During motion, the 2s quantities (s coordinates q and s velocities q̇) (for s degrees of freedom) can change with time. However, there are functions of these quantities that remain constant with time and only depend on the "initial" conditions. These functions are called *integrals of the motion*.
If there are s dof in a closed system, the number of *independent* integrals of motion is 2s-1. Why the -1? The solutions of the equations of motion give us 2s arbitrary constants. But these equations, if the system is closed, do not involve the explicit dependence on time, which means that the choice of the origin of time is arbitrary. So from the 2s functions
qᵢ(t+t₀,C₁,C₂,...,C₂ₛ₋₁) q̇ᵢ(t+t₀,C₁,C₂,...,C₂ₛ₋₁)
we can eliminate t₀ and express the constants C₁, C₂, etc as functions of qᵢ,q̇ᵢ
C₁ = f₁(qᵢ,q̇ᵢ) C₂ = f₂(qᵢ,q̇ᵢ) C₃ = f₃(qᵢ,q̇ᵢ) ...
These functions are integral of motion. But they are not all of equal importance. Some are truly important, as a consequence of symmetries like homogeneity and isotropy of space and time. In any case, the quantities that are represented by these functions are called conserved quantities.
Conserved quantities for systems that are not interacting with others are additive. So if there is a quantity Q that is conserved in closed systems A and B, we have Q_A and Q_B, both conserved and Q = Q_A + Q_B. This additivity is what gives the importance to some conserved quantities. Consider again a given conserved quantity Q. Consider now that the two systems, A and B, interact for a given interval of time. Before the interaction they had Q_A and Q_B. And after the interaction they have Q'_A and Q'_B. These values are different than before, but since the super-system A+B is closed, the quantity Q_A+Q_B must be equal to Q'_A+Q'_B. This is of paramount importance in physics.
Let's consider the homogeneity of time. This is a symmetry. Because of it, the Lagrangian of a closed system cannot explicitly depend on time. Then, the total derivative of L(q,q̇) lacks the ∂L/∂t term and becomes
dL/dt = ∑( (∂L/∂qᵢ)q̇ᵢ + (∂L/∂q̇ᵢ)q̈ᵢ ).
We don't bother to write ∑{i}. Since i is the only index, it is clear that ∑ runs over i.
The first term on the right contains ∂L/∂qᵢ, which can be substituted by (d/dt)∂L/∂q̇ᵢ according to E-L equations. We get
dL/dt = ∑( q̇ᵢ(d/dt)(∂L/∂q̇ᵢ) + (∂L/∂q̇ᵢ)q̈ᵢ ).
Now see how this has the form (U·dV + V·dU), which is just d(UV). So we rewrite as
dL/dt = ∑( d/dt)(q̇ᵢ(∂L/∂q̇ᵢ) ).
We rewrite again as
(d/dt) ( ∑(q̇ᵢ(∂L/∂q̇ᵢ) - L ) = 0.
This means that
∑(q̇ᵢ(∂L/∂q̇ᵢ) - L = constant ≡ E = energy.
So we define this constant as the energy of the system. Then, homogeneity of time implies the conservation of a quantity that we define as energy.
The additivity of energy follows from the additivity of L. Why? Because E is a linear function of L. Why? Notice that
E = ( ∑(q̇ᵢ(∂/∂q̇ᵢ) - 1)L .
The operator acting on L to obtain E is linear. This is clear from the form of the latter expression, which consists in a constant and a derivative. An operator f is linear on L=a·L₁+b·L₂ if f(L)=f(a·L₁+b·L₂)=af(L₁)+b·f(L₂), which is clearly the case.
The conservation of energy is not only valid in closed systems, but also in those under a uniform field, since such fields are independent of time. In general, energy is conserved in a system when its Lagrangian does not explicitly contain time. Mechanical systems for which its energy is conserved are called *conservative* systems.
A conservative system has a Lagrangian of the type L=T(q,q̇)-U(q), where T depends on the velocities in quadratic form, as we know. Let's use this expression to learn something very interesting.
For this, we need to know a theorem on homogeneous functions by Euler. But first, what is an homogeneous function?
A function f is homogeneous of degree k if f(ax) = aᵏ·f(x). For example, f=2x³ is homogeneous of degree 3, since f(ax) = 2·a³x³ = a³·f(x).
Once we know f is homogeneous of degree k, Euler's theorem says that
<r|∇>f(|r>) = kf(|r>).
Or, in the language of partial derivatives
∑qᵢ(∂f(qᵢ)/∂qᵢ) = kf(qᵢ).
And the dependence does not to be q: it can be q̇ if we want:
∑q̇ᵢ(∂f(q̇ᵢ)/∂q̇ᵢ) = kf(q̇ᵢ).
Once we know all this, we can use L as function of q̇, but since only T is depending on q̇,
∑q̇ᵢ(∂L/∂q̇ᵢ) = ∑q̇ᵢ(∂T/∂q̇ᵢ) = 2T.
The number 2 in 2T comes from the fact that we know T to be an homogeneous function of the velocities, and we also know that it is of degree 2.
We return now to our definition of energy
E = ∑(q̇ᵢ(∂L/∂q̇ᵢ) - L = 2T - L = 2T - (T-U) = T + U.
This should sound familiar from secondary school: the total mechanical energy of a system is the sum of its kinetic energy plus its potential energy:
E = T(q,q̇) + U(q).
In Cartesian coordinates, the kinetic term depends only on the velocities (in generalised coordinates depends on velocities and positions). The potential energy always depends on the positions only. But Cartesian coordinates are nice in that they separate the dependence in two parts:
E = ∑(mₐ/2)vₐ² + U(|r₁>,|r₂>,...).