Physics Formulae/Classical Mechanics Formulae

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Classical Mechanics.

Mass and Inertia
Mass can be considered to be inertial or gravitational.

Inertial mass is the mass associated with the inertia of a body. By Newton's 3rd Law of Motion, the acceleration of a body is proportional to the force applied to it. Force divided by acceleration is the inertial mass.

Gravitational mass is that mass associated with gravitational attraction. By Newton's Law of Gravity, the gravitational force exerted by or on a body is proportional to its gravitational mass.

By Einstein's Principle of Equivalence, inertial and gravitational mass are always equal.


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 * $$ M_\mathrm{inertial}\mathbf{a} = M_\mathrm{gravity}\mathbf{g} \,\!$$
 * }

Often, masses occur in discrete or continuous distributions. "Discrete mass" and "continuum mass" are not different concepts, but the physical situation may demand the calculation either as summation (discrete) or integration (continuous). Centre of mass is not to be confused with centre of gravity (see Gravitation section).

Note the convenient generalisation of mass density through an n-space, since mass density is simply the amount of mass per unit length, area or volume; there is only a change in dimension number between them.

Moment of Inertia Theorems
Often the calculations for the M.O.I. of a body are not easy; fortunatley there are theorems which can simplify the calculation.

Galilean Transforms
The transformation law from one inertial frame (reference frame travelling at constant velocity - including zero) to another is the Galilean transform. It is only true for classical (Galilei-Newtonian) mechanics.

Unprimed quantites refer to position, velocity and acceleration in one frame F; primed quantites refer to position, velocity and acceleration in another frame  F'  moving at velocity V relative to F. Conversely F moves at velocity (—V) relative to  F' .

Laws of Classical Mechanics
The following general approaches to classical mechanics are summarized below in the order of establishment. They are equivalent formulations, Newton's is very commonly used due to simplicity, but Hamilton's and Lagrange's equations are more general, and their range can extend into other branches of physics with suitable modifications.

Newton's Formulation (1687)
Force, acceleration, and the momentum rate of change are all equated neatly in  Newton's Laws .




 * In applications to a dynamical system of bodies the two equations (effectively)

combine into one. pi = momentum of body i, and FE =

resultant external force (due to any agent not part of system). Body i does not

exert a force on itself.
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 * $$ \frac{\mathrm{d}\mathbf{p}_\mathrm{i}}{\mathrm{d}t} = \mathbf{F}_{E} + \sum_{\mathrm{i} \neq \mathrm{j}} \mathbf{F}_\mathrm{ij} \,\!$$
 * }
 * }

Euler-Lagrange Formulation (1750s)



 * Written as a single equation:




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 * $$ \frac{\mathrm{d}}{\mathrm{d} t} \left ( \frac{\partial L}{\partial \dot{q}_\alpha } \right ) = -\frac{\partial L}{\partial q_{\alpha}} $$
 * }




 * }

Hamilton's Formulation (1833)



 * The Hamiltonian as a function of generalized coordinates and momenta has the

general form:


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 * $$H = \left [ \mathbf{q}(t), \mathbf{p}(t), t \right ] $$
 * }


 * }

— The value of the Hamiltonian H is the total energy of the dynamical system. For an isolated system, it generally equals the total kinetic T and potential energy V.

— Hamiltonians can be used to analyze energy changes of many classical systems; as diverse as the simplist one-body motion to complex many-body systems. They also apply in non-relativistic quantum mechanics; in the relativistic formulation the hamiltonian can be modified to be relativistic like many other quantities.

Derived Kinematic Quantities
For rotation the vectors are axial vectors (also known as pseudovectors), the direction is perpendicular to the plane of the position vector and tangential direction of rotation, and the sense of rotation is determined by a right hand screw system.

For the inclusion of the scalar angle of rotational position $$ \theta \,\!$$, it is nessercary to include a normal vector $$ \mathbf{\hat{n}} \,\!$$ to the plane containing and defined by the position vector and tangential direction of rotation, so that the vector equations to hold.

Using the basis vectors for polar coordinates, which are $$ \boldsymbol{\hat{r}}, \boldsymbol{\hat{\theta}}, \boldsymbol{\hat{\phi}} \,\!$$, the unit normal is $$ \mathbf{\hat{n}} = \boldsymbol{\hat{\theta}} \times \boldsymbol{\hat{r}} \,\!$$.

By vector geometry it can be found that:

$$ \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} \,\!$$

and hence the corollary using the above definitions:

$$ \mathbf{a} = \boldsymbol{\alpha} \times \mathbf{r} + \boldsymbol{\omega} \times \mathbf{v} \,\!$$

Translational Collisions
For conservation of mass and momentum see ../Conservation and Continuity Equations/.

General Planar Motion
The plane of motion is considered in a the cartesian x-y plane using basis vectors (i, j), or alternativley the polar plane containing the (r, θ) coordinates using the basis vectors $$ \left ( \mathbf{\hat{r}}, \boldsymbol{\hat{\theta}} \right )\,\!$$.

For any object moving in any path $$ \mathbf{r}=\mathbf{r}\left ( t \right ) \,\!$$ in a plane, the following are general kinematic and dynamic results :

They can be readily derived by vector geometry and using kinematic/dynamic definitions, and prove to be very useful. Corollaries of momentum, angular momentum etc can immediatley follow by applying the definitions.

Common special cases are:

— the angular components are constant, so these represent equations of motion in a streight line — the radial components i.e. $$ \left | \mathbf{r} \right |\,\!$$ is constant, representing circular motion, so these represent equations of motion in a rotating path (not neccersarily a circle, osscilations on an arc of a circle are possible) — $$ \omega\,\!$$ and $$ \left | \mathbf{r} \right |\,\!$$ are both constant, and $$ \alpha = 0\,\!$$, representing uniform circular motion — $$ \omega=0\,\!$$ and $$ \frac{\mathrm{d}^2r}{\mathrm{d}t^2} \,\!$$ is constant, representing uniform acceleration in a streight line

Energy Theorems and Principles
Work-Energy Equations

The change in translational and/or kinetic energy of a body is equal to the work done by a resultant force and/or torque acting on the body. The force/torque is exerted across a path C, this type of integration is a typical example of a line integral.

For formulae on energy conservation see Conservation and Continuity Equations.

Potential Energy and Work
Every conservative force has an associated potential energy (often incorrectly termed as "potential", which is related to energy but not exactly the same quantity):

By following two principles a non-relative value to U can be consistently assigned:

— Wherever the force is zero, its potential energy is defined to be zero as well.

— Whenever the force does positive work, potential energy decreases (becomes more negative), and vice versa.

Transport Mechanics
Here $$ \mathbf{A} \,\!$$ is a unit vector normal to the cross-section surface at the cross section considered.