describing the system’s geometry and configuration,
  2. specifying the way the system can move, including constraints on its motion
  3. describing the external forces and moments on the system,
  4. combining the above information according to Newton’s second law (\(\mathbf{F}=m\mathbf{a}\)), and
  5. organizing the resulting equations so that they can be integrated to obtain the system’s trajectory through time.

Together with the rest of SymPy, this module performs steps 4 and 5, provided that the user can perform 1 through 3 for the module. That is to say, the user must provide a complete representation of the free body diagrams that themselves represent the system, with which this code can provide equations of motion in a form amenable to numerical integration. Step 5 above amounts to arduous algebra for even fairly simple multi-body systems. Thus, it is desirable to use a symbolic math package, such as Sympy, to perform this step. It is for this reason that this module is a part of Sympy. Step 4 amounts to this specific module, sympy.physics.mechanics.

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