Math 2.0

Mechanics of particles and systems Foundations of particle mechanics Theorem

Conservation of energy

Builds on Newton's laws of motionKinetic energyPotential energyWork–energy theorem

Statement

Let NN point particles have masses mi>0m_i > 0 and positions xi(t)∈ℝ3x_i(t) \in \R^3. Let U⊆ℝ3NU \subseteq \R^{3N} be open and let V ⁣:U→ℝV \colon U \to \R be of class C1C^1. Suppose that the motion x=(x1,…,xN)x = (x_1, \dots, x_N) takes values in UU and obeys Newton's second law with the forces Fi=−∇xiVF_i = -\nabla_{x_i} V, that is, mix¨i(t)=−∇xiV(x(t)),i=1,…,N.m_i \ddot x_i(t) = -\nabla_{x_i} V\big(x(t)\big), \qquad i = 1, \dots, N. Then the total mechanical energy E(t)=∑i=1N12mi∣x˙i(t)∣2+V(x(t)),E(t) = \sum_{i=1}^{N} \tfrac12 m_i \abs{\dot x_i(t)}^2 + V\big(x(t)\big), the sum of the kinetic energy and the potential energy, is constant in tt.

Remarks

By the chain rule, E˙=∑imix¨i⋅x˙i+∑i∇xiV⋅x˙i=∑i(mix¨i+∇xiV)⋅x˙i=0\dot E = \sum_i m_i \ddot x_i \cdot \dot x_i + \sum_i \nabla_{x_i} V \cdot \dot x_i = \sum_i (m_i \ddot x_i + \nabla_{x_i} V) \cdot \dot x_i = 0. Equivalently, by the work–energy theorem, the gain of kinetic energy equals the work of the forces, which equals the loss of potential energy. Example: a stone thrown upward at speed v0v_0 with V=mgzV = m g z reaches its greatest height hh, where it is at rest, with 12mv02=mgh\frac12 m v_0^2 = m g h, that is, h=v02/(2g)h = v_0^2/(2g). The hypothesis that the forces come from a time-independent potential is needed: friction, or a potential that depends explicitly on time, can change EE. In physics, conservation of energy is a wider principle that also counts heat and other forms of energy; this statement is its mechanical part.

Going deeper

For a student

Energy is a first integral of the equations of motion, which reduces their order. For one degree of freedom, 12mq˙2+V(q)=E\frac12 m \dot q^2 + V(q) = E can be solved for q˙\dot q and integrated by quadrature. In Lagrangian mechanics this conservation follows from the invariance of the Lagrangian under time shifts, an instance of Noether's theorem.

Reading path

To understand this page from the ground up, read these 82 pages in order. The foundations come first.

  1. Derivative concept
  2. Differential equation concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Function concept
  6. Interval concept
  7. Limit point of a set definition
  8. Local maximum definition
  9. Ordinary differential equation concept
  10. Partition of an interval definition
  11. Sequence definition
  12. Velocity concept
  13. Fréchet derivative definition
  14. Ordinary differential equation definition
  15. Function of several variables concept
  16. Vector-valued function concept
  17. Lipschitz condition definition
  18. Classical mechanics concept
  19. Limit of a function definition
  20. Local minimum definition
  21. Mesh of a partition definition
  22. Riemann sum definition
  23. Real sequence definition
  24. Subsequence definition
  25. Limit of a function of several variables concept
  26. Momentum concept
  27. Parametrised curve concept
  28. Partial derivative concept
  29. Vector field concept
  30. Solution of an ordinary differential equation definition
  31. Inertial frame of reference concept
  32. Continuity at a point definition
  33. Derivative definition
  34. Riemann integrable function definition
  35. Bounded sequence definition
  36. Limit of a sequence definition
  37. Monotone sequence definition
  38. Continuity in R^n definition
  39. Differentiable function of several variables definition
  40. Partial derivative definition
  41. Line integral of a vector field concept
  42. Initial value problem definition
  43. Newton's laws of motion statement
  44. Continuous function definition
  45. Fermat's theorem on stationary points statement
  46. Antiderivative definition
  47. Riemann integral definition
  48. Convergent sequence definition
  49. Monotone convergence theorem for sequences statement
  50. Every sequence has a monotone subsequence statement
  51. Differentiable on an open set definition
  52. Gradient definition
  53. Higher partial derivatives definition
  54. Work done by a force field concept
  55. Picard–Lindelöf hypotheses definition
  56. Intermediate value theorem statement
  57. Definite integral definition
  58. Bolzano–Weierstrass theorem statement
  59. Iterated derivative definition
  60. Picard operator definition
  61. Additivity of the integral over intervals statement
  62. Boundedness theorem statement
  63. Monotonicity of the integral statement
  64. C^k function characterised by continuous derivatives statement
  65. An iterate of the Picard operator is a contraction statement
  66. Integral equation form of an initial value problem statement
  67. Extreme value theorem statement
  68. Vector field definition
  69. Picard–Lindelöf theorem statement
  70. Mean value theorem for integrals statement
  71. Rolle's theorem statement
  72. Conservative vector field definition
  73. Mean value theorem statement
  74. Classical mechanics definition
  75. First fundamental theorem of calculus statement
  76. Potential energy definition
  77. Momentum definition
  78. Second fundamental theorem of calculus statement
  79. Kinetic energy definition
  80. Fundamental theorem of calculus statement
  81. Work–energy theorem statement
  82. Conservation of energy statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Conservation_of_energy
About this page
TagAV2WRWW1TN
URIm2://AV2WRWW1TN
Kindstatement, theorem
WritingW0 stub
MSC70A05, 70F10
StatusThis copy of the site has no build data. Run m2 status AV2WRWW1TN.
AboutKinetic energy, Potential energy
UsesNewton's laws of motion, Kinetic energy, Potential energy, Work–energy theorem
Changed3 November 2026

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